Theorem ghmmhm 19268

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.

Step	Hyp	Ref	Expression
1		ghmgrp1 19260	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2	1	grpmndd 18990	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
3		ghmgrp2 19261	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
4	3	grpmndd 18990	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
5		eqid 2764	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2764	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	ghmf 19262	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8		eqid 2764	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
9		eqid 2764	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
10	5, 8, 9	ghmlin 19263	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
11	10	3expb 1134	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
12	11	ralrimivva 3207	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
13		eqid 2764	. . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆)
14		eqid 2764	. . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	ghmid 19264	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
16	7, 12, 15	3jca 1142	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))
17	5, 6, 8, 9, 13, 14	ismhm 18821	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))))
18	2, 4, 16, 17	syl21anbrc 1359	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  0_gc0g 17470  Mndcmnd 18770   MndHom cmhm 18817   GrpHom cghm 19255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-map 8812  df-0g 17472  df-mgm 18676  df-sgrp 18755  df-mnd 18771  df-mhm 18819  df-grp 18980  df-ghm 19256

This theorem is referenced by:  ghmmhmb  19269  ghmmulg  19270  resghm2  19275  ghmco  19278  ghmeql  19281  symgtrinv  19514  frgpup3lem  19819  gsummulglem  19983  gsumzinv  19987  gsuminv  19988  gsummulc1  20366  gsummulc2  20367  pwsco2rhm  20554  gsumvsmul  20995  rhmpreimaidl  21349  zrhpsgnmhm  21638  evlslem2  22134  evlsgsumadd  22151  rhmcomulmpl  22179  selvcllem4  22193  selvvvval  22197  selvadd  22198  selvmul  22199  evls1gsumadd  22389  rhmmpl  22445  rhmply1vsca  22450  mat2pmatmul  22793  pm2mp  22887  cayhamlem4  22950  tsmsinv  24210  plypf1  26274  amgmlem  27056  lgseisenlem4  27444  gsumvsmul1  33233  gsummulgc2  33248  fxpsubg  33355  algextdeglem8  34023  rhmcomulpsr  43169  rhmpsr  43170  evlselv  43176  mendring  43770  amgmwlem  50428  amgmlemALT  50429