Theorem ghmmhmb 13324

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 13323	. . 3 |- ( f e. ( S GrpHom T ) -> f e. ( S MndHom T ) )
2		eqid 2193	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2193	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4		eqid 2193	. . . . 5 |- ( +g ` S ) = ( +g ` S )
5		eqid 2193	. . . . 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 528	. . . . 5 |- ( ( ( S e. Grp /\ T e. Grp ) /\ f e. ( S MndHom T ) ) -> T e. Grp )
8	2, 3	mhmf 13037	. . . . . 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 13039	. . . . . . 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 1206	. . . . . 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 13322	. . . 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 2191	1 |- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   -->wf 5250   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   MndHom cmhm 13029   Grpcgrp 13072   GrpHom cghm 13310

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031  df-grp 13075  df-ghm 13311

This theorem is referenced by:  ghmex  13325  0ghm  13328  resghm2  13331  resghm2b  13332  ghmco  13334  ghmpropd  13353