MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mhm0 Structured version   Visualization version   GIF version

Theorem mhm0 17259
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 2626 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2626 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 eqid 2626 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2626 . . . 4 (+g𝑇) = (+g𝑇)
5 mhm0.z . . . 4 0 = (0g𝑆)
6 mhm0.y . . . 4 𝑌 = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 17253 . . 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 480 . 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 1073 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Mndcmnd 17210   MndHom cmhm 17249
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-mhm 17251
This theorem is referenced by:  mhmf1o  17261  resmhm  17275  resmhm2  17276  resmhm2b  17277  mhmco  17278  mhmima  17279  mhmeql  17280  pwsco2mhm  17287  gsumwmhm  17298  mhmmulg  17499  gsumzmhm  18253  rhm1  18646  madetsumid  20181  mdetunilem7  20338  pm2mp  20544  dchrzrh1  24864  dchrmulcl  24869  dchrn0  24870  dchrinvcl  24873  dchrfi  24875  dchrabs  24880  sumdchr2  24890  rpvmasum2  25096
  Copyright terms: Public domain W3C validator