Theorem ghmmhm 18449

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 18441	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		grpmnd 18190	. . 3 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
3	1, 2	syl 17	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
4		ghmgrp2 18442	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
5		grpmnd 18190	. . 3 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
6	4, 5	syl 17	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
7		eqid 2758	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
8		eqid 2758	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
9	7, 8	ghmf 18443	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
10		eqid 2758	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
11		eqid 2758	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
12	7, 10, 11	ghmlin 18444	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
13	12	3expb 1117	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
14	13	ralrimivva 3120	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
15		eqid 2758	. . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆)
16		eqid 2758	. . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇)
17	15, 16	ghmid 18445	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
18	9, 14, 17	3jca 1125	. 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)))
19	7, 8, 10, 11, 15, 16	ismhm 18038	. 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))))
20	3, 6, 18, 19	syl21anbrc 1341	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  +_gcplusg 16637  0_gc0g 16785  Mndcmnd 17991   MndHom cmhm 18034  Grpcgrp 18183   GrpHom cghm 18436

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8424  df-0g 16787  df-mgm 17932  df-sgrp 17981  df-mnd 17992  df-mhm 18036  df-grp 18186  df-ghm 18437

This theorem is referenced by:  ghmmhmb  18450  ghmmulg  18451  resghm2  18456  ghmco  18459  ghmeql  18462  symgtrinv  18681  frgpup3lem  18984  gsummulglem  19143  gsumzinv  19147  gsuminv  19148  gsummulc1  19441  gsummulc2  19442  pwsco2rhm  19576  gsumvsmul  19780  zrhpsgnmhm  20363  evlslem2  20856  evlsgsumadd  20868  evls1gsumadd  21057  mat2pmatmul  21445  pm2mp  21539  cayhamlem4  21602  tsmsinv  22862  plypf1  24922  amgmlem  25688  lgseisenlem4  26075  gsumvsmul1  30850  rhmpreimaidl  31137  mendring  40554  amgmwlem  45827  amgmlemALT  45828