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Theorem mhm0 17166
 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 2610 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2610 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2610 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2610 . . . 4 (+g𝑇) = (+g𝑇)
5 mhm0.z . . . 4 0 = (0g𝑆)
6 mhm0.y . . . 4 𝑌 = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 17160 . . 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 1068 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117   MndHom cmhm 17156 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-mhm 17158 This theorem is referenced by:  mhmf1o  17168  resmhm  17182  resmhm2  17183  resmhm2b  17184  mhmco  17185  mhmima  17186  mhmeql  17187  pwsco2mhm  17194  gsumwmhm  17205  mhmmulg  17406  gsumzmhm  18160  rhm1  18553  madetsumid  20086  mdetunilem7  20243  pm2mp  20449  dchrzrh1  24769  dchrmulcl  24774  dchrn0  24775  dchrinvcl  24778  dchrfi  24780  dchrabs  24785  sumdchr2  24795  rpvmasum2  25001
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