MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmpreima Structured version   Visualization version   GIF version

Theorem ghmpreima 17995
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 5702 . . 3 (𝐹𝑉) ⊆ dom 𝐹
2 eqid 2799 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2799 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 17977 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54adantr 473 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
61, 5fssdm 6272 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ⊆ (Base‘𝑆))
7 ghmgrp1 17975 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
87adantr 473 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
9 eqid 2799 . . . . . 6 (0g𝑆) = (0g𝑆)
102, 9grpidcl 17766 . . . . 5 (𝑆 ∈ Grp → (0g𝑆) ∈ (Base‘𝑆))
118, 10syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
12 eqid 2799 . . . . . . 7 (0g𝑇) = (0g𝑇)
139, 12ghmid 17979 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1413adantr 473 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
1512subg0cl 17915 . . . . . 6 (𝑉 ∈ (SubGrp‘𝑇) → (0g𝑇) ∈ 𝑉)
1615adantl 474 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑇) ∈ 𝑉)
1714, 16eqeltrd 2878 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) ∈ 𝑉)
185ffnd 6257 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
19 elpreima 6563 . . . . 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 705 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (𝐹𝑉))
2221ne0d 4122 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ≠ ∅)
23 elpreima 6563 . . . . 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 6563 . . . . . . . . . 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 473 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
287ad2antrr 718 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
29 simprll 798 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑎 ∈ (Base‘𝑆))
30 simprrl 800 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑏 ∈ (Base‘𝑆))
31 eqid 2799 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
322, 31grpcl 17746 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
3328, 29, 30, 32syl3anc 1491 . . . . . . . . . 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 784 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
35 eqid 2799 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
362, 31, 35ghmlin 17978 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
3734, 29, 30, 36syl3anc 1491 . . . . . . . . . . 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 786 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑉 ∈ (SubGrp‘𝑇))
39 simprlr 799 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑎) ∈ 𝑉)
40 simprrr 801 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑏) ∈ 𝑉)
4135subgcl 17917 . . . . . . . . . . . 12 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉 ∧ (𝐹𝑏) ∈ 𝑉) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4238, 39, 40, 41syl3anc 1491 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4337, 42eqeltrd 2878 . . . . . . . . . 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 6563 . . . . . . . . . . . 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 473 . . . . . . . . . 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 705 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
4847expr 449 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
4927, 48sylbid 232 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
5049ralrimiv 3146 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
518adantr 473 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑆 ∈ Grp)
52 simprl 788 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
53 eqid 2799 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
542, 53grpinvcl 17783 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
5551, 52, 54syl2anc 580 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
56 eqid 2799 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
572, 53, 56ghminv 17980 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
5857ad2ant2r 754 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
5956subginvcl 17916 . . . . . . . . 9 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6059ad2ant2l 753 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6158, 60eqeltrd 2878 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)
62 elpreima 6563 . . . . . . . . 9 (𝐹 Fn (Base‘𝑆) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6318, 62syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6463adantr 473 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6555, 61, 64mpbir2and 705 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))
6650, 65jca 508 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
6766ex 402 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
6824, 67sylbid 232 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (𝐹𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
6968ralrimiv 3146 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
702, 31, 53issubg2 17922 . . 3 (𝑆 ∈ Grp → ((𝐹𝑉) ∈ (SubGrp‘𝑆) ↔ ((𝐹𝑉) ⊆ (Base‘𝑆) ∧ (𝐹𝑉) ≠ ∅ ∧ ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))))
718, 70syl 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)))))
726, 22, 69, 71mpbir3and 1443 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wss 3769  c0 4115  ccnv 5311  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  0gc0g 16415  Grpcgrp 17738  invgcminusg 17739  SubGrpcsubg 17901   GrpHom cghm 17970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192  df-plusg 16280  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-subg 17904  df-ghm 17971
This theorem is referenced by:  ghmnsgpreima  17998  subggim  18021  gicsubgen  18033  lmhmpreima  19369
  Copyright terms: Public domain W3C validator