Theorem ghmnsgima 19119

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 1136	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		nsgsubg 19037	. . . 4 ⊢ (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
3	2	3ad2ant2 1134	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4		ghmima 19116	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
5	1, 3, 4	syl2anc 584	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
6	1	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7		ghmgrp1 19097	. . . . . . . . 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 770	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑆))
10		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
11	10	subgss 19006	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14		simprr 772	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
15	13, 14	sseldd 3936	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑆))
16		eqid 2729	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
17	10, 16	grpcl 18820	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
18	8, 9, 15, 17	syl3anc 1373	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆))
19		eqid 2729	. . . . . . . 8 ⊢ (-g‘𝑆) = (-g‘𝑆)
20		eqid 2729	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
21	10, 19, 20	ghmsub 19103	. . . . . . 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 1373	. . . . . 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 2729	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
24	10, 16, 23	ghmlin 19100	. . . . . . . 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 1373	. . . . . . 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 7364	. . . . . 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 2764	. . . . 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 19099	. . . . . . . . 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 480	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
32	31	ffnd 6653	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 Fn (Base‘𝑆))
33		simpl2 1193	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
34	10, 16, 19	nsgconj 19038	. . . . . . 7 ⊢ ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈)
35	33, 9, 14, 34	syl3anc 1373	. . . . . 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 7169	. . . . . 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 1373	. . . . 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 2829	. . . 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 3172	. . 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 6653	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41		oveq1 7356	. . . . . . . . 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 7367	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) = (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)))
44	43	eleq1d 2813	. . . . . . 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 3152	. . . . . 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 7022	. . . . 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 1138	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
49	48	raleqdv 3289	. . . 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 7357	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥)))
51	50	oveq1d 7364	. . . . . . . 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 2813	. . . . . . 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 7173	. . . . . 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 584	. . . . 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 3152	. . . 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 309	. . 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 257	. 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 19039	. 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 583	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  ran crn 5620   “ cima 5622   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  Grpcgrp 18812  -_gcsg 18814  SubGrpcsubg 18999  NrmSGrpcnsg 19000   GrpHom cghm 19091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-grp 18815  df-minusg 18816  df-sbg 18817  df-subg 19002  df-nsg 19003  df-ghm 19092

This theorem is referenced by: (None)