Theorem ghmeqker 17687

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 4188	. . . . . 6 |- { .0. } = { ( 0g ` T ) }
4	3	imaeq2i 5464	. . . . 5 |- ( `' F " { .0. } ) = ( `' F " { ( 0g ` T ) } )
5	1, 4	eqtri 2644	. . . 4 |- K = ( `' F " { ( 0g ` T ) } )
6	5	eleq2i 2693	. . 3 |- ( ( U .- V ) e. K <-> ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) )
7		ghmeqker.b	. . . . . . 7 |- B = ( Base ` S )
8		eqid 2622	. . . . . . 7 |- ( Base ` T ) = ( Base ` T )
9	7, 8	ghmf 17664	. . . . . 6 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
10		ffn 6045	. . . . . 6 |- ( F : B --> ( Base ` T ) -> F Fn B )
11	9, 10	syl 17	. . . . 5 |- ( F e. ( S GrpHom T ) -> F Fn B )
12	11	3ad2ant1 1082	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F Fn B )
13		fniniseg 6338	. . . 4 |- ( F Fn B -> ( ( U .- V ) e. ( `' F " { ( 0g ` T ) } ) <-> ( ( U .- V ) e. B /\ ( F ` ( U .- V ) ) = ( 0g ` T ) ) ) )
14	12, 13	syl 17	. . 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 ) ) ) )
15	6, 14	syl5bb 272	. 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 ) ) ) )
16		ghmgrp1 17662	. . . . 5 |- ( F e. ( S GrpHom T ) -> S e. Grp )
17		ghmeqker.m	. . . . . 6 |- .- = ( -g ` S )
18	7, 17	grpsubcl 17495	. . . . 5 |- ( ( S e. Grp /\ U e. B /\ V e. B ) -> ( U .- V ) e. B )
19	16, 18	syl3an1 1359	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( U .- V ) e. B )
20	19	biantrurd 529	. . 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 ) ) ) )
21		eqid 2622	. . . . 5 |- ( -g ` T ) = ( -g ` T )
22	7, 17, 21	ghmsub 17668	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) ( -g ` T ) ( F ` V ) ) )
23	22	eqeq1d 2624	. . 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 ) ) )
24	20, 23	bitr3d 270	. 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 ) ) )
25		ghmgrp2 17663	. . . 4 |- ( F e. ( S GrpHom T ) -> T e. Grp )
26	25	3ad2ant1 1082	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> T e. Grp )
27	9	3ad2ant1 1082	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> F : B --> ( Base ` T ) )
28		simp2 1062	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> U e. B )
29	27, 28	ffvelrnd 6360	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` U ) e. ( Base ` T ) )
30		simp3 1063	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
31	27, 30	ffvelrnd 6360	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
32		eqid 2622	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
33	8, 32, 21	grpsubeq0 17501	. . 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 ) ) )
34	26, 29, 31, 33	syl3anc 1326	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( ( F ` U ) ( -g ` T ) ( F ` V ) ) = ( 0g ` T ) <-> ( F ` U ) = ( F ` V ) ) )
35	15, 24, 34	3bitrrd 295	1 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) = ( F ` V ) <-> ( U .- V ) e. K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   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-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658

This theorem is referenced by:  kerf1hrm  18743  kercvrlsm  37653