Submonoids: CompleteLattice structure

This file defines a CompleteLattice structure on Submonoids, define the closure of a set as the minimal submonoid that includes this set, and prove a few results about extending properties from a dense set (i.e. a set with closure s = ⊤) to the whole monoid, see Submonoid.dense_induction and MonoidHom.ofClosureEqTopLeft/MonoidHom.ofClosureEqTopRight.

Main definitions

For each of the following definitions in the Submonoid namespace, there is a corresponding definition in the AddSubmonoid namespace.

• Submonoid.copy : copy of a submonoid with carrier replaced by a set that is equal but possibly not definitionally equal to the carrier of the original Submonoid.
• Submonoid.closure : monoid closure of a set, i.e., the least submonoid that includes the set.
• Submonoid.gi : closure : Set M → Submonoid M and coercion coe : Submonoid M → Set M form a GaloisInsertion;
• MonoidHom.eqLocus: the submonoid of elements x : M such that f x = g x;
• MonoidHom.ofClosureEqTopRight: if a map f : M → N between two monoids satisfies f 1 = 1 and f (x * y) = f x * f y for y from some dense set s, then f is a monoid homomorphism. E.g., if f : ℕ → M satisfies f 0 = 0 and f (x + 1) = f x + f 1, then f is an additive monoid homomorphism.

Implementation notes

Submonoid inclusion is denoted ≤ rather than ⊆, although ∈ is defined as membership of a submonoid's underlying set.

Note that Submonoid M does not actually require Monoid M, instead requiring only the weaker MulOneClass M.

This file is designed to have very few dependencies. In particular, it should not use natural numbers. Submonoid is implemented by extending Subsemigroup requiring one_mem'.

Tags

submonoid, submonoids

instance AddSubmonoid.instInfSet {M : Type u_1} [AddZeroClass M] : InfSet (AddSubmonoid M)

Equations

• AddSubmonoid.instInfSet = { sInf := fun (s : Set (AddSubmonoid M)) => { carrier := ⋂ t ∈ s, ↑t, add_mem' := ⋯, zero_mem' := ⋯ } }

theorem AddSubmonoid.instInfSet.proof_1 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) : ∀ {a b : M}, a ∈ ⋂ t ∈ s, ↑t → b ∈ ⋂ t ∈ s, ↑t → a + b ∈ ⋂ x ∈ s, ↑x

theorem AddSubmonoid.instInfSet.proof_2 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) : 0 ∈ ⋂ x ∈ s, ↑x

instance Submonoid.instInfSet {M : Type u_1} [MulOneClass M] : InfSet (Submonoid M)

Equations

• Submonoid.instInfSet = { sInf := fun (s : Set (Submonoid M)) => { carrier := ⋂ t ∈ s, ↑t, mul_mem' := ⋯, one_mem' := ⋯ } }

@[simp]

theorem AddSubmonoid.coe_sInf {M : Type u_1} [AddZeroClass M] (S : Set (AddSubmonoid M)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

@[simp]

theorem Submonoid.coe_sInf {M : Type u_1} [MulOneClass M] (S : Set (Submonoid M)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

theorem AddSubmonoid.mem_sInf {M : Type u_1} [AddZeroClass M] {S : Set (AddSubmonoid M)} {x : M} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

theorem Submonoid.mem_sInf {M : Type u_1} [MulOneClass M] {S : Set (Submonoid M)} {x : M} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

theorem AddSubmonoid.mem_iInf {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} {S : ι → AddSubmonoid M} {x : M} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

theorem Submonoid.mem_iInf {M : Type u_1} [MulOneClass M] {ι : Sort u_4} {S : ι → Submonoid M} {x : M} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

@[simp]

theorem AddSubmonoid.coe_iInf {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} {S : ι → AddSubmonoid M} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

