Theorem ghmeqker 13477

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 3635	. . . . . 6 |- { .0. } = { ( 0g ` T ) }
4	3	imaeq2i 5008	. . . . 5 |- ( `' F " { .0. } ) = ( `' F " { ( 0g ` T ) } )
5	1, 4	eqtri 2217	. . . 4 |- K = ( `' F " { ( 0g ` T ) } )
6	5	eleq2i 2263	. . 3 |- ( ( U .- V ) e. K <-> ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) )
7		ghmeqker.b	. . . . . . 7 |- B = ( Base ` S )
8		eqid 2196	. . . . . . 7 |- ( Base ` T ) = ( Base ` T )
9	7, 8	ghmf 13453	. . . . . 6 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
10	9	ffnd 5411	. . . . 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 5685	. . . 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 13451	. . . . 5 |- ( F e. ( S GrpHom T ) -> S e. Grp )
16		ghmeqker.m	. . . . . 6 |- .- = ( -g ` S )
17	7, 16	grpsubcl 13282	. . . . 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 2196	. . . . 5 |- ( -g ` T ) = ( -g ` T )
21	7, 16, 20	ghmsub 13457	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) ( -g ` T ) ( F ` V ) ) )
22	21	eqeq1d 2205	. . 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 13452	. . . 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 5701	. . 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 5701	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
31		eqid 2196	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
32	8, 31, 20	grpsubeq0 13288	. . 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 2167   {csn 3623   `'ccnv 4663   "cima 4667   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   Basecbs 12703   0gc0g 12958   Grpcgrp 13202   -gcsg 13204   GrpHom cghm 13446

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-ghm 13447

This theorem is referenced by:  kerf1ghm  13480