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group unit lemmas #2189

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Jan 4, 2025
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group unit lemmas #2189

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8 changes: 3 additions & 5 deletions theories/Algebra/AbGroups/Centralizer.v
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Original file line number Diff line number Diff line change
Expand Up @@ -11,10 +11,8 @@ Local Open Scope mc_mult_scope.
Definition centralizer {G : Group} (g : G)
:= fun h => g * h = h * g.

Definition centralizer_unit {G : Group} (g : G) : centralizer g mon_unit.
Proof.
exact (grp_unit_r _ @ (grp_unit_l _)^).
Defined.
Definition centralizer_unit {G : Group} (g : G) : centralizer g mon_unit
:= grp_g1_1g _ _ idpath.

Definition centralizer_sgop {G : Group} (g h k : G)
(p : centralizer g h) (q : centralizer g k)
Expand All @@ -33,7 +31,7 @@ Definition centralizer_inverse {G : Group} (g h : G)
Proof.
unfold centralizer in *.
symmetry.
refine ((grp_unit_r _)^ @ _ @ grp_unit_l _).
apply grp_g1_1g.
refine (ap ((_ * g) *.) (grp_inv_r h)^ @ _ @ ap (.* (g * _)) (grp_inv_l h)).
refine (grp_assoc _ _ _ @ _ @ (grp_assoc _ _ _)^).
refine (ap (.* h^) _).
Expand Down
11 changes: 8 additions & 3 deletions theories/Algebra/Groups/Group.v
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Expand Up @@ -465,6 +465,12 @@ Section GroupEquations.
Definition grp_inv_gV_g : (x * y^) * y = x
:= (grp_assoc _ _ _)^ @ ap (x *.) (grp_inv_l y) @ grp_unit_r x.

Definition grp_1g_g1 : x = y <~> 1 * x = y * 1
:= equiv_concat_r (grp_unit_r _)^ _ oE equiv_concat_l (grp_unit_l _) _.

Definition grp_g1_1g : x = y <~> x * 1 = 1 * y
:= equiv_concat_r (grp_unit_l _)^ _ oE equiv_concat_l (grp_unit_r _) _.

End GroupEquations.

(** ** Cancelation lemmas *)
Expand Down Expand Up @@ -678,7 +684,7 @@ Definition grp_pow_commutes {G : Group} (n : Int) (g h : G)
: h * (grp_pow g n) = (grp_pow g n) * h.
Proof.
induction n.
- exact (grp_unit_r _ @ (grp_unit_l _)^).
- by apply grp_g1_1g.
- rewrite grp_pow_succ.
nrapply grp_commutes_op; assumption.
- rewrite grp_pow_pred.
Expand Down Expand Up @@ -1168,8 +1174,7 @@ Definition grp_conj_unit {G : Group} : grp_conj (G:=G) 1 $== Id _.
Proof.
intros x.
apply grp_moveR_gV.
rhs nrapply grp_unit_r.
nrapply grp_unit_l.
by nrapply grp_1g_g1.
Defined.

(** Conjugation commutes with group homomorphisms. *)
Expand Down
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