@[simp]

theorem Submonoid.coe_iInf {M : Type u_1} [MulOneClass M] {ι : Sort u_4} {S : ι → Submonoid M} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

theorem AddSubmonoid.instCompleteLattice.proof_5 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) (b : AddSubmonoid M) : a ≤ b → b ≤ a → a = b

theorem AddSubmonoid.instCompleteLattice.proof_6 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) (b : AddSubmonoid M) : a ≤ a ⊔ b

theorem AddSubmonoid.instCompleteLattice.proof_2 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) : a ≤ a

theorem AddSubmonoid.instCompleteLattice.proof_13 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) (a : AddSubmonoid M) : (∀ b ∈ s, b ≤ a) → sSup s ≤ a

theorem AddSubmonoid.instCompleteLattice.proof_15 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) (a : AddSubmonoid M) : (∀ b ∈ s, a ≤ b) → a ≤ sInf s

theorem AddSubmonoid.instCompleteLattice.proof_12 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) (a : AddSubmonoid M) : a ∈ s → a ≤ sSup s

instance AddSubmonoid.instCompleteLattice {M : Type u_1} [AddZeroClass M] : CompleteLattice (AddSubmonoid M)

The AddSubmonoids of an AddMonoid form a complete lattice.

Equations

• AddSubmonoid.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem AddSubmonoid.instCompleteLattice.proof_10 {M : Type u_1} [AddZeroClass M] : ∀ (x x_1 : AddSubmonoid M) (x_2 : M), x_2 ∈ ↑x ∧ x_2 ∈ ↑x_1 → x_2 ∈ ↑x_1

theorem AddSubmonoid.instCompleteLattice.proof_8 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) (b : AddSubmonoid M) (c : AddSubmonoid M) : a ≤ c → b ≤ c → a ⊔ b ≤ c

theorem AddSubmonoid.instCompleteLattice.proof_16 {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) : ∀ x ∈ ⊥, x ∈ S

theorem AddSubmonoid.instCompleteLattice.proof_14 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) (a : AddSubmonoid M) : a ∈ s → sInf s ≤ a

theorem AddSubmonoid.instCompleteLattice.proof_9 {M : Type u_1} [AddZeroClass M] : ∀ (x x_1 : AddSubmonoid M) (x_2 : M), x_2 ∈ ↑x ∧ x_2 ∈ ↑x_1 → x_2 ∈ ↑x

theorem AddSubmonoid.instCompleteLattice.proof_4 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) (b : AddSubmonoid M) : a < b ↔ a ≤ b ∧ ¬b ≤ a

theorem AddSubmonoid.instCompleteLattice.proof_7 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) (b : AddSubmonoid M) : b ≤ a ⊔ b

theorem AddSubmonoid.instCompleteLattice.proof_1 {M : Type u_1} [AddZeroClass M] : ∀ (x : Set (AddSubmonoid M)), IsGLB x (sInf x)

theorem AddSubmonoid.instCompleteLattice.proof_3 {M : Type u_1} [AddZeroClass M] (a : AddSubmonoid M) (b : AddSubmonoid M) (c : AddSubmonoid M) : a ≤ b → b ≤ c → a ≤ c

theorem AddSubmonoid.instCompleteLattice.proof_11 {M : Type u_1} [AddZeroClass M] : ∀ (x x_1 x_2 : AddSubmonoid M), x ≤ x_1 → x ≤ x_2 → ∀ x_3 ∈ x, x_3 ∈ ↑x_1 ∧ x_3 ∈ ↑x_2

instance Submonoid.instCompleteLattice {M : Type u_1} [MulOneClass M] : CompleteLattice (Submonoid M)

Submonoids of a monoid form a complete lattice.

Equations

• Submonoid.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

def AddSubmonoid.closure {M : Type u_1} [AddZeroClass M] (s : Set M) : AddSubmonoid M

The AddSubmonoid generated by a set

Equations

• AddSubmonoid.closure s = sInf {S : AddSubmonoid M | s ⊆ ↑S}

def Submonoid.closure {M : Type u_1} [MulOneClass M] (s : Set M) : Submonoid M

The Submonoid generated by a set.

