Theorem ghmsub 14004

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 13998	. . . . . 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 2234	. . . . . 6 |- ( invg ` S ) = ( invg ` S )
6	4, 5	grpinvcl 13803	. . . . 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 2234	. . . . 5 |- ( +g ` S ) = ( +g ` S )
9		eqid 2234	. . . . 5 |- ( +g ` T ) = ( +g ` T )
10	4, 8, 9	ghmlin 14001	. . . 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 2234	. . . . . 6 |- ( invg ` T ) = ( invg ` T )
13	4, 5, 12	ghminv 14003	. . . . 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 6074	. . 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 2267	. 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 13801	. . . 4 |- ( ( U e. B /\ V e. B ) -> ( U .- V ) = ( U ( +g ` S ) ( ( invg ` S ) ` V ) ) )
19	18	fveq2d 5679	. . 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 2234	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
22	4, 21	ghmf 14000	. . . . 5 |- ( F e. ( S GrpHom T ) -> F : B --> ( Base ` T ) )
23		ffvelcdm 5815	. . . . . 6 |- ( ( F : B --> ( Base ` T ) /\ U e. B ) -> ( F ` U ) e. ( Base ` T ) )
24		ffvelcdm 5815	. . . . . 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 13801	. . 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 2277	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 2205   -->wf 5353   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   Grpcgrp 13755   invgcminusg 13756   -gcsg 13757   GrpHom cghm 13993

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-ghm 13994

This theorem is referenced by:  ghmnsgima  14021  ghmnsgpreima  14022  ghmeqker  14024  ghmf1  14026