Theorem ghmnsgima 13398

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 999	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		nsgsubg 13335	. . . 4 ⊢ (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
3	2	3ad2ant2 1021	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4		ghmima 13395	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
5	1, 3, 4	syl2anc 411	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
6	1	adantr 276	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7		ghmgrp1 13375	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑆 ∈ Grp)
9		simprl 529	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑆))
10		eqid 2196	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10	subgss 13304	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ (Base‘𝑆))
12	3, 11	syl 14	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ⊆ (Base‘𝑆))
13	12	adantr 276	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14		simprr 531	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
15	13, 14	sseldd 3184	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑆))
16		eqid 2196	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17	10, 16	grpcl 13140	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
18	8, 9, 15, 17	syl3anc 1249	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
19		eqid 2196	. . . . . . . 8 ⊢ (-g‘𝑆) = (-g‘𝑆)
20		eqid 2196	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
21	10, 19, 20	ghmsub 13381	. . . . . . 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 1249	. . . . . 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 2196	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
24	10, 16, 23	ghmlin 13378	. . . . . . . 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 1249	. . . . . . 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 5937	. . . . . 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 2229	. . . . 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 13377	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶𝑌)
30	1, 29	syl 14	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹:(Base‘𝑆)⟶𝑌)
31	30	adantr 276	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
32	31	ffnd 5408	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 Fn (Base‘𝑆))
33		simpl2 1003	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
34	10, 16, 19	nsgconj 13336	. . . . . . 7 ⊢ ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈)
35	33, 9, 14, 34	syl3anc 1249	. . . . . 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 5797	. . . . . 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 1249	. . . . 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 2274	. . . 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 2579	. . 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 5408	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41		oveq1 5929	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)𝑦))
42		id 19	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → 𝑥 = (𝐹‘𝑧))
43	41, 42	oveq12d 5940	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) = (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)))
44	43	eleq1d 2265	. . . . . . 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 2497	. . . . . 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 5700	. . . . 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 14	. . . 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 1001	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
49	48	raleqdv 2699	. . . 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 5930	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥)))
51	50	oveq1d 5937	. . . . . . . 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 2265	. . . . . . 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 5802	. . . . . 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 411	. . . . 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 2497	. . . 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 218	. . 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 167	. 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 13337	. 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 417	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ran crn 4664   “ cima 4666   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  Grpcgrp 13132  -_gcsg 13134  SubGrpcsubg 13297  NrmSGrpcnsg 13298   GrpHom cghm 13370

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-sbg 13137  df-subg 13300  df-nsg 13301  df-ghm 13371

This theorem is referenced by: (None)