Theorem ghmmhm 13459

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 13451	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	grpmndd 13215	. 2 |- ( F e. ( S GrpHom T ) -> S e. Mnd )
3		ghmgrp2 13452	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
4	3	grpmndd 13215	. 2 |- ( F e. ( S GrpHom T ) -> T e. Mnd )
5		eqid 2196	. . . 4 |- ( Base ` S ) = ( Base ` S )
6		eqid 2196	. . . 4 |- ( Base ` T ) = ( Base ` T )
7	5, 6	ghmf 13453	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8		eqid 2196	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
9		eqid 2196	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
10	5, 8, 9	ghmlin 13454	. . . . 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 ) ) )
11	10	3expb 1206	. . . 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 ) ) )
12	11	ralrimivva 2579	. . 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 ) ) )
13		eqid 2196	. . . 4 |- ( 0g ` S ) = ( 0g ` S )
14		eqid 2196	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
15	13, 14	ghmid 13455	. . 3 |- ( F e. ( S GrpHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
16	7, 12, 15	3jca 1179	. 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 ) ) )
17	5, 6, 8, 9, 13, 14	ismhm 13163	. 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 ) ) ) )
18	2, 4, 16, 17	syl21anbrc 1184	1 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   -->wf 5255   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958   Mndcmnd 13118   MndHom cmhm 13159   GrpHom cghm 13446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-mhm 13161  df-grp 13205  df-ghm 13447

This theorem is referenced by:  ghmmhmb  13460  ghmmulg  13462  resghm2  13467  ghmco  13470  ghmeql  13473  lgseisenlem4  15398