Theorem ghmmhm 17670

Description: A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
ghmmhm	|- ( F e. ( S GrpHom T ) -> F e. ( 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 17662	. . . 4 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		grpmnd 17429	. . . 4 |- ( S e. Grp -> S e. Mnd )
3	1, 2	syl 17	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Mnd )
4		ghmgrp2 17663	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
5		grpmnd 17429	. . . 4 |- ( T e. Grp -> T e. Mnd )
6	4, 5	syl 17	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Mnd )
7	3, 6	jca 554	. 2 |- ( F e. ( S GrpHom T ) -> ( S e. Mnd /\ T e. Mnd ) )
8		eqid 2622	. . . 4 |- ( Base ` S ) = ( Base ` S )
9		eqid 2622	. . . 4 |- ( Base ` T ) = ( Base ` T )
10	8, 9	ghmf 17664	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
11		eqid 2622	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
12		eqid 2622	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
13	8, 11, 12	ghmlin 17665	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
14	13	3expb 1266	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
15	14	ralrimivva 2971	. . 3 |- ( F e. ( S GrpHom T ) -> A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
16		eqid 2622	. . . 4 |- ( 0g ` S ) = ( 0g ` S )
17		eqid 2622	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
18	16, 17	ghmid 17666	. . 3 |- ( F e. ( S GrpHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
19	10, 15, 18	3jca 1242	. 2 |- ( F e. ( S GrpHom T ) -> ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) )
20	8, 9, 11, 12, 16, 17	ismhm 17337	. 2 |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) ) )
21	7, 19, 20	sylanbrc 698	1 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   MndHom cmhm 17333   Grpcgrp 17422   GrpHom cghm 17657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658

This theorem is referenced by:  ghmmhmb  17671  ghmmulg  17672  resghm2  17677  ghmco  17680  ghmeql  17683  symgtrinv  17892  frgpup3lem  18190  gsummulglem  18341  gsumzinv  18345  gsuminv  18346  gsummulc1  18606  gsummulc2  18607  pwsco2rhm  18739  gsumvsmul  18927  evlslem2  19512  evls1gsumadd  19689  zrhpsgnmhm  19930  mat2pmatmul  20536  pm2mp  20630  cayhamlem4  20693  tsmsinv  21951  plypf1  23968  amgmlem  24716  lgseisenlem4  25103  mendring  37762  amgmwlem  42548  amgmlemALT  42549