Equations

• Submonoid.closure s = sInf {S : Submonoid M | s ⊆ ↑S}

theorem AddSubmonoid.mem_closure {M : Type u_1} [AddZeroClass M] {s : Set M} {x : M} : x ∈ AddSubmonoid.closure s ↔ ∀ (S : AddSubmonoid M), s ⊆ ↑S → x ∈ S

theorem Submonoid.mem_closure {M : Type u_1} [MulOneClass M] {s : Set M} {x : M} : x ∈ Submonoid.closure s ↔ ∀ (S : Submonoid M), s ⊆ ↑S → x ∈ S

@[simp]

theorem AddSubmonoid.subset_closure {M : Type u_1} [AddZeroClass M] {s : Set M} : s ⊆ ↑(AddSubmonoid.closure s)

The AddSubmonoid generated by a set includes the set.

@[simp]

theorem Submonoid.subset_closure {M : Type u_1} [MulOneClass M] {s : Set M} : s ⊆ ↑(Submonoid.closure s)

The submonoid generated by a set includes the set.

theorem AddSubmonoid.not_mem_of_not_mem_closure {M : Type u_1} [AddZeroClass M] {s : Set M} {P : M} (hP : P ∉ AddSubmonoid.closure s) : P ∉ s

theorem Submonoid.not_mem_of_not_mem_closure {M : Type u_1} [MulOneClass M] {s : Set M} {P : M} (hP : P ∉ Submonoid.closure s) : P ∉ s

@[simp]

theorem AddSubmonoid.closure_le {M : Type u_1} [AddZeroClass M] {s : Set M} {S : AddSubmonoid M} : AddSubmonoid.closure s ≤ S ↔ s ⊆ ↑S

An additive submonoid S includes closure s if and only if it includes s

@[simp]

theorem Submonoid.closure_le {M : Type u_1} [MulOneClass M] {s : Set M} {S : Submonoid M} : Submonoid.closure s ≤ S ↔ s ⊆ ↑S

A submonoid S includes closure s if and only if it includes s.

theorem AddSubmonoid.closure_mono {M : Type u_1} [AddZeroClass M] ⦃s : Set M⦄ ⦃t : Set M⦄ (h : s ⊆ t) : AddSubmonoid.closure s ≤ AddSubmonoid.closure t

Additive submonoid closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t

theorem Submonoid.closure_mono {M : Type u_1} [MulOneClass M] ⦃s : Set M⦄ ⦃t : Set M⦄ (h : s ⊆ t) : Submonoid.closure s ≤ Submonoid.closure t

Submonoid closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

theorem AddSubmonoid.closure_eq_of_le {M : Type u_1} [AddZeroClass M] {s : Set M} {S : AddSubmonoid M} (h₁ : s ⊆ ↑S) (h₂ : S ≤ AddSubmonoid.closure s) : AddSubmonoid.closure s = S

theorem Submonoid.closure_eq_of_le {M : Type u_1} [MulOneClass M] {s : Set M} {S : Submonoid M} (h₁ : s ⊆ ↑S) (h₂ : S ≤ Submonoid.closure s) : Submonoid.closure s = S

theorem AddSubmonoid.closure_induction {M : Type u_1} [AddZeroClass M] {s : Set M} {p : (x : M) → x ∈ AddSubmonoid.closure s → Prop} (mem : ∀ (x : M) (h : x ∈ s), p x ⋯) (one : p 0 ⋯) (mul : ∀ (x y : M) (hx : x ∈ AddSubmonoid.closure s) (hy : y ∈ AddSubmonoid.closure s), p x hx → p y hy → p (x + y) ⋯) {x : M} (hx : x ∈ AddSubmonoid.closure s) : p x hx

