Theorem ghmmhm 13970

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 13962	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	grpmndd 13726	. 2 |- ( F e. ( S GrpHom T ) -> S e. Mnd )
3		ghmgrp2 13963	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
4	3	grpmndd 13726	. 2 |- ( F e. ( S GrpHom T ) -> T e. Mnd )
5		eqid 2232	. . . 4 |- ( Base ` S ) = ( Base ` S )
6		eqid 2232	. . . 4 |- ( Base ` T ) = ( Base ` T )
7	5, 6	ghmf 13964	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8		eqid 2232	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
9		eqid 2232	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
10	5, 8, 9	ghmlin 13965	. . . . 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 1231	. . . 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 2624	. . 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 2232	. . . 4 |- ( 0g ` S ) = ( 0g ` S )
14		eqid 2232	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
15	13, 14	ghmid 13966	. . 3 |- ( F e. ( S GrpHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
16	7, 12, 15	3jca 1204	. 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 13674	. 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 1209	1 |- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   -->wf 5348   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Mndcmnd 13629   MndHom cmhm 13670   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:  ghmmhmb  13971  ghmmulg  13973  resghm2  13978  ghmco  13981  ghmeql  13984  lgseisenlem4  15946