Theorem ghmmhm 14006

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 13998	. . 3 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	grpmndd 13768	. 2 |- ( F e. ( S GrpHom T ) -> S e. Mnd )
3		ghmgrp2 13999	. . 3 |- ( F e. ( S GrpHom T ) -> T e. Grp )
4	3	grpmndd 13768	. 2 |- ( F e. ( S GrpHom T ) -> T e. Mnd )
5		eqid 2234	. . . 4 |- ( Base ` S ) = ( Base ` S )
6		eqid 2234	. . . 4 |- ( Base ` T ) = ( Base ` T )
7	5, 6	ghmf 14000	. . 3 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8		eqid 2234	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
9		eqid 2234	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
10	5, 8, 9	ghmlin 14001	. . . . 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 2626	. . 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 2234	. . . 4 |- ( 0g ` S ) = ( 0g ` S )
14		eqid 2234	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
15	13, 14	ghmid 14002	. . 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 13716	. 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 2205   A.wral 2522   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Mndcmnd 13677   MndHom cmhm 13712   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:  ghmmhmb  14007  ghmmulg  14009  resghm2  14014  ghmco  14017  ghmeql  14020  lgseisenlem4  16072