Theorem ghmmhm 19190

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 19182	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2	1	grpmndd 18911	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
3		ghmgrp2 19183	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
4	3	grpmndd 18911	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
5		eqid 2735	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2735	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	ghmf 19184	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8		eqid 2735	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
9		eqid 2735	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
10	5, 8, 9	ghmlin 19185	. . . . 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 1121	. . . 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 3178	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
13		eqid 2735	. . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆)
14		eqid 2735	. . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	ghmid 19186	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
16	7, 12, 15	3jca 1129	. 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 18742	. 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 1346	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3049  ⟶wf 6483  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  +_gcplusg 17209  0_gc0g 17391  Mndcmnd 18691   MndHom cmhm 18738   GrpHom cghm 19176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8764  df-0g 17393  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-mhm 18740  df-grp 18901  df-ghm 19177

This theorem is referenced by:  ghmmhmb  19191  ghmmulg  19192  resghm2  19197  ghmco  19200  ghmeql  19203  symgtrinv  19436  frgpup3lem  19741  gsummulglem  19905  gsumzinv  19909  gsuminv  19910  gsummulc1  20284  gsummulc2  20285  pwsco2rhm  20469  gsumvsmul  20910  rhmpreimaidl  21264  zrhpsgnmhm  21553  evlslem2  22046  evlsgsumadd  22063  evls1gsumadd  22277  rhmcomulmpl  22335  rhmmpl  22336  rhmply1vsca  22341  mat2pmatmul  22684  pm2mp  22778  cayhamlem4  22841  tsmsinv  24101  plypf1  26165  amgmlem  26941  lgseisenlem4  27329  gsumvsmul1  33100  gsummulgc2  33115  fxpsubg  33222  algextdeglem8  33856  rhmcomulpsr  42982  rhmpsr  42983  selvcllem4  43002  selvvvval  43006  evlselv  43008  selvadd  43009  selvmul  43010  mendring  43604  amgmwlem  50265  amgmlemALT  50266