Theorem ghmmhmb 13791

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 13790	. . 3 |- ( f e. ( S GrpHom T ) -> f e. ( S MndHom T ) )
2		eqid 2229	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2229	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4		eqid 2229	. . . . 5 |- ( +g ` S ) = ( +g ` S )
5		eqid 2229	. . . . 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 13498	. . . . . 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 13500	. . . . . . 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 1228	. . . . . 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 13789	. . . 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 2227	1 |- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   -->wf 5314   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   MndHom cmhm 13490   Grpcgrp 13533   GrpHom cghm 13777

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-mhm 13492  df-grp 13536  df-ghm 13778

This theorem is referenced by:  ghmex  13792  0ghm  13795  resghm2  13798  resghm2b  13799  ghmco  13801  ghmpropd  13820