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

Theorem ghmfghm 18589
Description: The function fulfilling the conditions of ghmgrp 17893 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
Hypotheses
Ref Expression
ghmabl.x 𝑋 = (Base‘𝐺)
ghmabl.y 𝑌 = (Base‘𝐻)
ghmabl.p + = (+g𝐺)
ghmabl.q = (+g𝐻)
ghmabl.f ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
ghmabl.1 (𝜑𝐹:𝑋onto𝑌)
ghmfghm.3 (𝜑𝐺 ∈ Grp)
Assertion
Ref Expression
ghmfghm (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
Distinct variable groups:   𝑥, + ,𝑦   𝑥, ,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Proof of Theorem ghmfghm
StepHypRef Expression
1 ghmabl.x . 2 𝑋 = (Base‘𝐺)
2 ghmabl.y . 2 𝑌 = (Base‘𝐻)
3 ghmabl.p . 2 + = (+g𝐺)
4 ghmabl.q . 2 = (+g𝐻)
5 ghmfghm.3 . 2 (𝜑𝐺 ∈ Grp)
6 ghmabl.f . . 3 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
7 ghmabl.1 . . 3 (𝜑𝐹:𝑋onto𝑌)
86, 1, 2, 3, 4, 7, 5ghmgrp 17893 . 2 (𝜑𝐻 ∈ Grp)
9 fof 6353 . . 3 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
107, 9syl 17 . 2 (𝜑𝐹:𝑋𝑌)
1163expb 1153 . 2 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
121, 2, 3, 4, 5, 8, 10, 11isghmd 18020 1 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1111   = wceq 1656  wcel 2164  wf 6119  ontowfo 6121  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  Grpcgrp 17776   GrpHom cghm 18008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-ghm 18009
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator