Theorem ghmsub 13321

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 13315	. . . . . 6 |- ( F e. ( S GrpHom T ) -> S e. Grp )
2	1	3ad2ant1 1020	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> S e. Grp )
3		simp3 1001	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> V e. B )
4		ghmsub.b	. . . . . 6 |- B = ( Base ` S )
5		eqid 2193	. . . . . 6 |- ( invg ` S ) = ( invg ` S )
6	4, 5	grpinvcl 13120	. . . . 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 2193	. . . . 5 |- ( +g ` S ) = ( +g ` S )
9		eqid 2193	. . . . 5 |- ( +g ` T ) = ( +g ` T )
10	4, 8, 9	ghmlin 13318	. . . 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 1294	. . 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 2193	. . . . . 6 |- ( invg ` T ) = ( invg ` T )
13	4, 5, 12	ghminv 13320	. . . . 5 |- ( ( F e. ( S GrpHom T ) /\ V e. B ) -> ( F ` ( ( invg ` S ) ` V ) ) = ( ( invg ` T ) ` ( F ` V ) ) )
14	13	3adant2 1018	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( ( invg ` S ) ` V ) ) = ( ( invg ` T ) ` ( F ` V ) ) )
15	14	oveq2d 5934	. . 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 2226	. 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 13118	. . . 4 |- ( ( U e. B /\ V e. B ) -> ( U .- V ) = ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) )
19	18	fveq2d 5558	. . 3 |- ( ( U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) )
20	19	3adant1 1017	. 2 |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( F ` ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) ) )
21		eqid 2193	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
22	4, 21	ghmf 13317	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
23		ffvelcdm 5691	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ U e. B ) -> ( F ` U ) e. ( Base ` T ) )
24		ffvelcdm 5691	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ V e. B ) -> ( F ` V ) e. ( Base ` T ) )
25	23, 24	anim12dan 600	. . . . 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 1201	. . 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 13118	. . 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 2236	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 980   = wceq 1364   e. wcel 2164   -->wf 5250   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   invgcminusg 13073   -gcsg 13074   GrpHom cghm 13310

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-ghm 13311

This theorem is referenced by:  ghmnsgima  13338  ghmnsgpreima  13339  ghmeqker  13341  ghmf1  13343