Theorem conjghm 13808

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 109	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ Grp)
4	3	adantr 276	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
5	1, 2	grpcl 13536	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
6	5	3expa 1227	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝐴 + 𝑥) ∈ 𝑋)
7		simplr 528	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8		conjghm.m	. . . . . 6 ⊢ − = (-g‘𝐺)
9	1, 8	grpsubcl 13608	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
10	4, 6, 7, 9	syl3anc 1271	. . . 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 5788	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶𝑋)
13	3	adantr 276	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐺 ∈ Grp)
14		simplr 528	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
15		simprl 529	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
16	1, 2, 13, 14, 15	grpcld 13542	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐴 + 𝑦) ∈ 𝑋)
17	1, 8	grpsubcl 13608	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑦) − 𝐴) ∈ 𝑋)
18	13, 16, 14, 17	syl3anc 1271	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) − 𝐴) ∈ 𝑋)
19		simprr 531	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
20	1, 8	grpsubcl 13608	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑧 − 𝐴) ∈ 𝑋)
21	13, 19, 14, 20	syl3anc 1271	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 − 𝐴) ∈ 𝑋)
22	1, 2	grpass 13537	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (((𝐴 + 𝑦) − 𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑧 − 𝐴) ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = (((𝐴 + 𝑦) − 𝐴) + (𝐴 + (𝑧 − 𝐴))))
23	13, 18, 14, 21, 22	syl13anc 1273	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = (((𝐴 + 𝑦) − 𝐴) + (𝐴 + (𝑧 − 𝐴))))
24	1, 2, 8	grpnpcan 13620	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑦) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐴 + 𝑦) − 𝐴) + 𝐴) = (𝐴 + 𝑦))
25	13, 16, 14, 24	syl3anc 1271	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) − 𝐴) + 𝐴) = (𝐴 + 𝑦))
26	25	oveq1d 6015	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
27	1, 2, 8	grpaddsubass 13618	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
28	13, 16, 19, 14, 27	syl13anc 1273	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + 𝑦) + (𝑧 − 𝐴)))
29	1, 2	grpass 13537	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
30	13, 14, 15, 19, 29	syl13anc 1273	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑦) + 𝑧) = (𝐴 + (𝑦 + 𝑧)))
31	30	oveq1d 6015	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) + 𝑧) − 𝐴) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
32	26, 28, 31	3eqtr2rd 2269	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) = ((((𝐴 + 𝑦) − 𝐴) + 𝐴) + (𝑧 − 𝐴)))
33	1, 2, 8	grpaddsubass 13618	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) = (𝐴 + (𝑧 − 𝐴)))
34	13, 14, 19, 14, 33	syl13anc 1273	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) = (𝐴 + (𝑧 − 𝐴)))
35	34	oveq2d 6016	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)) = (((𝐴 + 𝑦) − 𝐴) + (𝐴 + (𝑧 − 𝐴))))
36	23, 32, 35	3eqtr4d 2272	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) = (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)))
37		oveq2 6008	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝑧)))
38	37	oveq1d 6015	. . . . 5 ⊢ (𝑥 = (𝑦 + 𝑧) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
39	1, 2, 13, 15, 19	grpcld 13542	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 + 𝑧) ∈ 𝑋)
40	1, 2, 13, 14, 39	grpcld 13542	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐴 + (𝑦 + 𝑧)) ∈ 𝑋)
41	1, 8	grpsubcl 13608	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + (𝑦 + 𝑧)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) ∈ 𝑋)
42	13, 40, 14, 41	syl3anc 1271	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + (𝑦 + 𝑧)) − 𝐴) ∈ 𝑋)
43	11, 38, 39, 42	fvmptd3 5727	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐴 + (𝑦 + 𝑧)) − 𝐴))
44		oveq2 6008	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 + 𝑥) = (𝐴 + 𝑦))
45	44	oveq1d 6015	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + 𝑦) − 𝐴))
46	11, 45, 15, 18	fvmptd3 5727	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = ((𝐴 + 𝑦) − 𝐴))
47		oveq2 6008	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐴 + 𝑥) = (𝐴 + 𝑧))
48	47	oveq1d 6015	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + 𝑧) − 𝐴))
