Theorem ghmmhmb 13971

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 13970	. . 3 |- ( f e. ( S GrpHom T ) -> f e. ( S MndHom T ) )
2		eqid 2232	. . . . 5 |- ( Base ` S ) = ( Base ` S )
3		eqid 2232	. . . . 5 |- ( Base ` T ) = ( Base ` T )
4		eqid 2232	. . . . 5 |- ( +g ` S ) = ( +g ` S )
5		eqid 2232	. . . . 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 13678	. . . . . 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 13680	. . . . . . 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 13969	. . . 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 2230	1 |- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   -->wf 5348   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   MndHom cmhm 13670   Grpcgrp 13713   GrpHom cghm 13957

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mhm 13672  df-grp 13716  df-ghm 13958

This theorem is referenced by:  ghmex  13972  0ghm  13975  resghm2  13978  resghm2b  13979  ghmco  13981  ghmpropd  14000