Theorem ghmpreima 19146

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 6042	. . 3 ⊢ (◡𝐹 “ 𝑉) ⊆ dom 𝐹
2		eqid 2729	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2729	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 19128	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6	1, 5	fssdm 6689	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ⊆ (Base‘𝑆))
7		ghmgrp1 19126	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
8	7	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9		eqid 2729	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
10	2, 9	grpidcl 18873	. . . . 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 2729	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
13	9, 12	ghmid 19130	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
14	13	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
15	12	subg0cl 19042	. . . . . 6 ⊢ (𝑉 ∈ (SubGrp‘𝑇) → (0g‘𝑇) ∈ 𝑉)
16	15	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑇) ∈ 𝑉)
17	14, 16	eqeltrd 2828	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g‘𝑆)) ∈ 𝑉)
18	5	ffnd 6671	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19		elpreima 7012	. . . . 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 713	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g‘𝑆) ∈ (◡𝐹 “ 𝑉))
22	21	ne0d 4301	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ≠ ∅)
23		elpreima 7012	. . . . 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 7012	. . . . . . . . . 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 480	. . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹‘𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
29		simprll 778	. . . . . . . . . . 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 780	. . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
32	2, 31	grpcl 18849	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
33	28, 29, 30, 32	syl3anc 1373	. . . . . . . . . 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 766	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
36	2, 31, 35	ghmlin 19129	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 768	. . . . . . . . . . . 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 779	. . . . . . . . . . . 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 781	. . . . . . . . . . . 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 19044	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉 ∧ (𝐹‘𝑏) ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑇)(𝐹‘𝑏)) ∈ 𝑉)
42	38, 39, 40, 41	syl3anc 1373	. . . . . . . . . . 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 2828	. . . . . . . . . 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 7012	. . . . . . . . . . . 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 480	. . . . . . . . . 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 713	. . . . . . . . 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 456	. . . . . . . 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 240	. . . . . . 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 3124	. . . . . 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 770	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
52		eqid 2729	. . . . . . . . 9 ⊢ (invg‘𝑆) = (invg‘𝑆)
53	2, 52	grpinvcl 18895	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
54	8, 51, 53	syl2an2r 685	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (Base‘𝑆))
55		eqid 2729	. . . . . . . . . 10 ⊢ (invg‘𝑇) = (invg‘𝑇)
56	2, 52, 55	ghminv 19131	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑎)) = ((invg‘𝑇)‘(𝐹‘𝑎)))
57	56	ad2ant2r 747	. . . . . . . 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 19043	. . . . . . . . 9 ⊢ ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹‘𝑎) ∈ 𝑉) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
59	58	ad2ant2l 746	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑇)‘(𝐹‘𝑎)) ∈ 𝑉)
60	57, 59	eqeltrd 2828	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → (𝐹‘((invg‘𝑆)‘𝑎)) ∈ 𝑉)
61		elpreima 7012	. . . . . . . . 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 480	. . . . . . 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 713	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹‘𝑎) ∈ 𝑉)) → ((invg‘𝑆)‘𝑎) ∈ (◡𝐹 “ 𝑉))
65	50, 64	jca 511	. . . . 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 412	. . . 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 240	. . 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 3124	. 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 19049	. . 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 1343	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3911  ∅c0 4292  ^◡ccnv 5630   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Grpcgrp 18841  inv_gcminusg 18842  SubGrpcsubg 19028   GrpHom cghm 19120

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-subg 19031  df-ghm 19121

This theorem is referenced by:  ghmnsgpreima  19149  subggim  19174  gicsubgen  19187  lmhmpreima  20931  evpmsubg  33077