Theorem conjghm 19224

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 482	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ Grp)
4	3	adantr 480	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
5	1, 2	grpcl 18917	. . . . . 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 769	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8		conjghm.m	. . . . . 6 ⊢ − = (-g‘𝐺)
9	1, 8	grpsubcl 18996	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
10	4, 6, 7, 9	syl3anc 1374	. . . 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 7066	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶𝑋)
13	3	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐺 ∈ Grp)
14		simplr 769	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
15		simprl 771	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
16	1, 2	grpcl 18917	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 + 𝑦) ∈ 𝑋)
17	13, 14, 15, 16	syl3anc 1374	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐴 + 𝑦) ∈ 𝑋)
18	1, 8	grpsubcl 18996	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑦) − 𝐴) ∈ 𝑋)
19	13, 17, 14, 18	syl3anc 1374	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) − 𝐴) ∈ 𝑋)
20		simprr 773	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
21	1, 8	grpsubcl 18996	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑧 − 𝐴) ∈ 𝑋)
22	13, 20, 14, 21	syl3anc 1374	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 − 𝐴) ∈ 𝑋)
23	1, 2	grpass 18918	. . . . . 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 1375	. . . . 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 19008	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐴 + 𝑦) − 𝐴) + 𝐴) = (𝐴 + 𝑦))
26	13, 17, 14, 25	syl3anc 1374	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) − 𝐴) + 𝐴) = (𝐴 + 𝑦))
27	26	oveq1d 7382	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
28	1, 2, 8	grpaddsubass 19006	. . . . . . 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 1375	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
30	1, 2	grpass 18918	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
31	13, 14, 15, 20, 30	syl13anc 1375	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
32	31	oveq1d 7382	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
33	27, 29, 32	3eqtr2rd 2778	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) = ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)))
34	1, 2, 8	grpaddsubass 19006	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) = (𝐴 + (𝑧 − 𝐴)))
35	13, 14, 20, 14, 34	syl13anc 1375	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) = (𝐴 + (𝑧 − 𝐴)))
36	35	oveq2d 7383	. . . . 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 2781	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) = (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)))
38	1, 2	grpcl 18917	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦 + 𝑧) ∈ 𝑋)
39	13, 15, 20, 38	syl3anc 1374	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 + 𝑧) ∈ 𝑋)
40		oveq2 7375	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝑧)))
41	40	oveq1d 7382	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
42		ovex 7400	. . . . . 6 ⊢ ((𝐴 + (𝑦 + 𝑧)) − 𝐴) ∈ V
43	41, 11, 42	fvmpt 6947	. . . . 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 7375	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
46	45	oveq1d 7382	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + 𝑦) − 𝐴))
47		ovex 7400	. . . . . . 7 ⊢ ((𝐴 + 𝑦) − 𝐴) ∈ V
48	46, 11, 47	fvmpt 6947	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = ((𝐴 + 𝑦) − 𝐴))
49	48	ad2antrl 729	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = ((𝐴 + 𝑦) − 𝐴))
50		oveq2 7375	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 + 𝑥) = (𝐴 + 𝑧))
51	50	oveq1d 7382	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + 𝑧) − 𝐴))
52		ovex 7400	. . . . . . 7 ⊢ ((𝐴 + 𝑧) − 𝐴) ∈ V
53	51, 11, 52	fvmpt 6947	. . . . . 6 ⊢ (𝑧 ∈ 𝑋 → (𝐹‘𝑧) = ((𝐴 + 𝑧) − 𝐴))
54	53	ad2antll 730	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = ((𝐴 + 𝑧) − 𝐴))
55	49, 54	oveq12d 7385	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦) + (𝐹‘𝑧)) = (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)))
56	37, 44, 55	3eqtr4d 2781	. . 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 19200	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐺))
58	3	adantr 480	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ Grp)
59		eqid 2736	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
60	1, 59	grpinvcl 18963	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
61	60	adantr 480	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
62		simpr 484	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
63		simplr 769	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐴 ∈ 𝑋)
64	1, 2	grpcl 18917	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
65	58, 62, 63, 64	syl3anc 1374	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
66	1, 2	grpcl 18917	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑦 + 𝐴) ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
67	58, 61, 65, 66	syl3anc 1374	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
68	3	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
69	65	adantrl 717	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 + 𝐴) ∈ 𝑋)
70	6	adantrr 718	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐴 + 𝑥) ∈ 𝑋)
71	60	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
72	1, 2	grplcan 18976	. . . . . 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 1375	. . . . 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 2736	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
75	1, 2, 74, 59	grplinv 18965	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + 𝐴) = (0g‘𝐺))
76	75	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((invg‘𝐺)‘𝐴) + 𝐴) = (0g‘𝐺))
77	76	oveq1d 7382	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = ((0g‘𝐺) + 𝑥))
78		simplr 769	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
79		simprl 771	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
80	1, 2	grpass 18918	. . . . . . . 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 1375	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
82	1, 2, 74	grplid 18943	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((0g‘𝐺) + 𝑥) = 𝑥)
83	82	ad2ant2r 748	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((0g‘𝐺) + 𝑥) = 𝑥)
84	77, 81, 83	3eqtr3rd 2780	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
85	84	eqeq2d 2747	. . . . 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 773	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
87	1, 2, 8	grpsubadd 19004	. . . . . 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 1375	. . . . 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 2743	. . . 4 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥)
91		eqcom 2743	. . . 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 7621	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋–1-1-onto→𝑋)
94	57, 93	jca 511	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋–1-1-onto→𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5166  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Grpcgrp 18909  inv_gcminusg 18910  -_gcsg 18911   GrpHom cghm 19187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-ghm 19188

This theorem is referenced by:  conjsubg  19225  conjsubgen  19226