An induction principle for additive closure membership. If p holds for 0 and all elements of s, and is preserved under addition, then p holds for all elements of the additive closure of s.

theorem Submonoid.closure_induction {M : Type u_1} [MulOneClass M] {s : Set M} {p : (x : M) → x ∈ Submonoid.closure s → Prop} (mem : ∀ (x : M) (h : x ∈ s), p x ⋯) (one : p 1 ⋯) (mul : ∀ (x y : M) (hx : x ∈ Submonoid.closure s) (hy : y ∈ Submonoid.closure s), p x hx → p y hy → p (x * y) ⋯) {x : M} (hx : x ∈ Submonoid.closure s) : p x hx

An induction principle for closure membership. If p holds for 1 and all elements of s, and is preserved under multiplication, then p holds for all elements of the closure of s.

@[deprecated Submonoid.closure_induction]

theorem Submonoid.closure_induction' {M : Type u_1} [MulOneClass M] {s : Set M} {p : (x : M) → x ∈ Submonoid.closure s → Prop} (mem : ∀ (x : M) (h : x ∈ s), p x ⋯) (one : p 1 ⋯) (mul : ∀ (x y : M) (hx : x ∈ Submonoid.closure s) (hy : y ∈ Submonoid.closure s), p x hx → p y hy → p (x * y) ⋯) {x : M} (hx : x ∈ Submonoid.closure s) : p x hx

Alias of Submonoid.closure_induction.

---

An induction principle for closure membership. If p holds for 1 and all elements of s, and is preserved under multiplication, then p holds for all elements of the closure of s.

theorem AddSubmonoid.closure_induction₂ {M : Type u_1} [AddZeroClass M] {s : Set M} {p : (x y : M) → x ∈ AddSubmonoid.closure s → y ∈ AddSubmonoid.closure s → Prop} (mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯) (one_left : ∀ (x : M) (hx : x ∈ AddSubmonoid.closure s), p 0 x ⋯ hx) (one_right : ∀ (x : M) (hx : x ∈ AddSubmonoid.closure s), p x 0 hx ⋯) (mul_left : ∀ (x y z : M) (hx : x ∈ AddSubmonoid.closure s) (hy : y ∈ AddSubmonoid.closure s) (hz : z ∈ AddSubmonoid.closure s), p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz) (mul_right : ∀ (x y z : M) (hx : x ∈ AddSubmonoid.closure s) (hy : y ∈ AddSubmonoid.closure s) (hz : z ∈ AddSubmonoid.closure s), p z x hz hx → p z y hz hy → p z (x + y) hz ⋯) {x : M} {y : M} (hx : x ∈ AddSubmonoid.closure s) (hy : y ∈ AddSubmonoid.closure s) : p x y hx hy

An induction principle for additive closure membership for predicates with two arguments.

theorem Submonoid.closure_induction₂ {M : Type u_1} [MulOneClass M] {s : Set M} {p : (x y : M) → x ∈ Submonoid.closure s → y ∈ Submonoid.closure s → Prop} (mem : ∀ (x y : M) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯) (one_left : ∀ (x : M) (hx : x ∈ Submonoid.closure s), p 1 x ⋯ hx) (one_right : ∀ (x : M) (hx : x ∈ Submonoid.closure s), p x 1 hx ⋯) (mul_left : ∀ (x y z : M) (hx : x ∈ Submonoid.closure s) (hy : y ∈ Submonoid.closure s) (hz : z ∈ Submonoid.closure s), p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz) (mul_right : ∀ (x y z : M) (hx : x ∈ Submonoid.closure s) (hy : y ∈ Submonoid.closure s) (hz : z ∈ Submonoid.closure s), p z x hz hx → p z y hz hy → p z (x * y) hz ⋯) {x : M} {y : M} (hx : x ∈ Submonoid.closure s) (hy : y ∈ Submonoid.closure s) : p x y hx hy

