Theorem ghmmhm 19440

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 19432	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2	1	grpmndd 19157	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
3		ghmgrp2 19433	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
4	3	grpmndd 19157	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
5		eqid 2760	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2760	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	ghmf 19434	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8		eqid 2760	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
9		eqid 2760	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
10	5, 8, 9	ghmlin 19435	. . . . 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 1138	. . . 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 3205	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
13		eqid 2760	. . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆)
14		eqid 2760	. . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	ghmid 19436	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
16	7, 12, 15	3jca 1146	. 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 18980	. 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 1363	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  Mndcmnd 18923   MndHom cmhm 18976   GrpHom cghm 19427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-grp 19147  df-ghm 19428

This theorem is used by:  ghmmhmb  19441  ghmmulg  19442  resghm2  19447  ghmco  19450  ghmeql  19453  symgtrinv  19686  frgpup3lem  19991  gsummulglem  20155  gsumzinv  20159  gsuminv  20160  gsummulc1  20545  gsummulc2  20546  rhmadd  20718  rhm0  20723  pwsco2rhm  20742  gsumvsmul  21201  rhmpreimaidl  21571  zrhpsgnmhm  21890  evlslem2  22388  evlsgsumadd  22405  rhmcomulmpl  22433  selvcllem4  22447  selvvvval  22451  selvadd  22452  selvmul  22453  evls1gsumadd  22642  rhmmpl  22698  rhmply1vsca  22703  mat2pmatmul  23049  pm2mp  23143  cayhamlem4  23206  tsmsinv  24467  plypf1  26531  amgmlem  27317  lgseisenlem4  27705  gsumvsmul1  33612  gsummulgc2  33627  fxpsubg  33734  algextdeglem8  34356  rhmcomulpsr  43611  rhmpsr  43612  evlselv  43618  mendring  44190  amgmwlem  50987  amgmlemALT  50988