Theorem ghmeqker 14005

Description: Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghmeqker.b	|- B = ( Base ` S )
ghmeqker.z	|- .0. = ( 0g ` T )
ghmeqker.k	|- K = ( `' F " { .0. } )
ghmeqker.m	|- .- = ( -g ` S )

Assertion

Ref	Expression
ghmeqker	|- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) = ( F ` V ) <-> ( U .- V ) e. K ) )

Proof of Theorem ghmeqker

Step	Hyp	Ref	Expression
1		ghmeqker.k	. . . . 5 |- K = ( `' F " { .0. } )
2		ghmeqker.z	. . . . . . 7 |- .0. = ( 0g ` T )
3	2	sneqi 3703	. . . . . 6 |- { .0. } = { ( 0g ` T ) }
4	3	imaeq2i 5101	. . . . 5 |- ( `' F " { .0. } ) = ( `' F " { ( 0g ` T ) } )
5	1, 4	eqtri 2255	. . . 4 |- K = ( `' F " { ( 0g ` T ) } )
6	5	eleq2i 2301	. . 3 |- ( ( U .- V ) e. K <-> ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) )
7		ghmeqker.b	. . . . . . 7 |- B = ( Base ` S )
8		eqid 2234	. . . . . . 7 |- ( Base ` T ) = ( Base ` T )
9	7, 8	ghmf 13981	. . . . . 6 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
10	9	ffnd 5511	. . . . 5 |- ( F e. ( S GrpHom T ) -> F Fn B )
11	10	3ad2ant1 1045	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F Fn B )
12		fniniseg 5800	. . . 4 |- ( F Fn B -> ( ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) <-> ( ( U .- V ) e. B /\ ( F ` ( U .- V ) ) = ( 0g ` T ) ) ) )
13	11, 12	syl 14	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) <-> ( ( U .- V ) e. B /\ ( F ` ( U .- V ) ) = ( 0g ` T ) ) ) )
14	6, 13	bitrid 192	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( U .- V ) e. K <-> ( ( U .- V ) e. B /\ ( F ` ( U .- V ) ) = ( 0g ` T ) ) ) )
15		ghmgrp1 13979	. . . . 5 |- ( F e. ( S GrpHom T ) -> S e. Grp )
16		ghmeqker.m	. . . . . 6 |- .- = ( -g ` S )
17	7, 16	grpsubcl 13810	. . . . 5 |- ( ( S e. Grp /\ U e. B /\ V e. B ) -> ( U .- V ) e. B )
18	15, 17	syl3an1 1307	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( U .- V ) e. B )
19	18	biantrurd 305	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` ( U .- V ) ) = ( 0g ` T ) <-> ( ( U .- V ) e. B /\ ( F ` ( U .- V ) ) = ( 0g ` T ) ) ) )
20		eqid 2234	. . . . 5 |- ( -g ` T ) = ( -g ` T )
21	7, 16, 20	ghmsub 13985	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) ( -g ` T ) ( F ` V ) ) )
22	21	eqeq1d 2243	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` ( U .- V ) ) = ( 0g ` T ) <-> ( ( F ` U ) ( -g ` T ) ( F ` V ) ) = ( 0g ` T ) ) )
23	19, 22	bitr3d 190	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( ( U .- V ) e. B /\ ( F ` ( U .- V ) ) = ( 0g ` T ) ) <-> ( ( F ` U ) ( -g ` T ) ( F ` V ) ) = ( 0g ` T ) ) )
24		ghmgrp2 13980	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
25	24	3ad2ant1 1045	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> T e. Grp )
26	9	3ad2ant1 1045	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F : B --> ( Base ` T ) )
27		simp2 1025	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> U e. B )
28	26, 27	ffvelcdmd 5815	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` U ) e. ( Base ` T ) )
29		simp3 1026	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
30	26, 29	ffvelcdmd 5815	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
31		eqid 2234	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
32	8, 31, 20	grpsubeq0 13816	. . 3 |- ( ( T e. Grp /\ ( F ` U ) e. ( Base ` T ) /\ ( F ` V ) e. ( Base ` T ) ) -> ( ( ( F ` U ) ( -g ` T ) ( F ` V ) ) = ( 0g ` T ) <-> ( F ` U ) = ( F ` V ) ) )
33	25, 28, 30, 32	syl3anc 1274	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( ( F ` U ) ( -g ` T ) ( F ` V ) ) = ( 0g ` T ) <-> ( F ` U ) = ( F ` V ) ) )
34	14, 23, 33	3bitrrd 215	1 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) = ( F ` V ) <-> ( U .- V ) e. K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {csn 3691   `'ccnv 4750   "cima 4754   Fn wfn 5349   -->wf 5350   ` cfv 5354  (class class class)co 6052   Basecbs 13229   0gc0g 13486   Grpcgrp 13730   -gcsg 13732   GrpHom cghm 13974

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 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-sbg 13735  df-ghm 13975

This theorem is referenced by:  kerf1ghm  14008