An induction principle for closure membership for predicates with two arguments.

theorem AddSubmonoid.dense_induction {M : Type u_1} [AddZeroClass M] {p : M → Prop} (s : Set M) (closure : AddSubmonoid.closure s = ⊤) (mem : ∀ x ∈ s, p x) (one : p 0) (mul : ∀ (x y : M), p x → p y → p (x + y)) (x : M) : p x

If s is a dense set in an additive monoid M, AddSubmonoid.closure s = ⊤, then in order to prove that some predicate p holds for all x : M it suffices to verify p x for x ∈ s, verify p 0, and verify that p x and p y imply p (x + y).

theorem Submonoid.dense_induction {M : Type u_1} [MulOneClass M] {p : M → Prop} (s : Set M) (closure : Submonoid.closure s = ⊤) (mem : ∀ x ∈ s, p x) (one : p 1) (mul : ∀ (x y : M), p x → p y → p (x * y)) (x : M) : p x

If s is a dense set in a monoid M, Submonoid.closure s = ⊤, then in order to prove that some predicate p holds for all x : M it suffices to verify p x for x ∈ s, verify p 1, and verify that p x and p y imply p (x * y).

theorem Submonoid.closure_eq_one_union {M : Type u_1} [MulOneClass M] (s : Set M) : ↑(Submonoid.closure s) = {1} ∪ ↑(Subsemigroup.closure s)

The Submonoid.closure of a set is the union of {1} and its Subsemigroup.closure.

theorem AddSubmonoid.gi.proof_2 (M : Type u_1) [AddZeroClass M] : ∀ (x : Set M) (x_1 : ↑(AddSubmonoid.closure x) ≤ x), (fun (s : Set M) (x : ↑(AddSubmonoid.closure s) ≤ s) => AddSubmonoid.closure s) x x_1 = (fun (s : Set M) (x : ↑(AddSubmonoid.closure s) ≤ s) => AddSubmonoid.closure s) x x_1

def AddSubmonoid.gi (M : Type u_1) [AddZeroClass M] : GaloisInsertion AddSubmonoid.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• AddSubmonoid.gi M = { choice := fun (s : Set M) (x : ↑(AddSubmonoid.closure s) ≤ s) => AddSubmonoid.closure s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

theorem AddSubmonoid.gi.proof_1 (M : Type u_1) [AddZeroClass M] : ∀ (x : AddSubmonoid M), ↑x ⊆ ↑(AddSubmonoid.closure ↑x)

def Submonoid.gi (M : Type u_1) [MulOneClass M] : GaloisInsertion Submonoid.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• Submonoid.gi M = { choice := fun (s : Set M) (x : ↑(Submonoid.closure s) ≤ s) => Submonoid.closure s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

@[simp]

theorem AddSubmonoid.closure_eq {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) : AddSubmonoid.closure ↑S = S

Additive closure of an additive submonoid S equals S

@[simp]

theorem Submonoid.closure_eq {M : Type u_1} [MulOneClass M] (S : Submonoid M) : Submonoid.closure ↑S = S

Closure of a submonoid S equals S.

@[simp]

theorem AddSubmonoid.closure_empty {M : Type u_1} [AddZeroClass M] : AddSubmonoid.closure ∅ = ⊥

@[simp]

theorem Submonoid.closure_empty {M : Type u_1} [MulOneClass M] : Submonoid.closure ∅ = ⊥

@[simp]

theorem AddSubmonoid.closure_univ {M : Type u_1} [AddZeroClass M] : AddSubmonoid.closure Set.univ = ⊤

