Theorem ghmmhm 13326

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 13318	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	grpmndd 13088	. 2 |- ( F e. ( S GrpHom T ) -> S e. Mnd )
3		ghmgrp2 13319	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
4	3	grpmndd 13088	. 2 |- ( F e. ( S GrpHom T ) -> T e. Mnd )
5		eqid 2193	. . . 4 |- ( Base ` S ) = ( Base ` S )
6		eqid 2193	. . . 4 |- ( Base ` T ) = ( Base ` T )
7	5, 6	ghmf 13320	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8		eqid 2193	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
9		eqid 2193	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
10	5, 8, 9	ghmlin 13321	. . . . 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 2576	. . 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 2193	. . . 4 |- ( 0g ` S ) = ( 0g ` S )
14		eqid 2193	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
15	13, 14	ghmid 13322	. . 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 13036	. 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 2164   A.wral 2472   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Mndcmnd 13000   MndHom cmhm 13032   GrpHom cghm 13313

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-grp 13078  df-ghm 13314

This theorem is referenced by:  ghmmhmb  13327  ghmmulg  13329  resghm2  13334  ghmco  13337  ghmeql  13340  lgseisenlem4  15230