Theorem ghmnsgima 19198

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 1133	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		nsgsubg 19117	. . . 4 ⊢ (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
3	2	3ad2ant2 1131	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4		ghmima 19195	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
5	1, 3, 4	syl2anc 582	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
6	1	adantr 479	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7		ghmgrp1 19176	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑆 ∈ Grp)
9		simprl 769	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑆))
10		eqid 2725	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10	subgss 19086	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ (Base‘𝑆))
12	3, 11	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ⊆ (Base‘𝑆))
13	12	adantr 479	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14		simprr 771	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
15	13, 14	sseldd 3973	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑆))
16		eqid 2725	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17	10, 16	grpcl 18902	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
18	8, 9, 15, 17	syl3anc 1368	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
19		eqid 2725	. . . . . . . 8 ⊢ (-g‘𝑆) = (-g‘𝑆)
20		eqid 2725	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
21	10, 19, 20	ghmsub 19182	. . . . . . 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 1368	. . . . . 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 2725	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
24	10, 16, 23	ghmlin 19179	. . . . . . . 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 1368	. . . . . . 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 7431	. . . . . 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 2765	. . . . 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 19178	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶𝑌)
30	1, 29	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹:(Base‘𝑆)⟶𝑌)
31	30	adantr 479	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
32	31	ffnd 6718	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 Fn (Base‘𝑆))
33		simpl2 1189	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
34	10, 16, 19	nsgconj 19118	. . . . . . 7 ⊢ ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈)
35	33, 9, 14, 34	syl3anc 1368	. . . . . 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 7241	. . . . . 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 1368	. . . . 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 2826	. . . 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 3191	. . 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 6718	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41		oveq1 7423	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)𝑦))
42		id 22	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑧) → 𝑥 = (𝐹‘𝑧))
43	41, 42	oveq12d 7434	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) = (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)))
44	43	eleq1d 2810	. . . . . . 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 3168	. . . . . 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 7093	. . . . 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 17	. . . 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 1135	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
49	48	raleqdv 3315	. . . 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 7424	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥)))
51	50	oveq1d 7431	. . . . . . . 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 2810	. . . . . . 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 7246	. . . . . 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 582	. . . . 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 3168	. . . 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 308	. . 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 256	. 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 19119	. 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 581	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051   ⊆ wss 3939  ran crn 5673   “ cima 5675   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7416  Basecbs 17179  +_gcplusg 17232  Grpcgrp 18894  -_gcsg 18896  SubGrpcsubg 19079  NrmSGrpcnsg 19080   GrpHom cghm 19171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18897  df-minusg 18898  df-sbg 18899  df-subg 19082  df-nsg 19083  df-ghm 19172

This theorem is referenced by: (None)