@[simp]

theorem Submonoid.closure_univ {M : Type u_1} [MulOneClass M] : Submonoid.closure Set.univ = ⊤

theorem AddSubmonoid.closure_union {M : Type u_1} [AddZeroClass M] (s : Set M) (t : Set M) : AddSubmonoid.closure (s ∪ t) = AddSubmonoid.closure s ⊔ AddSubmonoid.closure t

theorem Submonoid.closure_union {M : Type u_1} [MulOneClass M] (s : Set M) (t : Set M) : Submonoid.closure (s ∪ t) = Submonoid.closure s ⊔ Submonoid.closure t

theorem AddSubmonoid.sup_eq_closure {M : Type u_1} [AddZeroClass M] (N : AddSubmonoid M) (N' : AddSubmonoid M) : N ⊔ N' = AddSubmonoid.closure (↑N ∪ ↑N')

theorem Submonoid.sup_eq_closure {M : Type u_1} [MulOneClass M] (N : Submonoid M) (N' : Submonoid M) : N ⊔ N' = Submonoid.closure (↑N ∪ ↑N')

theorem AddSubmonoid.closure_iUnion {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} (s : ι → Set M) : AddSubmonoid.closure (⋃ (i : ι), s i) = ⨆ (i : ι), AddSubmonoid.closure (s i)

theorem Submonoid.closure_iUnion {M : Type u_1} [MulOneClass M] {ι : Sort u_4} (s : ι → Set M) : Submonoid.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Submonoid.closure (s i)

theorem AddSubmonoid.closure_singleton_le_iff_mem {M : Type u_1} [AddZeroClass M] (m : M) (p : AddSubmonoid M) : AddSubmonoid.closure {m} ≤ p ↔ m ∈ p

theorem Submonoid.closure_singleton_le_iff_mem {M : Type u_1} [MulOneClass M] (m : M) (p : Submonoid M) : Submonoid.closure {m} ≤ p ↔ m ∈ p

theorem AddSubmonoid.mem_iSup {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} (p : ι → AddSubmonoid M) {m : M} : m ∈ ⨆ (i : ι), p i ↔ ∀ (N : AddSubmonoid M), (∀ (i : ι), p i ≤ N) → m ∈ N

theorem Submonoid.mem_iSup {M : Type u_1} [MulOneClass M] {ι : Sort u_4} (p : ι → Submonoid M) {m : M} : m ∈ ⨆ (i : ι), p i ↔ ∀ (N : Submonoid M), (∀ (i : ι), p i ≤ N) → m ∈ N

theorem AddSubmonoid.iSup_eq_closure {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} (p : ι → AddSubmonoid M) : ⨆ (i : ι), p i = AddSubmonoid.closure (⋃ (i : ι), ↑(p i))

theorem Submonoid.iSup_eq_closure {M : Type u_1} [MulOneClass M] {ι : Sort u_4} (p : ι → Submonoid M) : ⨆ (i : ι), p i = Submonoid.closure (⋃ (i : ι), ↑(p i))

theorem AddSubmonoid.disjoint_def {M : Type u_1} [AddZeroClass M] {p₁ : AddSubmonoid M} {p₂ : AddSubmonoid M} : Disjoint p₁ p₂ ↔ ∀ {x : M}, x ∈ p₁ → x ∈ p₂ → x = 0

theorem Submonoid.disjoint_def {M : Type u_1} [MulOneClass M] {p₁ : Submonoid M} {p₂ : Submonoid M} : Disjoint p₁ p₂ ↔ ∀ {x : M}, x ∈ p₁ → x ∈ p₂ → x = 1

theorem AddSubmonoid.disjoint_def' {M : Type u_1} [AddZeroClass M] {p₁ : AddSubmonoid M} {p₂ : AddSubmonoid M} : Disjoint p₁ p₂ ↔ ∀ {x y : M}, x ∈ p₁ → y ∈ p₂ → x = y → x = 0

theorem Submonoid.disjoint_def' {M : Type u_1} [MulOneClass M] {p₁ : Submonoid M} {p₂ : Submonoid M} : Disjoint p₁ p₂ ↔ ∀ {x y : M}, x ∈ p₁ → y ∈ p₂ → x = y → x = 1

theorem AddMonoidHom.eqOn_closureM {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {f : M →+ N} {g : M →+ N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(AddSubmonoid.closure s)

