Theorem ghmpreima 18382

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 5951	. . 3 ⊢ (◡𝐹 “ 𝑉) ⊆ dom 𝐹
2		eqid 2823	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2823	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 18364	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	adantr 483	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6	1, 5	fssdm 6532	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ⊆ (Base‘𝑆))
7		ghmgrp1 18362	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	7	adantr 483	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9		eqid 2823	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	2, 9	grpidcl 18133	. . . . 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 2823	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
13	9, 12	ghmid 18366	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
14	13	adantr 483	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
15	12	subg0cl 18289	. . . . . 6 ⊢ (𝑉 ∈ (SubGrp‘𝑇) → (0g‘𝑇) ∈ 𝑉)
16	15	adantl 484	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑇) ∈ 𝑉)
17	14, 16	eqeltrd 2915	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) ∈ 𝑉)
18	5	ffnd 6517	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19		elpreima 6830	. . . . 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 711	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑆) ∈ (◡𝐹 “ 𝑉))
22	21	ne0d 4303	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ≠ ∅)
23		elpreima 6830	. . . . 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 6830	. . . . . . . . . 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 483	. . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
29		simprll 777	. . . . . . . . . . 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 779	. . . . . . . . . . 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 2823	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	2, 31	grpcl 18113	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	28, 29, 30, 32	syl3anc 1367	. . . . . . . . . 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 765	. . . . . . . . . . . 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 2823	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
36	2, 31, 35	ghmlin 18365	. . . . . . . . . . . 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 1367	. . . . . . . . . . 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 767	. . . . . . . . . . . 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 778	. . . . . . . . . . . 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 780	. . . . . . . . . . . 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 18291	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉 ∧ (𝐹‘𝑏) ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)) ∈ 𝑉)
42	38, 39, 40, 41	syl3anc 1367	. . . . . . . . . . 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 2915	. . . . . . . . . 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 6830	. . . . . . . . . . . 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 483	. . . . . . . . . 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 711	. . . . . . . . 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 459	. . . . . . . 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 242	. . . . . . 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 3183	. . . . . 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 769	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
52		eqid 2823	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
53	2, 52	grpinvcl 18153	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
54	8, 51, 53	syl2an2r 683	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
55		eqid 2823	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
56	2, 52, 55	ghminv 18367	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑎)) = ((invg‘𝑇)‘(𝐹‘𝑎)))
57	56	ad2ant2r 745	. . . . . . . 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 18290	. . . . . . . . 9 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
59	58	ad2ant2l 744	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
60	57, 59	eqeltrd 2915	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)
61		elpreima 6830	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → (((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉) ↔ (((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)))
62	18, 61	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)))
63	62	adantr 483	. . . . . . 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 711	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉))
65	50, 64	jca 514	. . . . 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 415	. . . 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	24, 66	sylbid 242	. . 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 3183	. 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, 31, 52	issubg2 18296	. . 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)))))
70	8, 69	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)))))
71	6, 22, 68, 70	mpbir3and 1338	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140   ⊆ wss 3938  ∅c0 4293  ^◡ccnv 5556   “ cima 5560   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  0_gc0g 16715  Grpcgrp 18105  inv_gcminusg 18106  SubGrpcsubg 18275   GrpHom cghm 18357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-subg 18278  df-ghm 18358

This theorem is referenced by:  ghmnsgpreima  18385  subggim  18408  gicsubgen  18420  lmhmpreima  19822  evpmsubg  30791