Theorem ghmid 14052

Description: A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghmid.y	|- Y = ( 0g ` S )
ghmid.z	|- .0. = ( 0g ` T )

Assertion

Ref	Expression
ghmid	|- ( F e. ( S GrpHom T ) -> ( F ` Y ) = .0. )

Proof of Theorem ghmid

Step	Hyp	Ref	Expression
1		ghmgrp1 14048	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		eqid 2238	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
3		ghmid.y	. . . . . . 7 |- Y = ( 0g ` S )
4	2, 3	grpidcl 13834	. . . . . 6 |- ( S e. Grp -> Y e. ( Base ` S ) )
5	1, 4	syl 14	. . . . 5 |- ( F e. ( S GrpHom T ) -> Y e. ( Base ` S ) )
6		eqid 2238	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
7		eqid 2238	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
8	2, 6, 7	ghmlin 14051	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ Y e. ( Base ` S ) /\ Y e. ( Base ` S ) ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) )
9	5, 5, 8	mpd3an23 1380	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) )
10	2, 6, 3	grplid 13836	. . . . . 6 |- ( ( S e. Grp /\ Y e. ( Base ` S ) ) -> ( Y ( +g ` S ) Y ) = Y )
11	1, 5, 10	syl2anc 415	. . . . 5 |- ( F e. ( S GrpHom T ) -> ( Y ( +g ` S ) Y ) = Y )
12	11	fveq2d 5699	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( F ` Y ) )
13	9, 12	eqtr3d 2273	. . 3 |- ( F e. ( S GrpHom T ) -> ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) = ( F ` Y ) )
14		ghmgrp2 14049	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
15		eqid 2238	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
16	2, 15	ghmf 14050	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
17	16, 5	ffvelcdmd 5844	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` Y ) e. ( Base ` T ) )
18		ghmid.z	. . . . 5 |- .0. = ( 0g ` T )
19	15, 7, 18	grpid 13844	. . . 4 |- ( ( T e. Grp /\ ( F ` Y ) e. ( Base ` T ) ) -> ( ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) = ( F ` Y ) <-> .0. = ( F ` Y ) ) )
20	14, 17, 19	syl2anc 415	. . 3 |- ( F e. ( S GrpHom T ) -> ( ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) = ( F ` Y ) <-> .0. = ( F ` Y ) ) )
21	13, 20	mpbid 147	. 2 |- ( F e. ( S GrpHom T ) -> .0. = ( F ` Y ) )
22	21	eqcomd 2244	1 |- ( F e. ( S GrpHom T ) -> ( F ` Y ) = .0. )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   Grpcgrp 13805   GrpHom cghm 14043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-ghm 14044

This theorem is used by:  ghminv  14053  ghmmhm  14056  ghmpreima  14069  f1ghm0to0  14075  kerf1ghm  14077  zrh0  14960  zndvds0  14985