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Theorem mhm0 17390
Description: A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypotheses
Ref Expression
mhm0.z 0 = (0g𝑆)
mhm0.y 𝑌 = (0g𝑇)
Assertion
Ref Expression
mhm0 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)

Proof of Theorem mhm0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2651 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2651 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2651 . . . 4 (+g𝑇) = (+g𝑇)
5 mhm0.z . . . 4 0 = (0g𝑆)
6 mhm0.y . . . 4 𝑌 = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 17384 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
87simprbi 479 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹0 ) = 𝑌))
98simp3d 1095 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341   MndHom cmhm 17380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-mhm 17382
This theorem is referenced by:  mhmf1o  17392  resmhm  17406  resmhm2  17407  resmhm2b  17408  mhmco  17409  mhmima  17410  mhmeql  17411  pwsco2mhm  17418  gsumwmhm  17429  mhmmulg  17630  gsumzmhm  18383  rhm1  18778  madetsumid  20315  mdetunilem7  20472  pm2mp  20678  dchrzrh1  25014  dchrmulcl  25019  dchrn0  25020  dchrinvcl  25023  dchrfi  25025  dchrabs  25030  sumdchr2  25040  rpvmasum2  25246
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