If two monoid homomorphisms are equal on a set, then they are equal on its submonoid closure.

theorem MonoidHom.eqOn_closureM {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {f : M →* N} {g : M →* N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Submonoid.closure s)

If two monoid homomorphisms are equal on a set, then they are equal on its submonoid closure.

theorem AddMonoidHom.eq_of_eqOn_denseM {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {s : Set M} (hs : AddSubmonoid.closure s = ⊤) {f : M →+ N} {g : M →+ N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem MonoidHom.eq_of_eqOn_denseM {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {s : Set M} (hs : Submonoid.closure s = ⊤) {f : M →* N} {g : M →* N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem IsAddUnit.addSubmonoid.proof_2 (M : Type u_1) [AddMonoid M] : IsAddUnit 0

theorem IsAddUnit.addSubmonoid.proof_1 (M : Type u_1) [AddMonoid M] {a : M} {b : M} (ha : a ∈ setOf IsAddUnit) (hb : b ∈ setOf IsAddUnit) : a + b ∈ setOf IsAddUnit

def IsAddUnit.addSubmonoid (M : Type u_4) [AddMonoid M] : AddSubmonoid M

The additive submonoid consisting of the additive units of an additive monoid

Equations

• IsAddUnit.addSubmonoid M = { carrier := setOf IsAddUnit, add_mem' := ⋯, zero_mem' := ⋯ }

def IsUnit.submonoid (M : Type u_4) [Monoid M] : Submonoid M

The submonoid consisting of the units of a monoid

Equations

• IsUnit.submonoid M = { carrier := setOf IsUnit, mul_mem' := ⋯, one_mem' := ⋯ }

theorem IsAddUnit.mem_addSubmonoid_iff {M : Type u_4} [AddMonoid M] (a : M) : a ∈ IsAddUnit.addSubmonoid M ↔ IsAddUnit a

theorem IsUnit.mem_submonoid_iff {M : Type u_4} [Monoid M] (a : M) : a ∈ IsUnit.submonoid M ↔ IsUnit a

def AddMonoidHom.ofClosureMEqTopLeft {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddMonoid N] {s : Set M} (f : M → N) (hs : AddSubmonoid.closure s = ⊤) (h1 : f 0 = 0) (hmul : ∀ x ∈ s, ∀ (y : M), f (x + y) = f x + f y) : M →+ N

Let s be a subset of an additive monoid M such that the closure of s is the whole monoid. Then AddMonoidHom.ofClosureEqTopLeft defines an additive monoid homomorphism from M asking for a proof of f (x + y) = f x + f y only for x ∈ s.

Equations

• AddMonoidHom.ofClosureMEqTopLeft f hs h1 hmul = { toFun := f, map_zero' := h1, map_add' := ⋯ }

theorem AddMonoidHom.ofClosureMEqTopLeft.proof_1 {M : Type u_2} {N : Type u_1} [AddMonoid M] [AddMonoid N] {s : Set M} (f : M → N) (hs : AddSubmonoid.closure s = ⊤) (h1 : f 0 = 0) (hmul : ∀ x ∈ s, ∀ (y : M), f (x + y) = f x + f y) (x : M) (y : M) : f (x + y) = f x + f y

def MonoidHom.ofClosureMEqTopLeft {M : Type u_4} {N : Type u_5} [Monoid M] [Monoid N] {s : Set M} (f : M → N) (hs : Submonoid.closure s = ⊤) (h1 : f 1 = 1) (hmul : ∀ x ∈ s, ∀ (y : M), f (x * y) = f x * f y) : M →* N

Let s be a subset of a monoid M such that the closure of s is the whole monoid. Then MonoidHom.ofClosureEqTopLeft defines a monoid homomorphism from M asking for a proof of f (x * y) = f x * f y only for x ∈ s.

