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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 2730 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2730 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2730 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2730 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 19130 | . . 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 496 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp)) |
7 | 6 | simpld 493 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 Grpcgrp 18855 GrpHom cghm 19127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-ghm 19128 |
This theorem is referenced by: ghmid 19136 ghminv 19137 ghmsub 19138 ghmmhm 19140 ghmmulg 19142 ghmrn 19143 resghm2 19147 resghm2b 19148 ghmco 19150 ghmpreima 19152 ghmeql 19153 ghmnsgima 19154 ghmnsgpreima 19155 ghmeqker 19157 f1ghm0to0 19159 ghmf1 19160 kerf1ghm 19161 ghmf1o 19162 ghmpropd 19170 isgim 19176 giclcl 19187 lactghmga 19314 invghm 19742 ghmplusg 19755 evl1addd 22080 evl1subd 22081 ghmcnp 23839 ghmquskerlem1 32802 ghmquskerlem2 32804 ghmquskerlem3 32805 ghmqusker 32806 evlsaddval 41442 evladdval 41449 gicabl 42143 |
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