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Theorem ghmfghm 19792
Description: The function fulfilling the conditions of ghmgrp 19029 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 19029 . 2 (𝜑𝐻 ∈ Grp)
9 fof 6816 . . 3 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
107, 9syl 17 . 2 (𝜑𝐹:𝑋𝑌)
1163expb 1117 . 2 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
121, 2, 3, 4, 5, 8, 10, 11isghmd 19186 1 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wf 6549  ontowfo 6551  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  Grpcgrp 18897   GrpHom cghm 19174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-ghm 19175
This theorem is referenced by: (None)
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