Theorem conjghm 19123

Description: Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	⊢ 𝑋 = (Base‘𝐺)
conjghm.p	⊢ + = (+g‘𝐺)
conjghm.m	⊢ − = (-g‘𝐺)
conjghm.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴))

Assertion

Ref	Expression
conjghm	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋–1-1-onto→𝑋))

Distinct variable groups:   𝑥, −   𝑥, +   𝑥,𝐴   𝑥,𝐺   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem conjghm

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		conjghm.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		conjghm.p	. . 3 ⊢ + = (+g‘𝐺)
3		simpl 484	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ Grp)
4	3	adantr 482	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
5	1, 2	grpcl 18827	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
6	5	3expa 1119	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
7		simplr 768	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8		conjghm.m	. . . . . 6 ⊢ − = (-g‘𝐺)
9	1, 8	grpsubcl 18903	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
10	4, 6, 7, 9	syl3anc 1372	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
11		conjghm.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴))
12	10, 11	fmptd 7114	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶𝑋)
13	3	adantr 482	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐺 ∈ Grp)
14		simplr 768	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
15		simprl 770	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
16	1, 2	grpcl 18827	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 + 𝑦) ∈ 𝑋)
17	13, 14, 15, 16	syl3anc 1372	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐴 + 𝑦) ∈ 𝑋)
18	1, 8	grpsubcl 18903	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑦) − 𝐴) ∈ 𝑋)
19	13, 17, 14, 18	syl3anc 1372	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) − 𝐴) ∈ 𝑋)
20		simprr 772	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
21	1, 8	grpsubcl 18903	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑧 − 𝐴) ∈ 𝑋)
22	13, 20, 14, 21	syl3anc 1372	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 − 𝐴) ∈ 𝑋)
23	1, 2	grpass 18828	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (((𝐴 + 𝑦) − 𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑧 − 𝐴) ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = (((𝐴 + 𝑦) − 𝐴) + (𝐴 + (𝑧 − 𝐴))))
24	13, 19, 14, 22, 23	syl13anc 1373	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = (((𝐴 + 𝑦) − 𝐴) + (𝐴 + (𝑧 − 𝐴))))
25	1, 2, 8	grpnpcan 18915	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐴 + 𝑦) − 𝐴) + 𝐴) = (𝐴 + 𝑦))
26	13, 17, 14, 25	syl3anc 1372	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) − 𝐴) + 𝐴) = (𝐴 + 𝑦))
27	26	oveq1d 7424	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
28	1, 2, 8	grpaddsubass 18913	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
29	13, 17, 20, 14, 28	syl13anc 1373	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
30	1, 2	grpass 18828	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
31	13, 14, 15, 20, 30	syl13anc 1373	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
32	31	oveq1d 7424	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
33	27, 29, 32	3eqtr2rd 2780	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) = ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)))
34	1, 2, 8	grpaddsubass 18913	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) = (𝐴 + (𝑧 − 𝐴)))
35	13, 14, 20, 14, 34	syl13anc 1373	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) = (𝐴 + (𝑧 − 𝐴)))
36	35	oveq2d 7425	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)) = (((𝐴 + 𝑦) − 𝐴) + (𝐴 + (𝑧 − 𝐴))))
37	24, 33, 36	3eqtr4d 2783	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) = (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)))
38	1, 2	grpcl 18827	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦 + 𝑧) ∈ 𝑋)
39	13, 15, 20, 38	syl3anc 1372	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 + 𝑧) ∈ 𝑋)
40		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝑧)))
41	40	oveq1d 7424	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
42		ovex 7442	. . . . . 6 ⊢ ((𝐴 + (𝑦 + 𝑧)) − 𝐴) ∈ V
43	41, 11, 42	fvmpt 6999	. . . . 5 ⊢ ((𝑦 + 𝑧) ∈ 𝑋 → (𝐹‘(𝑦 + 𝑧)) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
44	39, 43	syl 17	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
45		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
46	45	oveq1d 7424	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + 𝑦) − 𝐴))
47		ovex 7442	. . . . . . 7 ⊢ ((𝐴 + 𝑦) − 𝐴) ∈ V
48	46, 11, 47	fvmpt 6999	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = ((𝐴 + 𝑦) − 𝐴))