Equations

• MonoidHom.ofClosureMEqTopLeft f hs h1 hmul = { toFun := f, map_one' := h1, map_mul' := ⋯ }

@[simp]

theorem AddMonoidHom.coe_ofClosureMEqTopLeft {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] {s : Set M} (f : M → N) (hs : AddSubmonoid.closure s = ⊤) (h1 : f 0 = 0) (hmul : ∀ x ∈ s, ∀ (y : M), f (x + y) = f x + f y) : ⇑(AddMonoidHom.ofClosureMEqTopLeft f hs h1 hmul) = f

@[simp]

theorem MonoidHom.coe_ofClosureMEqTopLeft {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {s : Set M} (f : M → N) (hs : Submonoid.closure s = ⊤) (h1 : f 1 = 1) (hmul : ∀ x ∈ s, ∀ (y : M), f (x * y) = f x * f y) : ⇑(MonoidHom.ofClosureMEqTopLeft f hs h1 hmul) = f

def AddMonoidHom.ofClosureMEqTopRight {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddMonoid N] {s : Set M} (f : M → N) (hs : AddSubmonoid.closure s = ⊤) (h1 : f 0 = 0) (hmul : ∀ (x y : M), y ∈ s → f (x + y) = f x + f y) : M →+ N

Let s be a subset of an additive monoid M such that the closure of s is the whole monoid. Then AddMonoidHom.ofClosureEqTopRight defines an additive monoid homomorphism from M asking for a proof of f (x + y) = f x + f y only for y ∈ s.

Equations

• AddMonoidHom.ofClosureMEqTopRight f hs h1 hmul = { toFun := f, map_zero' := h1, map_add' := ⋯ }

theorem AddMonoidHom.ofClosureMEqTopRight.proof_1 {M : Type u_2} {N : Type u_1} [AddMonoid M] [AddMonoid N] {s : Set M} (f : M → N) (hs : AddSubmonoid.closure s = ⊤) (h1 : f 0 = 0) (hmul : ∀ (x y : M), y ∈ s → f (x + y) = f x + f y) (x : M) (y : M) : { toFun := f, map_zero' := h1 }.toFun (x + y) = { toFun := f, map_zero' := h1 }.toFun x + { toFun := f, map_zero' := h1 }.toFun y

def MonoidHom.ofClosureMEqTopRight {M : Type u_4} {N : Type u_5} [Monoid M] [Monoid N] {s : Set M} (f : M → N) (hs : Submonoid.closure s = ⊤) (h1 : f 1 = 1) (hmul : ∀ (x y : M), y ∈ s → f (x * y) = f x * f y) : M →* N

Let s be a subset of a monoid M such that the closure of s is the whole monoid. Then MonoidHom.ofClosureEqTopRight defines a monoid homomorphism from M asking for a proof of f (x * y) = f x * f y only for y ∈ s.

Equations

• MonoidHom.ofClosureMEqTopRight f hs h1 hmul = { toFun := f, map_one' := h1, map_mul' := ⋯ }

@[simp]

theorem AddMonoidHom.coe_ofClosureMEqTopRight {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] {s : Set M} (f : M → N) (hs : AddSubmonoid.closure s = ⊤) (h1 : f 0 = 0) (hmul : ∀ (x y : M), y ∈ s → f (x + y) = f x + f y) : ⇑(AddMonoidHom.ofClosureMEqTopRight f hs h1 hmul) = f

@[simp]

theorem MonoidHom.coe_ofClosureMEqTopRight {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {s : Set M} (f : M → N) (hs : Submonoid.closure s = ⊤) (h1 : f 1 = 1) (hmul : ∀ (x y : M), y ∈ s → f (x * y) = f x * f y) : ⇑(MonoidHom.ofClosureMEqTopRight f hs h1 hmul) = f