Theorem ghmeqker 13857

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 3681	. . . . . 6 |- { .0. } = { ( 0g ` T ) }
4	3	imaeq2i 5074	. . . . 5 |- ( `' F " { .0. } ) = ( `' F " { ( 0g ` T ) } )
5	1, 4	eqtri 2252	. . . 4 |- K = ( `' F " { ( 0g ` T ) } )
6	5	eleq2i 2298	. . 3 |- ( ( U .- V ) e. K <-> ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) )
7		ghmeqker.b	. . . . . . 7 |- B = ( Base ` S )
8		eqid 2231	. . . . . . 7 |- ( Base ` T ) = ( Base ` T )
9	7, 8	ghmf 13833	. . . . . 6 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
10	9	ffnd 5483	. . . . 5 |- ( F e. ( S GrpHom T ) -> F Fn B )
11	10	3ad2ant1 1044	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F Fn B )
12		fniniseg 5767	. . . 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 13831	. . . . 5 |- ( F e. ( S GrpHom T ) -> S e. Grp )
16		ghmeqker.m	. . . . . 6 |- .- = ( -g ` S )
17	7, 16	grpsubcl 13662	. . . . 5 |- ( ( S e. Grp /\ U e. B /\ V e. B ) -> ( U .- V ) e. B )
18	15, 17	syl3an1 1306	. . . 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 2231	. . . . 5 |- ( -g ` T ) = ( -g ` T )
21	7, 16, 20	ghmsub 13837	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) ( -g ` T ) ( F ` V ) ) )
22	21	eqeq1d 2240	. . 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 13832	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
25	24	3ad2ant1 1044	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> T e. Grp )
26	9	3ad2ant1 1044	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F : B --> ( Base ` T ) )
27		simp2 1024	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> U e. B )
28	26, 27	ffvelcdmd 5783	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` U ) e. ( Base ` T ) )
29		simp3 1025	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
30	26, 29	ffvelcdmd 5783	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
31		eqid 2231	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
32	8, 31, 20	grpsubeq0 13668	. . 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 1273	. 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 1004   = wceq 1397   e. wcel 2202   {csn 3669   `'ccnv 4724   "cima 4728   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   0gc0g 13338   Grpcgrp 13582   -gcsg 13584   GrpHom cghm 13826

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-sbg 13587  df-ghm 13827

This theorem is referenced by:  kerf1ghm  13860