49	48	ad2antrl 727	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = ((𝐴 + 𝑦) − 𝐴))
50		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 + 𝑥) = (𝐴 + 𝑧))
51	50	oveq1d 7424	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + 𝑧) − 𝐴))
52		ovex 7442	. . . . . . 7 ⊢ ((𝐴 + 𝑧) − 𝐴) ∈ V
53	51, 11, 52	fvmpt 6999	. . . . . 6 ⊢ (𝑧 ∈ 𝑋 → (𝐹‘𝑧) = ((𝐴 + 𝑧) − 𝐴))
54	53	ad2antll 728	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = ((𝐴 + 𝑧) − 𝐴))
55	49, 54	oveq12d 7427	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦) + (𝐹‘𝑧)) = (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)))
56	37, 44, 55	3eqtr4d 2783	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) + (𝐹‘𝑧)))
57	1, 1, 2, 2, 3, 3, 12, 56	isghmd 19101	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐺))
58	3	adantr 482	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ Grp)
59		eqid 2733	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
60	1, 59	grpinvcl 18872	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
61	60	adantr 482	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
62		simpr 486	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
63		simplr 768	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐴 ∈ 𝑋)
64	1, 2	grpcl 18827	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
65	58, 62, 63, 64	syl3anc 1372	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
66	1, 2	grpcl 18827	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑦 + 𝐴) ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
67	58, 61, 65, 66	syl3anc 1372	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
68	3	adantr 482	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
69	65	adantrl 715	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 + 𝐴) ∈ 𝑋)
70	6	adantrr 716	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐴 + 𝑥) ∈ 𝑋)
71	60	adantr 482	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
72	1, 2	grplcan 18885	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ((𝑦 + 𝐴) ∈ 𝑋 ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)) ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
73	68, 69, 70, 71, 72	syl13anc 1373	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)) ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
74		eqid 2733	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
75	1, 2, 74, 59	grplinv 18874	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + 𝐴) = (0g‘𝐺))
76	75	adantr 482	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((invg‘𝐺)‘𝐴) + 𝐴) = (0g‘𝐺))
77	76	oveq1d 7424	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = ((0g‘𝐺) + 𝑥))
78		simplr 768	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
79		simprl 770	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
80	1, 2	grpass 18828	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
81	68, 71, 78, 79, 80	syl13anc 1373	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
82	1, 2, 74	grplid 18852	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((0g‘𝐺) + 𝑥) = 𝑥)
83	82	ad2ant2r 746	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((0g‘𝐺) + 𝑥) = 𝑥)
84	77, 81, 83	3eqtr3rd 2782	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
85	84	eqeq2d 2744	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥 ↔ (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥))))
86		simprr 772	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
87	1, 2, 8	grpsubadd 18911	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐴 + 𝑥) − 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
88	68, 70, 78, 86, 87	syl13anc 1373	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐴 + 𝑥) − 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
89	73, 85, 88	3bitr4d 311	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥 ↔ ((𝐴 + 𝑥) − 𝐴) = 𝑦))
90		eqcom 2740	. . . 4 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥)
91		eqcom 2740	. . . 4 ⊢ (𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ ((𝐴 + 𝑥) − 𝐴) = 𝑦)
92	89, 90, 91	3bitr4g 314	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ 𝑦 = ((𝐴 + 𝑥) − 𝐴)))
93	11, 10, 67, 92	f1o2d 7660	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋–1-1-onto→𝑋)
94	57, 93	jca 513	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋–1-1-onto→𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5232  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Grpcgrp 18819  inv_gcminusg 18820  -_gcsg 18821   GrpHom cghm 19089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-ghm 19090

This theorem is referenced by:  conjsubg  19124  conjsubgen  19125