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group_theory.submonoid.inverses

Submonoid of inverses #

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Given a submonoid N of a monoid M, we define the submonoid N.left_inv as the submonoid of left inverses of N. When M is commutative, we may define from_comm_left_inv : N.left_inv →* N since the inverses are unique. When N ≤ is_unit.submonoid M, this is precisely the pointwise inverse of N, and we may define left_inv_equiv : S.left_inv ≃* S.

For the pointwise inverse of submonoids of groups, please refer to group_theory.submonoid.pointwise.

TODO #

Define the submonoid of right inverses and two-sided inverses. See the comments of #10679 for a possible implementation.

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S.left_neg is the additive submonoid containing all the left additive inverses of S.

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def submonoid.left_inv {M : Type u_1} [monoid M] (S : submonoid M) :

S.left_inv is the submonoid containing all the left inverses of S.

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theorem submonoid.unit_mem_left_inv {M : Type u_1} [monoid M] (S : submonoid M) (x : Mˣ) (hx : x S) :
theorem add_submonoid.add_unit_mem_left_neg {M : Type u_1} [add_monoid M] (S : add_submonoid M) (x : add_units M) (hx : x S) :
noncomputable def add_submonoid.from_left_neg {M : Type u_1} [add_monoid M] (S : add_submonoid M) :

The function from S.left_add to S sending an element to its right additive inverse in S. This is an add_monoid_hom when M is commutative.

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noncomputable def submonoid.from_left_inv {M : Type u_1} [monoid M] (S : submonoid M) :

The function from S.left_inv to S sending an element to its right inverse in S. This is a monoid_hom when M is commutative.

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theorem add_submonoid.add_from_left_neg {M : Type u_1} [add_monoid M] (S : add_submonoid M) (x : (S.left_neg)) :
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theorem submonoid.mul_from_left_inv {M : Type u_1} [monoid M] (S : submonoid M) (x : (S.left_inv)) :
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theorem submonoid.from_left_inv_one {M : Type u_1} [monoid M] (S : submonoid M) :
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theorem submonoid.from_left_inv_mul {M : Type u_1} [comm_monoid M] (S : submonoid M) (x : (S.left_inv)) :
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theorem add_submonoid.from_left_neg_eq_iff {M : Type u_1} [add_comm_monoid M] (S : add_submonoid M) (a : (S.left_neg)) (b : M) :
(S.from_left_neg a) = b a + b = 0
theorem submonoid.from_left_inv_eq_iff {M : Type u_1} [comm_monoid M] (S : submonoid M) (a : (S.left_inv)) (b : M) :
(S.from_left_inv a) = b a * b = 1
noncomputable def add_submonoid.from_comm_left_neg {M : Type u_1} [add_comm_monoid M] (S : add_submonoid M) :

The add_monoid_hom from S.left_neg to S sending an element to its right additive inverse in S.

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noncomputable def submonoid.from_comm_left_inv {M : Type u_1} [comm_monoid M] (S : submonoid M) :

The monoid_hom from S.left_inv to S sending an element to its right inverse in S.

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noncomputable def submonoid.left_inv_equiv {M : Type u_1} [comm_monoid M] (S : submonoid M) (hS : S is_unit.submonoid M) :

The submonoid of pointwise inverse of S is mul_equiv to S.

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The additive submonoid of pointwise additive inverse of S is add_equiv to S.

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theorem submonoid.left_inv_equiv_apply {M : Type u_1} [comm_monoid M] (S : submonoid M) (hS : S is_unit.submonoid M) (ᾰ : (S.left_inv)) :
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theorem submonoid.left_inv_equiv_mul {M : Type u_1} [comm_monoid M] (S : submonoid M) (hS : S is_unit.submonoid M) (x : (S.left_inv)) :
((S.left_inv_equiv hS) x) * x = 1
theorem submonoid.mul_left_inv_equiv {M : Type u_1} [comm_monoid M] (S : submonoid M) (hS : S is_unit.submonoid M) (x : (S.left_inv)) :
x * ((S.left_inv_equiv hS) x) = 1
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theorem submonoid.left_inv_equiv_symm_mul {M : Type u_1} [comm_monoid M] (S : submonoid M) (hS : S is_unit.submonoid M) (x : S) :
(((S.left_inv_equiv hS).symm) x) * x = 1
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theorem submonoid.mul_left_inv_equiv_symm {M : Type u_1} [comm_monoid M] (S : submonoid M) (hS : S is_unit.submonoid M) (x : S) :
x * (((S.left_inv_equiv hS).symm) x) = 1
theorem submonoid.left_inv_eq_inv {M : Type u_1} [group M] (S : submonoid M) :
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theorem submonoid.from_left_inv_eq_inv {M : Type u_1} [group M] (S : submonoid M) (x : (S.left_inv)) :
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theorem submonoid.left_inv_equiv_symm_eq_inv {M : Type u_1} [comm_group M] (S : submonoid M) (hS : S is_unit.submonoid M) (x : S) :