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Theorem ghmpreima 19116
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)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))

Proof of Theorem ghmpreima
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 6080 . . 3 (𝐹𝑉) ⊆ dom 𝐹
2 eqid 2732 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2732 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 19098 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54adantr 481 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
61, 5fssdm 6737 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ⊆ (Base‘𝑆))
7 ghmgrp1 19096 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
87adantr 481 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9 eqid 2732 . . . . . 6 (0g𝑆) = (0g𝑆)
102, 9grpidcl 18852 . . . . 5 (𝑆 ∈ Grp → (0g𝑆) ∈ (Base‘𝑆))
118, 10syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
12 eqid 2732 . . . . . . 7 (0g𝑇) = (0g𝑇)
139, 12ghmid 19100 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1413adantr 481 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
1512subg0cl 19016 . . . . . 6 (𝑉 ∈ (SubGrp‘𝑇) → (0g𝑇) ∈ 𝑉)
1615adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑇) ∈ 𝑉)
1714, 16eqeltrd 2833 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) ∈ 𝑉)
185ffnd 6718 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19 elpreima 7059 . . . . 5 (𝐹 Fn (Base‘𝑆) → ((0g𝑆) ∈ (𝐹𝑉) ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) ∈ 𝑉)))
2018, 19syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((0g𝑆) ∈ (𝐹𝑉) ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) ∈ 𝑉)))
2111, 17, 20mpbir2and 711 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (𝐹𝑉))
2221ne0d 4335 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ≠ ∅)
23 elpreima 7059 . . . . 5 (𝐹 Fn (Base‘𝑆) → (𝑎 ∈ (𝐹𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)))
2418, 23syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (𝐹𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)))
25 elpreima 7059 . . . . . . . . . 10 (𝐹 Fn (Base‘𝑆) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
2618, 25syl 17 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
2726adantr 481 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
287ad2antrr 724 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
29 simprll 777 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑎 ∈ (Base‘𝑆))
30 simprrl 779 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑏 ∈ (Base‘𝑆))
31 eqid 2732 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
322, 31grpcl 18829 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
3328, 29, 30, 32syl3anc 1371 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
34 simpll 765 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
35 eqid 2732 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
362, 31, 35ghmlin 19099 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3734, 29, 30, 36syl3anc 1371 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
38 simplr 767 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑉 ∈ (SubGrp‘𝑇))
39 simprlr 778 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑎) ∈ 𝑉)
40 simprrr 780 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑏) ∈ 𝑉)
4135subgcl 19018 . . . . . . . . . . . 12 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉 ∧ (𝐹𝑏) ∈ 𝑉) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4238, 39, 40, 41syl3anc 1371 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4337, 42eqeltrd 2833 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)
44 elpreima 7059 . . . . . . . . . . . 12 (𝐹 Fn (Base‘𝑆) → ((𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ↔ ((𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)))
4518, 44syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ↔ ((𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)))
4645adantr 481 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → ((𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ↔ ((𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)))
4733, 43, 46mpbir2and 711 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
4847expr 457 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
4927, 48sylbid 239 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
5049ralrimiv 3145 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
51 simprl 769 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
52 eqid 2732 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
532, 52grpinvcl 18874 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
548, 51, 53syl2an2r 683 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
55 eqid 2732 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
562, 52, 55ghminv 19101 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
5756ad2ant2r 745 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
5855subginvcl 19017 . . . . . . . . 9 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
5958ad2ant2l 744 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6057, 59eqeltrd 2833 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)
61 elpreima 7059 . . . . . . . . 9 (𝐹 Fn (Base‘𝑆) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6218, 61syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6362adantr 481 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6454, 60, 63mpbir2and 711 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))
6550, 64jca 512 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
6665ex 413 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
6724, 66sylbid 239 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (𝐹𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
6867ralrimiv 3145 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
692, 31, 52issubg2 19023 . . 3 (𝑆 ∈ Grp → ((𝐹𝑉) ∈ (SubGrp‘𝑆) ↔ ((𝐹𝑉) ⊆ (Base‘𝑆) ∧ (𝐹𝑉) ≠ ∅ ∧ ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))))
708, 69syl 17 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝐹𝑉) ∈ (SubGrp‘𝑆) ↔ ((𝐹𝑉) ⊆ (Base‘𝑆) ∧ (𝐹𝑉) ≠ ∅ ∧ ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))))
716, 22, 68, 70mpbir3and 1342 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  wss 3948  c0 4322  ccnv 5675  cima 5679   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  0gc0g 17387  Grpcgrp 18821  invgcminusg 18822  SubGrpcsubg 19002   GrpHom cghm 19091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825  df-subg 19005  df-ghm 19092
This theorem is referenced by:  ghmnsgpreima  19119  subggim  19142  gicsubgen  19154  lmhmpreima  20664  evpmsubg  32347
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