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Theorem ghmfghm 18443
Description: The function fulfilling the conditions of ghmgrp 17747 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 17747 . 2 (𝜑𝐻 ∈ Grp)
9 fof 6256 . . 3 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
107, 9syl 17 . 2 (𝜑𝐹:𝑋𝑌)
1163expb 1113 . 2 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
121, 2, 3, 4, 5, 8, 10, 11isghmd 17877 1 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wf 6027  ontowfo 6029  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630   GrpHom cghm 17865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-ghm 17866
This theorem is referenced by: (None)
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