Theorem ghmid 13794

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 13790	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		eqid 2229	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
3		ghmid.y	. . . . . . 7 |- Y = ( 0g ` S )
4	2, 3	grpidcl 13570	. . . . . 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 2229	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
7		eqid 2229	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
8	2, 6, 7	ghmlin 13793	. . . . 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 1373	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) )
10	2, 6, 3	grplid 13572	. . . . . 6 |- ( ( S e. Grp /\ Y e. ( Base ` S ) ) -> ( Y ( +g ` S ) Y ) = Y )
11	1, 5, 10	syl2anc 411	. . . . 5 |- ( F e. ( S GrpHom T ) -> ( Y ( +g ` S ) Y ) = Y )
12	11	fveq2d 5633	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( F ` Y ) )
13	9, 12	eqtr3d 2264	. . 3 |- ( F e. ( S GrpHom T ) -> ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) = ( F ` Y ) )
14		ghmgrp2 13791	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
15		eqid 2229	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
16	2, 15	ghmf 13792	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
17	16, 5	ffvelcdmd 5773	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` Y ) e. ( Base ` T ) )
18		ghmid.z	. . . . 5 |- .0. = ( 0g ` T )
19	15, 7, 18	grpid 13580	. . . 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 411	. . 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 2235	1 |- ( F e. ( S GrpHom T ) -> ( F ` Y ) = .0. )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   Basecbs 13040   +g cplusg 13118   0gc0g 13297   Grpcgrp 13541   GrpHom cghm 13785

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 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

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 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-inn 9119  df-2 9177  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-ghm 13786

This theorem is referenced by:  ghminv  13795  ghmmhm  13798  ghmpreima  13811  f1ghm0to0  13817  kerf1ghm  13819  zrh0  14597  zndvds0  14622