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Mirrors > Home > MPE Home > Th. List > ghmgrp1 | Structured version Visualization version GIF version |
Description: A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
ghmgrp1 | ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2737 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2737 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2737 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 18967 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑦 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)(𝐹‘(𝑦(+g‘𝑆)𝑥)) = ((𝐹‘𝑦)(+g‘𝑇)(𝐹‘𝑥))))) |
6 | 5 | simplbi 498 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp)) |
7 | 6 | simpld 495 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 Basecbs 17043 +gcplusg 17093 Grpcgrp 18708 GrpHom cghm 18964 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-ghm 18965 |
This theorem is referenced by: ghmid 18973 ghminv 18974 ghmsub 18975 ghmmhm 18977 ghmmulg 18979 ghmrn 18980 resghm2 18984 resghm2b 18985 ghmco 18987 ghmpreima 18989 ghmeql 18990 ghmnsgima 18991 ghmnsgpreima 18992 ghmeqker 18994 ghmf1 18996 ghmf1o 18997 ghmpropd 19005 isgim 19011 giclcl 19021 lactghmga 19146 invghm 19571 ghmplusg 19583 f1ghm0to0 20127 kerf1ghm 20130 evl1addd 21659 evl1subd 21660 ghmcnp 23418 evlsaddval 40672 gicabl 41335 |
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