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

Theorem ghmmhm 17442
Description: A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
ghmmhm (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Proof of Theorem ghmmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp1 17434 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 grpmnd 17201 . . . 4 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
4 ghmgrp2 17435 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
5 grpmnd 17201 . . . 4 (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
64, 5syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
73, 6jca 553 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
8 eqid 2610 . . . 4 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2610 . . . 4 (Base‘𝑇) = (Base‘𝑇)
108, 9ghmf 17436 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
11 eqid 2610 . . . . . 6 (+g𝑆) = (+g𝑆)
12 eqid 2610 . . . . . 6 (+g𝑇) = (+g𝑇)
138, 11, 12ghmlin 17437 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
14133expb 1258 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
1514ralrimivva 2954 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
16 eqid 2610 . . . 4 (0g𝑆) = (0g𝑆)
17 eqid 2610 . . . 4 (0g𝑇) = (0g𝑇)
1816, 17ghmid 17438 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1910, 15, 183jca 1235 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇)))
208, 9, 11, 12, 16, 17ismhm 17109 . 2 (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)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
217, 19, 20sylanbrc 695 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wf 5786  cfv 5790  (class class class)co 6527  Basecbs 15644  +gcplusg 15717  0gc0g 15872  Mndcmnd 17066   MndHom cmhm 17105  Grpcgrp 17194   GrpHom cghm 17429
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-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
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-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-mhm 17107  df-grp 17197  df-ghm 17430
This theorem is referenced by:  ghmmhmb  17443  ghmmulg  17444  resghm2  17449  ghmco  17452  ghmeql  17455  symgtrinv  17664  frgpup3lem  17962  gsummulglem  18113  gsumzinv  18117  gsuminv  18118  gsummulc1  18378  gsummulc2  18379  pwsco2rhm  18511  gsumvsmul  18699  evlslem2  19282  evls1gsumadd  19459  zrhpsgnmhm  19697  mat2pmatmul  20303  pm2mp  20397  cayhamlem4  20460  tsmsinv  21709  plypf1  23717  amgmlem  24461  lgseisenlem4  24848  mendring  36575  amgmwlem  42310  amgmlemALT  42311
  Copyright terms: Public domain W3C validator