Theorem ghmmhmb 14007

Description: Group homomorphisms and monoid homomorphisms coincide. (Thus, GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
ghmmhmb	|- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )

Proof of Theorem ghmmhmb

Dummy variables x y f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmmhm 14006	. . 3 |- ( f e. ( S GrpHom T ) -> f e. ( S MndHom T ) )
2		eqid 2234	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2234	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4		eqid 2234	. . . . 5 |- ( +g ` S ) = ( +g ` S )
5		eqid 2234	. . . . 5 |- ( +g ` T ) = ( +g ` T )
6		simpll 527	. . . . 5 |- ( ( ( S e. Grp /\ T e. Grp ) /\ f e. ( S MndHom T ) ) -> S e. Grp )
7		simplr 529	. . . . 5 |- ( ( ( S e. Grp /\ T e. Grp ) /\ f e. ( S MndHom T ) ) -> T e. Grp )
8	2, 3	mhmf 13720	. . . . . 6 |- ( f e. ( S MndHom T ) -> f : ( Base ` S ) --> ( Base ` T ) )
9	8	adantl 277	. . . . 5 |- ( ( ( S e. Grp /\ T e. Grp ) /\ f e. ( S MndHom T ) ) -> f : ( Base ` S ) --> ( Base ` T ) )
10	2, 4, 5	mhmlin 13722	. . . . . . 7 |- ( ( f e. ( S MndHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( f ` ( x ( +g ` S ) y ) ) = ( ( f ` x ) ( +g ` T ) ( f ` y ) ) )
11	10	3expb 1231	. . . . . 6 |- ( ( f e. ( S MndHom T ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( f ` ( x ( +g ` S ) y ) ) = ( ( f ` x ) ( +g ` T ) ( f ` y ) ) )
12	11	adantll 476	. . . . 5 |- ( ( ( ( S e. Grp /\ T e. Grp ) /\ f e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( f ` ( x ( +g ` S ) y ) ) = ( ( f ` x ) ( +g ` T ) ( f ` y ) ) )
13	2, 3, 4, 5, 6, 7, 9, 12	isghmd 14005	. . . 4 |- ( ( ( S e. Grp /\ T e. Grp ) /\ f e. ( S MndHom T ) ) -> f e. ( S GrpHom T ) )
14	13	ex 115	. . 3 |- ( ( S e. Grp /\ T e. Grp ) -> ( f e. ( S MndHom T ) -> f e. ( S GrpHom T ) ) )
15	1, 14	impbid2 143	. 2 |- ( ( S e. Grp /\ T e. Grp ) -> ( f e. ( S GrpHom T ) <-> f e. ( S MndHom T ) ) )
16	15	eqrdv 2232	1 |- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   MndHom cmhm 13712   Grpcgrp 13755   GrpHom cghm 13993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714  df-grp 13758  df-ghm 13994

This theorem is referenced by:  ghmex  14008  0ghm  14011  resghm2  14014  resghm2b  14015  ghmco  14017  ghmpropd  14036