Theorem ghmsub 13457

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 13451	. . . . . 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 2196	. . . . . 6 |- ( invg ` S ) = ( invg ` S )
6	4, 5	grpinvcl 13250	. . . . 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 2196	. . . . 5 |- ( +g ` S ) = ( +g ` S )
9		eqid 2196	. . . . 5 |- ( +g ` T ) = ( +g ` T )
10	4, 8, 9	ghmlin 13454	. . . 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 2196	. . . . . 6 |- ( invg ` T ) = ( invg ` T )
13	4, 5, 12	ghminv 13456	. . . . 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 5941	. . 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 2229	. 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 13248	. . . 4 |- ( ( U e. B /\ V e. B ) -> ( U .- V ) = ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) )
19	18	fveq2d 5565	. . 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 2196	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
22	4, 21	ghmf 13453	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
23		ffvelcdm 5698	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ U e. B ) -> ( F ` U ) e. ( Base ` T ) )
24		ffvelcdm 5698	. . . . . 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 13248	. . 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 2239	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 2167   -->wf 5255   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   Grpcgrp 13202   invgcminusg 13203   -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:  ghmnsgima  13474  ghmnsgpreima  13475  ghmeqker  13477  ghmf1  13479