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Theorem ghmmhm 17493
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 17485 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 grpmnd 17252 . . . 4 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
4 ghmgrp2 17486 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
5 grpmnd 17252 . . . 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 17487 . . 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 17488 . . . . 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 17489 . . 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 17160 . 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 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117   MndHom cmhm 17156  Grpcgrp 17245   GrpHom cghm 17480
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 4699  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-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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481
This theorem is referenced by:  ghmmhmb  17494  ghmmulg  17495  resghm2  17500  ghmco  17503  ghmeql  17506  symgtrinv  17715  frgpup3lem  18013  gsummulglem  18164  gsumzinv  18168  gsuminv  18169  gsummulc1  18429  gsummulc2  18430  pwsco2rhm  18562  gsumvsmul  18750  evlslem2  19333  evls1gsumadd  19510  zrhpsgnmhm  19749  mat2pmatmul  20355  pm2mp  20449  cayhamlem4  20512  tsmsinv  21761  plypf1  23772  amgmlem  24516  lgseisenlem4  24903  mendring  36781  amgmwlem  42357  amgmlemALT  42358
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