Theorem ghmeqker 13344

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 3631	. . . . . 6 |- { .0. } = { ( 0g ` T ) }
4	3	imaeq2i 5004	. . . . 5 |- ( `' F " { .0. } ) = ( `' F " { ( 0g ` T ) } )
5	1, 4	eqtri 2214	. . . 4 |- K = ( `' F " { ( 0g ` T ) } )
6	5	eleq2i 2260	. . 3 |- ( ( U .- V ) e. K <-> ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) )
7		ghmeqker.b	. . . . . . 7 |- B = ( Base ` S )
8		eqid 2193	. . . . . . 7 |- ( Base ` T ) = ( Base ` T )
9	7, 8	ghmf 13320	. . . . . 6 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
10	9	ffnd 5405	. . . . 5 |- ( F e. ( S GrpHom T ) -> F Fn B )
11	10	3ad2ant1 1020	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F Fn B )
12		fniniseg 5679	. . . 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 13318	. . . . 5 |- ( F e. ( S GrpHom T ) -> S e. Grp )
16		ghmeqker.m	. . . . . 6 |- .- = ( -g ` S )
17	7, 16	grpsubcl 13155	. . . . 5 |- ( ( S e. Grp /\ U e. B /\ V e. B ) -> ( U .- V ) e. B )
18	15, 17	syl3an1 1282	. . . 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 2193	. . . . 5 |- ( -g ` T ) = ( -g ` T )
21	7, 16, 20	ghmsub 13324	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) ( -g ` T ) ( F ` V ) ) )
22	21	eqeq1d 2202	. . 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 13319	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
25	24	3ad2ant1 1020	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> T e. Grp )
26	9	3ad2ant1 1020	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F : B --> ( Base ` T ) )
27		simp2 1000	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> U e. B )
28	26, 27	ffvelcdmd 5695	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` U ) e. ( Base ` T ) )
29		simp3 1001	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
30	26, 29	ffvelcdmd 5695	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
31		eqid 2193	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
32	8, 31, 20	grpsubeq0 13161	. . 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 1249	. 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 980   = wceq 1364   e. wcel 2164   {csn 3619   `'ccnv 4659   "cima 4663   Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   0gc0g 12870   Grpcgrp 13075   -gcsg 13077   GrpHom cghm 13313

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-sbg 13080  df-ghm 13314

This theorem is referenced by:  kerf1ghm  13347