49	1, 2, 13, 14, 19	grpcld 13542	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐴 + 𝑧) ∈ 𝑋)
50	1, 8	grpsubcl 13608	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑧) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑧) − 𝐴) ∈ 𝑋)
51	13, 49, 14, 50	syl3anc 1271	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐴 + 𝑧) − 𝐴) ∈ 𝑋)
52	11, 48, 19, 51	fvmptd3 5727	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = ((𝐴 + 𝑧) − 𝐴))
53	46, 52	oveq12d 6018	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦) + (𝐹‘𝑧)) = (((𝐴 + 𝑦) − 𝐴) + ((𝐴 + 𝑧) − 𝐴)))
54	36, 43, 53	3eqtr4d 2272	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦) + (𝐹‘𝑧)))
55	1, 1, 2, 2, 3, 3, 12, 54	isghmd 13784	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐺))
56	3	adantr 276	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ Grp)
57		eqid 2229	. . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺)
58	1, 57	grpinvcl 13576	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
59	58	adantr 276	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
60		simpr 110	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
61		simplr 528	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐴 ∈ 𝑋)
62	1, 2, 56, 60, 61	grpcld 13542	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑦 + 𝐴) ∈ 𝑋)
63	1, 2, 56, 59, 62	grpcld 13542	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ∈ 𝑋)
64	3	adantr 276	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐺 ∈ Grp)
65	62	adantrl 478	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 + 𝐴) ∈ 𝑋)
66	6	adantrr 479	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐴 + 𝑥) ∈ 𝑋)
67	58	adantr 276	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
68	1, 2	grplcan 13590	. . . . . 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)))
69	64, 65, 66, 67, 68	syl13anc 1273	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)) ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
70		eqid 2229	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
71	1, 2, 70, 57	grplinv 13578	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + 𝐴) = (0g‘𝐺))
72	71	adantr 276	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((invg‘𝐺)‘𝐴) + 𝐴) = (0g‘𝐺))
73	72	oveq1d 6015	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = ((0g‘𝐺) + 𝑥))
74		simplr 528	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
75		simprl 529	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
76	1, 2	grpass 13537	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
77	64, 67, 74, 75, 76	syl13anc 1273	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝐴) + 𝑥) = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
78	1, 2, 70	grplid 13559	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((0g‘𝐺) + 𝑥) = 𝑥)
79	78	ad2ant2r 509	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((0g‘𝐺) + 𝑥) = 𝑥)
80	73, 77, 79	3eqtr3rd 2271	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 = (((invg‘𝐺)‘𝐴) + (𝐴 + 𝑥)))
81	80	eqeq2d 2241	. . . . 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))))
82		simprr 531	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
83	1, 2, 8	grpsubadd 13616	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ((𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐴 + 𝑥) − 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
84	64, 66, 74, 82, 83	syl13anc 1273	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐴 + 𝑥) − 𝐴) = 𝑦 ↔ (𝑦 + 𝐴) = (𝐴 + 𝑥)))
85	69, 81, 84	3bitr4d 220	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥 ↔ ((𝐴 + 𝑥) − 𝐴) = 𝑦))
86		eqcom 2231	. . . 4 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) = 𝑥)
87		eqcom 2231	. . . 4 ⊢ (𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ ((𝐴 + 𝑥) − 𝐴) = 𝑦)
88	85, 86, 87	3bitr4g 223	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑦 + 𝐴)) ↔ 𝑦 = ((𝐴 + 𝑥) − 𝐴)))
89	11, 10, 63, 88	f1o2d 6209	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋–1-1-onto→𝑋)
90	55, 89	jca 306	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋–1-1-onto→𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ↦ cmpt 4144  –1-1-onto→wf1o 5316  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284  Grpcgrp 13528  inv_gcminusg 13529  -_gcsg 13530   GrpHom cghm 13772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-ghm 13773

This theorem is referenced by:  conjsubg  13809  conjsubgen  13810