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Mirrors > Home > MPE Home > Th. List > ghmf | Structured version Visualization version GIF version |
Description: A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
ghmf.x | ⊢ 𝑋 = (Base‘𝑆) |
ghmf.y | ⊢ 𝑌 = (Base‘𝑇) |
Ref | Expression |
---|---|
ghmf | ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmf.x | . . . 4 ⊢ 𝑋 = (Base‘𝑆) | |
2 | ghmf.y | . . . 4 ⊢ 𝑌 = (Base‘𝑇) | |
3 | eqid 2732 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2732 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 19091 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘(𝑦(+g‘𝑆)𝑥)) = ((𝐹‘𝑦)(+g‘𝑇)(𝐹‘𝑥))))) |
6 | 5 | simprbi 497 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘(𝑦(+g‘𝑆)𝑥)) = ((𝐹‘𝑦)(+g‘𝑇)(𝐹‘𝑥)))) |
7 | 6 | simpld 495 | 1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +gcplusg 17196 Grpcgrp 18818 GrpHom cghm 19088 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-ghm 19089 |
This theorem is referenced by: ghmid 19097 ghminv 19098 ghmsub 19099 ghmmhm 19101 ghmmulg 19103 ghmrn 19104 resghm 19107 ghmpreima 19113 ghmeql 19114 ghmnsgima 19115 ghmnsgpreima 19116 ghmeqker 19118 ghmf1 19120 ghmf1o 19121 gimcnv 19140 lactghmga 19272 frgpup3lem 19644 frgpup3 19645 ghmplusg 19713 rhmf 20262 isrhm2d 20264 kerf1ghm 20281 lmhmf 20644 lmhmpropd 20683 frgpcyg 21128 psgninv 21134 zrhpsgninv 21137 evpmss 21138 psgnevpmb 21139 psgnodpm 21140 zrhpsgnevpm 21143 zrhpsgnodpm 21144 evlslem2 21641 nmoi 24244 nmoix 24245 nmoi2 24246 nmoleub 24247 nmoeq0 24252 nmoco 24253 nmotri 24255 nmods 24260 nghmcn 24261 ghmquskerlem1 32523 ghmquskerco 32524 ghmquskerlem3 32526 ghmqusker 32527 isrnghmmul 46681 rnghmf 46687 |
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