MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmgrp1 Structured version   Visualization version   GIF version

Theorem ghmgrp1 17863
Description: A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmgrp1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)

Proof of Theorem ghmgrp1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2760 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2760 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2760 . . . 4 (+g𝑇) = (+g𝑇)
51, 2, 3, 4isghm 17861 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑦 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)(𝐹‘(𝑦(+g𝑆)𝑥)) = ((𝐹𝑦)(+g𝑇)(𝐹𝑥)))))
65simplbi 478 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
76simpld 477 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Grpcgrp 17623   GrpHom cghm 17858
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 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 3055  df-rex 3056  df-reu 3057  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-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  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-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-ghm 17859
This theorem is referenced by:  ghmid  17867  ghminv  17868  ghmsub  17869  ghmmhm  17871  ghmmulg  17873  ghmrn  17874  resghm2  17878  resghm2b  17879  ghmco  17881  ghmpreima  17883  ghmeql  17884  ghmnsgima  17885  ghmnsgpreima  17886  ghmeqker  17888  ghmf1  17890  ghmf1o  17891  ghmpropd  17899  isgim  17905  giclcl  17915  lactghmga  18024  invghm  18439  ghmplusg  18449  evl1addd  19907  evl1subd  19908  ghmcnp  22119  gicabl  38171
  Copyright terms: Public domain W3C validator