Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgmhmf Structured version   Visualization version   GIF version

Theorem mgmhmf 44058
Description: A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
mgmhmf.b 𝐵 = (Base‘𝑆)
mgmhmf.c 𝐶 = (Base‘𝑇)
Assertion
Ref Expression
mgmhmf (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵𝐶)

Proof of Theorem mgmhmf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmf.b . . 3 𝐵 = (Base‘𝑆)
2 mgmhmf.c . . 3 𝐶 = (Base‘𝑇)
3 eqid 2824 . . 3 (+g𝑆) = (+g𝑆)
4 eqid 2824 . . 3 (+g𝑇) = (+g𝑇)
51, 2, 3, 4ismgmhm 44057 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))))
6 simprl 769 . 2 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))) → 𝐹:𝐵𝐶)
75, 6sylbi 219 1 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  wral 3141  wf 6354  cfv 6358  (class class class)co 7159  Basecbs 16486  +gcplusg 16568  Mgmcmgm 17853   MgmHom cmgmhm 44051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-map 8411  df-mgmhm 44053
This theorem is referenced by:  mgmhmf1o  44061  resmgmhm  44072  resmgmhm2  44073  resmgmhm2b  44074  mgmhmco  44075  mgmhmima  44076  mgmhmeql  44077
  Copyright terms: Public domain W3C validator