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Theorem ghmpreima 19228
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 6083 . . 3 (𝐹𝑉) ⊆ dom 𝐹
2 eqid 2726 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2726 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 19210 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54adantr 479 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
61, 5fssdm 6739 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ⊆ (Base‘𝑆))
7 ghmgrp1 19208 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
87adantr 479 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9 eqid 2726 . . . . . 6 (0g𝑆) = (0g𝑆)
102, 9grpidcl 18955 . . . . 5 (𝑆 ∈ Grp → (0g𝑆) ∈ (Base‘𝑆))
118, 10syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
12 eqid 2726 . . . . . . 7 (0g𝑇) = (0g𝑇)
139, 12ghmid 19212 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1413adantr 479 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
1512subg0cl 19124 . . . . . 6 (𝑉 ∈ (SubGrp‘𝑇) → (0g𝑇) ∈ 𝑉)
1615adantl 480 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑇) ∈ 𝑉)
1714, 16eqeltrd 2826 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) ∈ 𝑉)
185ffnd 6721 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19 elpreima 7063 . . . . 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 7063 . . . . 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 7063 . . . . . . . . . 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 479 . . . . . . . 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 2726 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
322, 31grpcl 18931 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
3328, 29, 30, 32syl3anc 1368 . . . . . . . . . 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 2726 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
362, 31, 35ghmlin 19211 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3734, 29, 30, 36syl3anc 1368 . . . . . . . . . . 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 19126 . . . . . . . . . . . 12 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉 ∧ (𝐹𝑏) ∈ 𝑉) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4238, 39, 40, 41syl3anc 1368 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4337, 42eqeltrd 2826 . . . . . . . . . 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 7063 . . . . . . . . . . . 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 479 . . . . . . . . . 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 455 . . . . . . . 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 3135 . . . . . 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 2726 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
532, 52grpinvcl 18977 . . . . . . . 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 2726 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
562, 52, 55ghminv 19213 . . . . . . . . 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 19125 . . . . . . . . 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 2826 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)
61 elpreima 7063 . . . . . . . . 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 479 . . . . . . 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 510 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
6665ex 411 . . . 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 3135 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
692, 31, 52issubg2 19131 . . 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 1339 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wss 3946  c0 4322  ccnv 5673  cima 5677   Fn wfn 6541  wf 6542  cfv 6546  (class class class)co 7416  Basecbs 17208  +gcplusg 17261  0gc0g 17449  Grpcgrp 18923  invgcminusg 18924  SubGrpcsubg 19110   GrpHom cghm 19202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-minusg 18927  df-subg 19113  df-ghm 19203
This theorem is referenced by:  ghmnsgpreima  19231  subggim  19256  gicsubgen  19269  lmhmpreima  21022  evpmsubg  33029
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