Theorem ghmsub 13968

Description: Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghmsub.b	|- B = ( Base ` S )
ghmsub.m	|- .- = ( -g ` S )
ghmsub.n	|- N = ( -g ` T )

Assertion

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

Proof of Theorem ghmsub

Step	Hyp	Ref	Expression
1		ghmgrp1 13962	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	3ad2ant1 1045	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> S e. Grp )
3		simp3 1026	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
4		ghmsub.b	. . . . . 6 |- B = ( Base ` S )
5		eqid 2232	. . . . . 6 |- ( invg ` S ) = ( invg ` S )
6	4, 5	grpinvcl 13761	. . . . 5 |- ( ( S e. Grp /\ V e. B ) -> ( ( invg ` S ) ` V ) e. B )
7	2, 3, 6	syl2anc 411	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( invg ` S ) ` V ) e. B )
8		eqid 2232	. . . . 5 |- ( +g ` S ) = ( +g ` S )
9		eqid 2232	. . . . 5 |- ( +g ` T ) = ( +g ` T )
10	4, 8, 9	ghmlin 13965	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ ( ( invg ` S ) ` V ) e. B ) -> ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) = ( ( F ` U ) ( +g ` T ) ( F ` ( ( invg ` S ) ` V ) ) ) )
11	7, 10	syld3an3 1319	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) = ( ( F ` U ) ( +g ` T ) ( F ` ( ( invg ` S ) ` V ) ) ) )
12		eqid 2232	. . . . . 6 |- ( invg ` T ) = ( invg ` T )
13	4, 5, 12	ghminv 13967	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. B ) -> ( F ` ( ( invg ` S ) ` V ) ) = ( ( invg ` T ) ` ( F ` V ) ) )
14	13	3adant2 1043	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( ( invg ` S ) ` V ) ) = ( ( invg ` T ) ` ( F ` V ) ) )
15	14	oveq2d 6066	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) ( +g ` T ) ( F ` ( ( invg ` S ) ` V ) ) ) = ( ( F ` U ) ( +g ` T ) ( ( invg ` T ) ` ( F ` V ) ) ) )
16	11, 15	eqtrd 2265	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) = ( ( F ` U ) ( +g ` T ) ( ( invg ` T ) ` ( F ` V ) ) ) )
17		ghmsub.m	. . . . 5 |- .- = ( -g ` S )
18	4, 8, 5, 17	grpsubval 13759	. . . 4 |- ( ( U e. B /\ V e. B ) -> ( U .- V ) = ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) )
19	18	fveq2d 5674	. . 3 |- ( ( U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) )
20	19	3adant1 1042	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) )
21		eqid 2232	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
22	4, 21	ghmf 13964	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
23		ffvelcdm 5810	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ U e. B ) -> ( F ` U ) e. ( Base ` T ) )
24		ffvelcdm 5810	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
25	23, 24	anim12dan 604	. . . . 5 |- ( ( F : B --> ( Base ` T ) /\ ( U e. B /\ V e. B ) ) -> ( ( F ` U ) e. ( Base ` T ) /\ ( F ` V ) e. ( Base ` T ) ) )
26	22, 25	sylan 283	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ ( U e. B /\ V e. B ) ) -> ( ( F ` U ) e. ( Base ` T ) /\ ( F ` V ) e. ( Base ` T ) ) )
27	26	3impb 1226	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) e. ( Base ` T ) /\ ( F ` V ) e. ( Base ` T ) ) )
28		ghmsub.n	. . . 4 |- N = ( -g ` T )
29	21, 9, 12, 28	grpsubval 13759	. . 3 |- ( ( ( F ` U ) e. ( Base ` T ) /\ ( F ` V ) e. ( Base ` T ) ) -> ( ( F ` U ) N ( F ` V ) ) = ( ( F ` U ) ( +g ` T ) ( ( invg ` T ) ` ( F ` V ) ) ) )
30	27, 29	syl 14	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) N ( F ` V ) ) = ( ( F ` U ) ( +g ` T ) ( ( invg ` T ) ` ( F ` V ) ) ) )
31	16, 20, 30	3eqtr4d 2275	1 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) N ( F ` V ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   -->wf 5348   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   Grpcgrp 13713   invgcminusg 13714   -gcsg 13715   GrpHom cghm 13957

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-sbg 13718  df-ghm 13958

This theorem is referenced by:  ghmnsgima  13985  ghmnsgpreima  13986  ghmeqker  13988  ghmf1  13990