Theorem ghmnsgima 19370

Description: The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypothesis

Ref	Expression
ghmnsgima.1	⊢ 𝑌 = (Base‘𝑇)

Assertion

Ref	Expression
ghmnsgima	⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇))

Proof of Theorem ghmnsgima

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

Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		nsgsubg 19284	. . . 4 ⊢ (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
3	2	3ad2ant2 1152	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4		ghmima 19367	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
5	1, 3, 4	syl2anc 596	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
6	1	adantr 486	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7		ghmgrp1 19348	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	6, 7	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑆 ∈ Grp)
9		simprl 783	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑆))
10		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10	subgss 19253	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ (Base‘𝑆))
12	3, 11	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ⊆ (Base‘𝑆))
13	12	adantr 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14		simprr 785	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
15	13, 14	sseldd 3932	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑆))
16		eqid 2760	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17	10, 16	grpcl 19068	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
18	8, 9, 15, 17	syl3anc 1398	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
19		eqid 2760	. . . . . . . 8 ⊢ (-g‘𝑆) = (-g‘𝑆)
20		eqid 2760	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
21	10, 19, 20	ghmsub 19354	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) = ((𝐹‘(𝑧(+g‘𝑆)𝑥))(-g‘𝑇)(𝐹‘𝑧)))
22	6, 18, 9, 21	syl3anc 1398	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) = ((𝐹‘(𝑧(+g‘𝑆)𝑥))(-g‘𝑇)(𝐹‘𝑧)))
23		eqid 2760	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
24	10, 16, 23	ghmlin 19351	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘(𝑧(+g‘𝑆)𝑥)) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥)))
25	6, 9, 15, 24	syl3anc 1398	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘(𝑧(+g‘𝑆)𝑥)) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥)))
26	25	oveq1d 7429	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → ((𝐹‘(𝑧(+g‘𝑆)𝑥))(-g‘𝑇)(𝐹‘𝑧)) = (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)))
27	22, 26	eqtrd 2795	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) = (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)))
28		ghmnsgima.1	. . . . . . . . . 10 ⊢ 𝑌 = (Base‘𝑇)
29	10, 28	ghmf 19350	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶𝑌)
30	1, 29	syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹:(Base‘𝑆)⟶𝑌)
31	30	adantr 486	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
32	31	ffnd 6704	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 Fn (Base‘𝑆))
33		simpl2 1211	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
34	10, 16, 19	nsgconj 19285	. . . . . . 7 ⊢ ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈)
35	33, 9, 14, 34	syl3anc 1398	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈)
36		fnfvima 7233	. . . . . 6 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆) ∧ ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) ∈ (𝐹 “ 𝑈))
37	32, 13, 35, 36	syl3anc 1398	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) ∈ (𝐹 “ 𝑈))
38	27, 37	eqeltrrd 2861	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))
39	38	ralrimivva 3205	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑧 ∈ (Base‘𝑆)∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))
40	30	ffnd 6704	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41		oveq1 7421	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)𝑦))
42		id 23	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → 𝑥 = (𝐹‘𝑧))
43	41, 42	oveq12d 7432	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) = (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)))
44	43	eleq1d 2845	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑧) → (((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
45	44	ralbidv 3185	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
46	45	ralrn 7082	. . . . 5 ⊢ (𝐹 Fn (Base‘𝑆) → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
47	40, 46	syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
48		simp3 1156	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
49	48	raleqdv 3319	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈)))
50		oveq2 7422	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥)))
51	50	oveq1d 7429	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) = (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)))
52	51	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → ((((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
53	52	ralima 7237	. . . . . 6 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆)) → (∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
54	40, 12, 53	syl2anc 596	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
55	54	ralbidv 3185	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
56	47, 49, 55	3bitr3d 312	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)))
57	39, 56	mpbird 260	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈))
58	28, 23, 20	isnsg3 19286	. 2 ⊢ ((𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇) ↔ ((𝐹 “ 𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈)))
59	5, 57, 58	sylanbrc 595	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ran crn 5656   “ cima 5658   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  +_gcplusg 17345  Grpcgrp 19060  -_gcsg 19062  SubGrpcsubg 19246  NrmSGrpcnsg 19247   GrpHom cghm 19343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-sets 17259  df-slot 17277  df-ndx 17289  df-base 17305  df-ress 17326  df-plusg 17358  df-0g 17529  df-mgm 18733  df-sgrp 18824  df-mnd 18840  df-grp 19063  df-minusg 19064  df-sbg 19065  df-subg 19249  df-nsg 19250  df-ghm 19344

This theorem is used by: (None)