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Theorem ghmpreima 17903
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 5643 . . 3 (𝐹𝑉) ⊆ dom 𝐹
2 eqid 2760 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2760 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 17885 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54adantr 472 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6 fdm 6212 . . . 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 3794 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ⊆ (Base‘𝑆))
9 ghmgrp1 17883 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
109adantr 472 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → 𝑆 ∈ Grp)
11 eqid 2760 . . . . . 6 (0g𝑆) = (0g𝑆)
122, 11grpidcl 17671 . . . . 5 (𝑆 ∈ Grp → (0g𝑆) ∈ (Base‘𝑆))
1310, 12syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
14 eqid 2760 . . . . . . 7 (0g𝑇) = (0g𝑇)
1511, 14ghmid 17887 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1615adantr 472 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
1714subg0cl 17823 . . . . . 6 (𝑉 ∈ (SubGrp‘𝑇) → (0g𝑇) ∈ 𝑉)
1817adantl 473 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑇) ∈ 𝑉)
1916, 18eqeltrd 2839 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹‘(0g𝑆)) ∈ 𝑉)
20 ffn 6206 . . . . . 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 6501 . . . . 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 995 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (0g𝑆) ∈ (𝐹𝑉))
25 ne0i 4064 . . 3 ((0g𝑆) ∈ (𝐹𝑉) → (𝐹𝑉) ≠ ∅)
2624, 25syl 17 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ≠ ∅)
27 elpreima 6501 . . . . 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 6501 . . . . . . . . . 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 472 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) ↔ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉)))
329ad2antrr 764 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑆 ∈ Grp)
33 simprll 821 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑎 ∈ (Base‘𝑆))
34 simprrl 823 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑏 ∈ (Base‘𝑆))
35 eqid 2760 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
362, 35grpcl 17651 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
3732, 33, 34, 36syl3anc 1477 . . . . . . . . . 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 807 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
39 eqid 2760 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
402, 35, 39ghmlin 17886 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎(+g𝑆)𝑏)) = ((𝐹𝑎)(+g𝑇)(𝐹𝑏)))
4138, 33, 34, 40syl3anc 1477 . . . . . . . . . . 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 809 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → 𝑉 ∈ (SubGrp‘𝑇))
43 simprlr 822 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑎) ∈ 𝑉)
44 simprrr 824 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝐹𝑏) ∈ 𝑉)
4539subgcl 17825 . . . . . . . . . . . 12 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉 ∧ (𝐹𝑏) ∈ 𝑉) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4642, 43, 44, 45syl3anc 1477 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → ((𝐹𝑎)(+g𝑇)(𝐹𝑏)) ∈ 𝑉)
4741, 46eqeltrd 2839 . . . . . . . . . 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 6501 . . . . . . . . . . . 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 472 . . . . . . . . . 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 995 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ ((𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉) ∧ (𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉))) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
5251expr 644 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((𝑏 ∈ (Base‘𝑆) ∧ (𝐹𝑏) ∈ 𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
5331, 52sylbid 230 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝑏 ∈ (𝐹𝑉) → (𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉)))
5453ralrimiv 3103 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉))
5510adantr 472 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑆 ∈ Grp)
56 simprl 811 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → 𝑎 ∈ (Base‘𝑆))
57 eqid 2760 . . . . . . . . 9 (invg𝑆) = (invg𝑆)
582, 57grpinvcl 17688 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
5955, 56, 58syl2anc 696 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (Base‘𝑆))
60 eqid 2760 . . . . . . . . . 10 (invg𝑇) = (invg𝑇)
612, 57, 60ghminv 17888 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
6261ad2ant2r 800 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) = ((invg𝑇)‘(𝐹𝑎)))
6360subginvcl 17824 . . . . . . . . 9 ((𝑉 ∈ (SubGrp‘𝑇) ∧ (𝐹𝑎) ∈ 𝑉) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6463ad2ant2l 799 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑇)‘(𝐹𝑎)) ∈ 𝑉)
6562, 64eqeltrd 2839 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → (𝐹‘((invg𝑆)‘𝑎)) ∈ 𝑉)
66 elpreima 6501 . . . . . . . . 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 472 . . . . . . 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 995 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) ∧ (𝑎 ∈ (Base‘𝑆) ∧ (𝐹𝑎) ∈ 𝑉)) → ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))
7054, 69jca 555 . . . . 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 230 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝑎 ∈ (𝐹𝑉) → (∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉))))
7372ralrimiv 3103 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → ∀𝑎 ∈ (𝐹𝑉)(∀𝑏 ∈ (𝐹𝑉)(𝑎(+g𝑆)𝑏) ∈ (𝐹𝑉) ∧ ((invg𝑆)‘𝑎) ∈ (𝐹𝑉)))
742, 35, 57issubg2 17830 . . 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 1428 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wss 3715  c0 4058  ccnv 5265  dom cdm 5266  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  0gc0g 16322  Grpcgrp 17643  invgcminusg 17644  SubGrpcsubg 17809   GrpHom cghm 17878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-subg 17812  df-ghm 17879
This theorem is referenced by:  ghmnsgpreima  17906  subggim  17929  gicsubgen  17941  lmhmpreima  19270
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