Theorem ghmpreima 14046

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 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5145	. . 3 ⊢ (◡𝐹 “ 𝑉) ⊆ dom 𝐹
2		eqid 2238	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2238	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 14027	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6	1, 5	fssdm 5544	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ⊆ (Base‘𝑆))
7		ghmgrp1 14025	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	7	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9		eqid 2238	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	2, 9	grpidcl 13811	. . . . 5 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ (Base‘𝑆))
11	8, 10	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆))
12		eqid 2238	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
13	9, 12	ghmid 14029	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
14	13	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
15	12	subg0cl 13962	. . . . . 6 ⊢ (𝑉 ∈ (SubGrp‘𝑇) → (0g‘𝑇) ∈ 𝑉)
16	15	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑇) ∈ 𝑉)
17	14, 16	eqeltrd 2315	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) ∈ 𝑉)
18	5	ffnd 5529	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19		elpreima 5819	. . . . 5 ⊢ (𝐹 Fn (Base‘𝑆) → ((0g‘𝑆) ∈ (◡𝐹 “ 𝑉) ↔ ((0g‘𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g‘𝑆)) ∈ 𝑉)))
20	18, 19	syl 14	. . . 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 957	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑆) ∈ (◡𝐹 “ 𝑉))
22		elex2 2838	. . 3 ⊢ ((0g‘𝑆) ∈ (◡𝐹 “ 𝑉) → ∃𝑗 𝑗 ∈ (◡𝐹 “ 𝑉))
23	21, 22	syl 14	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∃𝑗 𝑗 ∈ (◡𝐹 “ 𝑉))
24		elpreima 5819	. . . . 5 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑎 ∈ (◡𝐹 “ 𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)))
25	18, 24	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (◡𝐹 “ 𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)))
26		elpreima 5819	. . . . . . . . . 10 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑏 ∈ (◡𝐹 “ 𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉)))
27	18, 26	syl 14	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑏 ∈ (◡𝐹 “ 𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉)))
28	27	adantr 276	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝑏 ∈ (◡𝐹 “ 𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉)))
29		eqid 2238	. . . . . . . . . . 11 ⊢ (+g‘𝑆) = (+g‘𝑆)
30	7	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
31		simprll 543	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑎 ∈ (Base‘𝑆))
32		simprrl 545	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑏 ∈ (Base‘𝑆))
33	2, 29, 30, 31, 32	grpcld 13796	. . . . . . . . . 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 531	. . . . . . . . . . . 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 2238	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
36	2, 29, 35	ghmlin 14028	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g‘𝑆)𝑏)) = ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)))
37	34, 31, 32, 36	syl3anc 1278	. . . . . . . . . . 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 533	. . . . . . . . . . . 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 544	. . . . . . . . . . . 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 546	. . . . . . . . . . . 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 13964	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉 ∧ (𝐹‘𝑏) ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)) ∈ 𝑉)
42	38, 39, 40, 41	syl3anc 1278	. . . . . . . . . . 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 2315	. . . . . . . . . 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 5819	. . . . . . . . . . . 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 14	. . . . . . . . . . 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 276	. . . . . . . . . 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 957	. . . . . . . . 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 375	. . . . . . . 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	28, 48	sylbid 150	. . . . . . 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 2622	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉))
51		simprl 535	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
52		eqid 2238	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
53	2, 52	grpinvcl 13830	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
54	8, 51, 53	syl2an2r 603	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
55		eqid 2238	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
56	2, 52, 55	ghminv 14030	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑎)) = ((invg‘𝑇)‘(𝐹‘𝑎)))
57	56	ad2ant2r 513	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝐹‘((invg‘𝑆)‘𝑎)) = ((invg‘𝑇)‘(𝐹‘𝑎)))
58	55	subginvcl 13963	. . . . . . . . 9 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
59	58	ad2ant2l 512	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
60	57, 59	eqeltrd 2315	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)
61		elpreima 5819	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → (((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉) ↔ (((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)))
62	18, 61	syl 14	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉) ↔ (((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)))
63	62	adantr 276	. . . . . . 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)))
64	54, 60, 63	mpbir2and 957	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉))
65	50, 64	jca 306	. . . . 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)))
66	65	ex 115	. . . 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))))
67	25, 66	sylbid 150	. . 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))))
68	67	ralrimiv 2622	. 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)))
69	2, 29, 52	issubg2m 13969	. . 3 ⊢ (𝑆 ∈ Grp → ((◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆) ↔ ((◡𝐹 “ 𝑉) ⊆ (Base‘𝑆) ∧ ∃𝑗 𝑗 ∈ (◡𝐹 “ 𝑉) ∧ ∀𝑎 ∈ (◡𝐹 “ 𝑉)(∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉)))))
70	8, 69	syl 14	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆) ↔ ((◡𝐹 “ 𝑉) ⊆ (Base‘𝑆) ∧ ∃𝑗 𝑗 ∈ (◡𝐹 “ 𝑉) ∧ ∀𝑎 ∈ (◡𝐹 “ 𝑉)(∀𝑏 ∈ (◡𝐹 “ 𝑉)(𝑎(+g‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑉) ∧ ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉)))))
71	6, 23, 68, 70	mpbir3and 1211	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ^◡ccnv 4768   “ cima 4772   Fn wfn 5367  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587  Grpcgrp 13782  inv_gcminusg 13783  SubGrpcsubg 13947   GrpHom cghm 14020

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-ghm 14021

This theorem is referenced by:  ghmnsgpreima  14049