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Theorem ghmfghm 18157
Description: The function fulfilling the conditions of ghmgrp 17460 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 17460 . 2 (𝜑𝐻 ∈ Grp)
9 fof 6072 . . 3 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
107, 9syl 17 . 2 (𝜑𝐹:𝑋𝑌)
1163expb 1263 . 2 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
121, 2, 3, 4, 5, 8, 10, 11isghmd 17590 1 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Grpcgrp 17343   GrpHom cghm 17578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-ghm 17579
This theorem is referenced by: (None)
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