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Theorem ghmpreima 17505
 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 5404 . . 3 (𝐹𝑉) ⊆ dom 𝐹
2 eqid 2610 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2610 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 17487 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6 fdm 5964 . . . 4 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → dom 𝐹 = (Base‘𝑆))
75, 6syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → dom 𝐹 = (Base‘𝑆))
81, 7syl5sseq 3616 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ⊆ (Base‘𝑆))
9 ghmgrp1 17485 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
109adantr 480 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
11 eqid 2610 . . . . . 6 (0g𝑆) = (0g𝑆)
122, 11grpidcl 17273 . . . . 5 (𝑆 ∈ Grp → (0g𝑆) ∈ (Base‘𝑆))
1310, 12syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
14 eqid 2610 . . . . . . 7 (0g𝑇) = (0g𝑇)
1511, 14ghmid 17489 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 480 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
1714subg0cl 17425 . . . . . 6 (𝑉 ∈ (SubGrp‘𝑇) → (0g𝑇) ∈ 𝑉)
1817adantl 481 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑇) ∈ 𝑉)
1916, 18eqeltrd 2688 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) ∈ 𝑉)
20 ffn 5958 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
215, 20syl 17 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹 Fn (Base‘𝑆))
22 elpreima 6245 . . . . 5 (𝐹 Fn (Base‘𝑆) → ((0g𝑆) ∈ (𝐹𝑉) ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) ∈ 𝑉)))
2321, 22syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((0g𝑆) ∈ (𝐹𝑉) ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) ∈ 𝑉)))
2413, 19, 23mpbir2and 959 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (𝐹𝑉))
25 ne0i 3880 . . 3 ((0g𝑆) ∈ (𝐹𝑉) → (𝐹𝑉) ≠ ∅)
2624, 25syl 17 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ≠ ∅)
27 elpreima 6245 . . . . 5 (𝐹 Fn (Base‘𝑆) → (𝑎 ∈ (𝐹𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)))
2821, 27syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (𝐹𝑉) ↔ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)))
29 elpreima 6245 . . . . . . . . . 10 (𝐹 Fn (Base‘𝑆) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
3021, 29syl 17 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
3130adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
329ad2antrr 758 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
33 simprll 798 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑎 ∈ (Base‘𝑆))
34 simprrl 800 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑏 ∈ (Base‘𝑆))
35 eqid 2610 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
362, 35grpcl 17253 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
3732, 33, 34, 36syl3anc 1318 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
38 simpll 786 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
39 eqid 2610 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
402, 35, 39ghmlin 17488 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4138, 33, 34, 40syl3anc 1318 . . . . . . . . . . 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)))
42 simplr 788 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑉 ∈ (SubGrp‘𝑇))
43 simprlr 799 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑎) ∈ 𝑉)
44 simprrr 801 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑏) ∈ 𝑉)
4539subgcl 17427 . . . . . . . . . . . 12 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉 ∧ (𝐹𝑏) ∈ 𝑉) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4642, 43, 44, 45syl3anc 1318 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4741, 46eqeltrd 2688 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)
48 elpreima 6245 . . . . . . . . . . . 12 (𝐹 Fn (Base‘𝑆) → ((𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ↔ ((𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)))
4921, 48syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ↔ ((𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎(+g𝑆)𝑏)) ∈ 𝑉)))
5049adantr 480 . . . . . . . . . 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)))
5137, 47, 50mpbir2and 959 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
5251expr 641 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
5331, 52sylbid 229 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
5453ralrimiv 2948 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
5510adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑆 ∈ Grp)
56 simprl 790 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
57 eqid 2610 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
582, 57grpinvcl 17290 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
5955, 56, 58syl2anc 691 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
60 eqid 2610 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
612, 57, 60ghminv 17490 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
6261ad2ant2r 779 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
6360subginvcl 17426 . . . . . . . . 9 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6463ad2ant2l 778 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6562, 64eqeltrd 2688 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)
66 elpreima 6245 . . . . . . . . 9 (𝐹 Fn (Base‘𝑆) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6721, 66syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6867adantr 480 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (((invg𝑆)‘𝑎) ∈ (𝐹𝑉) ↔ (((invg𝑆)‘𝑎) ∈ (Base‘𝑆) ∧ (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)))
6959, 65, 68mpbir2and 959 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))
7054, 69jca 553 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
7170ex 449 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
7228, 71sylbid 229 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (𝐹𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
7372ralrimiv 2948 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
742, 35, 57issubg2 17432 . . 3 (𝑆 ∈ Grp → ((𝐹𝑉) ∈ (SubGrp‘𝑆) ↔ ((𝐹𝑉) ⊆ (Base‘𝑆) ∧ (𝐹𝑉) ≠ ∅ ∧ ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))))
7510, 74syl 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)))))
768, 26, 73, 75mpbir3and 1238 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  SubGrpcsubg 17411   GrpHom cghm 17480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-ghm 17481 This theorem is referenced by:  ghmnsgpreima  17508  subggim  17531  gicsubgen  17544  lmhmpreima  18869
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