Theorem ghmsub 14054

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 14048	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	3ad2ant1 1049	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> S e. Grp )
3		simp3 1030	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
4		ghmsub.b	. . . . . 6 |- B = ( Base ` S )
5		eqid 2238	. . . . . 6 |- ( invg ` S ) = ( invg ` S )
6	4, 5	grpinvcl 13853	. . . . 5 |- ( ( S e. Grp /\ V e. B ) -> ( ( invg ` S ) ` V ) e. B )
7	2, 3, 6	syl2anc 415	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( invg ` S ) ` V ) e. B )
8		eqid 2238	. . . . 5 |- ( +g ` S ) = ( +g ` S )
9		eqid 2238	. . . . 5 |- ( +g ` T ) = ( +g ` T )
10	4, 8, 9	ghmlin 14051	. . . 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 1323	. . 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 2238	. . . . . 6 |- ( invg ` T ) = ( invg ` T )
13	4, 5, 12	ghminv 14053	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. B ) -> ( F ` ( ( invg ` S ) ` V ) ) = ( ( invg ` T ) ` ( F ` V ) ) )
14	13	3adant2 1047	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( ( invg ` S ) ` V ) ) = ( ( invg ` T ) ` ( F ` V ) ) )
15	14	oveq2d 6101	. . 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 2271	. 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 13851	. . . 4 |- ( ( U e. B /\ V e. B ) -> ( U .- V ) = ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) )
19	18	fveq2d 5699	. . 3 |- ( ( U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) )
20	19	3adant1 1046	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) )
21		eqid 2238	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
22	4, 21	ghmf 14050	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
23		ffvelcdm 5841	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ U e. B ) -> ( F ` U ) e. ( Base ` T ) )
24		ffvelcdm 5841	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
25	23, 24	anim12dan 608	. . . . 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 1230	. . 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 13851	. . 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 2281	1 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) N ( F ` V ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   -->wf 5373   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   Grpcgrp 13805   invgcminusg 13806   -gcsg 13807   GrpHom cghm 14043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-sbg 13810  df-ghm 14044

This theorem is used by:  ghmnsgima  14071  ghmnsgpreima  14072  ghmeqker  14074  ghmf1  14076