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Mirrors > Home > MPE Home > Th. List > ghmgrp2 | Structured version Visualization version GIF version |
Description: A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
ghmgrp2 | ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2728 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2728 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2728 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 19177 | . . 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 | simprd 494 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 Grpcgrp 18897 GrpHom cghm 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-ghm 19175 |
This theorem is referenced by: ghmid 19183 ghminv 19184 ghmmhm 19187 ghmmulg 19189 ghmrn 19190 resghm 19193 ghmco 19197 ghmker 19203 ghmeqker 19204 ghmf1 19207 ghmf1o 19209 ghmpropd 19217 isgim 19223 gicrcl 19235 ghmquskerlem1 19241 lactghmga 19367 ghmplusg 19808 ghmcyg 19858 ghmcnp 24039 abliso 32777 ghmqusnsglem1 33154 gicabl 42554 |
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