Theorem ghmid 17666

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 17662	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2		eqid 2622	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
3		ghmid.y	. . . . . . 7 |- Y = ( 0g ` S )
4	2, 3	grpidcl 17450	. . . . . 6 |- ( S e. Grp -> Y e. ( Base ` S ) )
5	1, 4	syl 17	. . . . 5 |- ( F e. ( S GrpHom T ) -> Y e. ( Base ` S ) )
6		eqid 2622	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
7		eqid 2622	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
8	2, 6, 7	ghmlin 17665	. . . . 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 1426	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) )
10	2, 6, 3	grplid 17452	. . . . . 6 |- ( ( S e. Grp /\ Y e. ( Base ` S ) ) -> ( Y ( +g ` S ) Y ) = Y )
11	1, 5, 10	syl2anc 693	. . . . 5 |- ( F e. ( S GrpHom T ) -> ( Y ( +g ` S ) Y ) = Y )
12	11	fveq2d 6195	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` ( Y ( +g ` S ) Y ) ) = ( F ` Y ) )
13	9, 12	eqtr3d 2658	. . 3 |- ( F e. ( S GrpHom T ) -> ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) = ( F ` Y ) )
14		ghmgrp2 17663	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
15		eqid 2622	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
16	2, 15	ghmf 17664	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
17	16, 5	ffvelrnd 6360	. . . 4 |- ( F e. ( S GrpHom T ) -> ( F ` Y ) e. ( Base ` T ) )
18		ghmid.z	. . . . 5 |- .0. = ( 0g ` T )
19	15, 7, 18	grpid 17457	. . . 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 693	. . 3 |- ( F e. ( S GrpHom T ) -> ( ( ( F ` Y ) ( +g ` T ) ( F ` Y ) ) = ( F ` Y ) <-> .0. = ( F ` Y ) ) )
21	13, 20	mpbid 222	. 2 |- ( F e. ( S GrpHom T ) -> .0. = ( F ` Y ) )
22	21	eqcomd 2628	1 |- ( F e. ( S GrpHom T ) -> ( F ` Y ) = .0. )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   GrpHom cghm 17657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658

This theorem is referenced by:  ghminv  17667  ghmmhm  17670  ghmpreima  17682  ghmf1  17689  lactghmga  17824  f1rhm0to0  18740  f1rhm0to0ALT  18741  kerf1hrm  18743  srng0  18860  islmhm2  19038  evlslem2  19512  evlslem6  19513  evlslem3  19514  zrh0  19862  chrrhm  19879  zndvds0  19899  ip0l  19981  0mat2pmat  20541  nmolb2d  22522  nmoi  22532  nmoix  22533  nmoleub  22535  nmoleub2lem2  22916  nmhmcn  22920  dchrptlem2  24990  psgnid  29847  nrhmzr  41873  zrinitorngc  42000