Theorem ghmpreima 17995

Description: The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Assertion

Ref	Expression
ghmpreima	⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))

Proof of Theorem ghmpreima

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5702	. . 3 ⊢ (◡𝐹 “ 𝑉) ⊆ dom 𝐹
2		eqid 2799	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2799	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 17977	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	adantr 473	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6	1, 5	fssdm 6272	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ⊆ (Base‘𝑆))
7		ghmgrp1 17975	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	7	adantr 473	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9		eqid 2799	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	2, 9	grpidcl 17766	. . . . 5 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ (Base‘𝑆))
11	8, 10	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆))
12		eqid 2799	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
13	9, 12	ghmid 17979	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
14	13	adantr 473	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
15	12	subg0cl 17915	. . . . . 6 ⊢ (𝑉 ∈ (SubGrp‘𝑇) → (0g‘𝑇) ∈ 𝑉)
16	15	adantl 474	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑇) ∈ 𝑉)
17	14, 16	eqeltrd 2878	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) ∈ 𝑉)
18	5	ffnd 6257	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19		elpreima 6563	. . . . 5 ⊢ (𝐹 Fn (Base‘𝑆) → ((0g‘𝑆) ∈ (◡𝐹 “ 𝑉) ↔ ((0g‘𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g‘𝑆)) ∈ 𝑉)))
20	18, 19	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((0g‘𝑆) ∈ (◡𝐹 “ 𝑉) ↔ ((0g‘𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g‘𝑆)) ∈ 𝑉)))
21	11, 17, 20	mpbir2and 705	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑆) ∈ (◡𝐹 “ 𝑉))
22	21	ne0d 4122	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ≠ ∅)
23		elpreima 6563	. . . . 5 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑎 ∈ (◡𝐹 “ 𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)))
24	18, 23	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (◡𝐹 “ 𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)))
25		elpreima 6563	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑏 ∈ (◡𝐹 “ 𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉)))
26	18, 25	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑏 ∈ (◡𝐹 “ 𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉)))
27	26	adantr 473	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝑏 ∈ (◡𝐹 “ 𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉)))
28	7	ad2antrr 718	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
29		simprll 798	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑎 ∈ (Base‘𝑆))
30		simprrl 800	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑏 ∈ (Base‘𝑆))
31		eqid 2799	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	2, 31	grpcl 17746	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	28, 29, 30, 32	syl3anc 1491	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
34		simpll 784	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
35		eqid 2799	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
36	2, 31, 35	ghmlin 17978	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
37	34, 29, 30, 36	syl3anc 1491	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
38		simplr 786	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑉 ∈ (SubGrp‘𝑇))
39		simprlr 799	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → (𝐹‘𝑎) ∈ 𝑉)
40		simprrr 801	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → (𝐹‘𝑏) ∈ 𝑉)
41	35	subgcl 17917	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉 ∧ (𝐹‘𝑏) ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)) ∈ 𝑉)
42	38, 39, 40, 41	syl3anc 1491	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)) ∈ 𝑉)
43	37, 42	eqeltrd 2878	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) ∈ 𝑉)
44		elpreima 6563	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (Base‘𝑆) → ((𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ↔ ((𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g‘𝑆)𝑏)) ∈ 𝑉)))
45	18, 44	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ↔ ((𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g‘𝑆)𝑏)) ∈ 𝑉)))
46	45	adantr 473	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → ((𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ↔ ((𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g‘𝑆)𝑏)) ∈ 𝑉)))
47	33, 43, 46	mpbir2and 705	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → (𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉))
48	47	expr 449	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉) → (𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉)))
49	27, 48	sylbid 232	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝑏 ∈ (◡𝐹 “ 𝑉) → (𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉)))
50	49	ralrimiv 3146	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉))
51	8	adantr 473	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → 𝑆 ∈ Grp)
52		simprl 788	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
53		eqid 2799	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
54	2, 53	grpinvcl 17783	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
55	51, 52, 54	syl2anc 580	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
56		eqid 2799	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
57	2, 53, 56	ghminv 17980	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑎)) = ((invg‘𝑇)‘(𝐹‘𝑎)))
58	57	ad2ant2r 754	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝐹‘((invg‘𝑆)‘𝑎)) = ((invg‘𝑇)‘(𝐹‘𝑎)))
59	56	subginvcl 17916	. . . . . . . . 9 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
60	59	ad2ant2l 753	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
61	58, 60	eqeltrd 2878	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)
62		elpreima 6563	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → (((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉) ↔ (((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)))
63	18, 62	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉) ↔ (((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)))
64	63	adantr 473	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉) ↔ (((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)))
65	55, 61, 64	mpbir2and 705	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉))
66	50, 65	jca 508	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉)))
67	66	ex 402	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) → (∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉))))
68	24, 67	sylbid 232	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (◡𝐹 “ 𝑉) → (∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉))))
69	68	ralrimiv 3146	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∀𝑎 ∈ (◡𝐹 “ 𝑉)(∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉)))
70	2, 31, 53	issubg2 17922	. . 3 ⊢ (𝑆 ∈ Grp → ((◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆) ↔ ((◡𝐹 “ 𝑉) ⊆ (Base‘𝑆) ∧ (◡𝐹 “ 𝑉) ≠ ∅ ∧ ∀𝑎 ∈ (◡𝐹 “ 𝑉)(∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉)))))
71	8, 70	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆) ↔ ((◡𝐹 “ 𝑉) ⊆ (Base‘𝑆) ∧ (◡𝐹 “ 𝑉) ≠ ∅ ∧ ∀𝑎 ∈ (◡𝐹 “ 𝑉)(∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉)))))
72	6, 22, 69, 71	mpbir3and 1443	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157   ≠ wne 2971  ∀wral 3089   ⊆ wss 3769  ∅c0 4115  ^◡ccnv 5311   “ cima 5315   Fn wfn 6096  ⟶wf 6097  ‘cfv 6101  (class class class)co 6878  Basecbs 16184  +_gcplusg 16267  0_gc0g 16415  Grpcgrp 17738  inv_gcminusg 17739  SubGrpcsubg 17901   GrpHom cghm 17970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192  df-plusg 16280  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-subg 17904  df-ghm 17971

This theorem is referenced by:  ghmnsgpreima  17998  subggim  18021  gicsubgen  18033  lmhmpreima  19369