Theorem lmodsubdir 14619

Description: Scalar multiplication distributive law for subtraction. (Contributed by NM, 2-Jul-2014.)

Hypotheses

Ref	Expression
lmodsubdir.v	|- V = ( Base ` W )
lmodsubdir.t	|- .x. = ( .s ` W )
lmodsubdir.f	|- F = (Scalar ` W )
lmodsubdir.k	|- K = ( Base ` F )
lmodsubdir.m	|- .- = ( -g ` W )
lmodsubdir.s	|- S = ( -g ` F )
lmodsubdir.w	|- ( ph -> W e. LMod )
lmodsubdir.a	|- ( ph -> A e. K )
lmodsubdir.b	|- ( ph -> B e. K )
lmodsubdir.x	|- ( ph -> X e. V )

Assertion

Ref	Expression
lmodsubdir	|- ( ph -> ( ( A S B ) .x. X ) = ( ( A .x. X ) .- ( B .x. X ) ) )

Proof of Theorem lmodsubdir

Step	Hyp	Ref	Expression
1		lmodsubdir.w	. . . 4 |- ( ph -> W e. LMod )
2		lmodsubdir.a	. . . 4 |- ( ph -> A e. K )
3		lmodsubdir.f	. . . . . . . 8 |- F = (Scalar ` W )
4	3	lmodring 14569	. . . . . . 7 |- ( W e. LMod -> F e. Ring )
5	1, 4	syl 14	. . . . . 6 |- ( ph -> F e. Ring )
6		ringgrp 14244	. . . . . 6 |- ( F e. Ring -> F e. Grp )
7	5, 6	syl 14	. . . . 5 |- ( ph -> F e. Grp )
8		lmodsubdir.b	. . . . 5 |- ( ph -> B e. K )
9		lmodsubdir.k	. . . . . 6 |- K = ( Base ` F )
10		eqid 2234	. . . . . 6 |- ( invg ` F ) = ( invg ` F )
11	9, 10	grpinvcl 13803	. . . . 5 |- ( ( F e. Grp /\ B e. K ) -> ( ( invg ` F ) ` B ) e. K )
12	7, 8, 11	syl2anc 411	. . . 4 |- ( ph -> ( ( invg ` F ) ` B ) e. K )
13		lmodsubdir.x	. . . 4 |- ( ph -> X e. V )
14		lmodsubdir.v	. . . . 5 |- V = ( Base ` W )
15		eqid 2234	. . . . 5 |- ( +g ` W ) = ( +g ` W )
16		lmodsubdir.t	. . . . 5 |- .x. = ( .s ` W )
17		eqid 2234	. . . . 5 |- ( +g ` F ) = ( +g ` F )
18	14, 15, 3, 16, 9, 17	lmodvsdir 14586	. . . 4 |- ( ( W e. LMod /\ ( A e. K /\ ( ( invg ` F ) ` B ) e. K /\ X e. V ) ) -> ( ( A ( +g ` F ) ( ( invg ` F ) ` B ) ) .x. X ) = ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` B ) .x. X ) ) )
19	1, 2, 12, 13, 18	syl13anc 1276	. . 3 |- ( ph -> ( ( A ( +g ` F ) ( ( invg ` F ) ` B ) ) .x. X ) = ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` B ) .x. X ) ) )
20		eqid 2234	. . . . . . 7 |- ( .r ` F ) = ( .r ` F )
21		eqid 2234	. . . . . . 7 |- ( 1r ` F ) = ( 1r ` F )
22	9, 20, 21, 10, 5, 8	ringnegl 14294	. . . . . 6 |- ( ph -> ( ( ( invg ` F ) ` ( 1r ` F ) ) ( .r ` F ) B ) = ( ( invg ` F ) ` B ) )
23	22	oveq1d 6073	. . . . 5 |- ( ph -> ( ( ( ( invg ` F ) ` ( 1r ` F ) ) ( .r ` F ) B ) .x. X ) = ( ( ( invg ` F ) ` B ) .x. X ) )
24	9, 21	ringidcl 14263	. . . . . . . 8 |- ( F e. Ring -> ( 1r ` F ) e. K )
25	5, 24	syl 14	. . . . . . 7 |- ( ph -> ( 1r ` F ) e. K )
26	9, 10	grpinvcl 13803	. . . . . . 7 |- ( ( F e. Grp /\ ( 1r ` F ) e. K ) -> ( ( invg ` F ) ` ( 1r ` F ) ) e. K )
27	7, 25, 26	syl2anc 411	. . . . . 6 |- ( ph -> ( ( invg ` F ) ` ( 1r ` F ) ) e. K )
28	14, 3, 16, 9, 20	lmodvsass 14587	. . . . . 6 |- ( ( W e. LMod /\ ( ( ( invg ` F ) ` ( 1r ` F ) ) e. K /\ B e. K /\ X e. V ) ) -> ( ( ( ( invg ` F ) ` ( 1r ` F ) ) ( .r ` F ) B ) .x. X ) = ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) )
29	1, 27, 8, 13, 28	syl13anc 1276	. . . . 5 |- ( ph -> ( ( ( ( invg ` F ) ` ( 1r ` F ) ) ( .r ` F ) B ) .x. X ) = ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) )
30	23, 29	eqtr3d 2269	. . . 4 |- ( ph -> ( ( ( invg ` F ) ` B ) .x. X ) = ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) )
31	30	oveq2d 6074	. . 3 |- ( ph -> ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` B ) .x. X ) ) = ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) ) )
32	19, 31	eqtrd 2267	. 2 |- ( ph -> ( ( A ( +g ` F ) ( ( invg ` F ) ` B ) ) .x. X ) = ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) ) )
33		lmodsubdir.s	. . . . 5 |- S = ( -g ` F )
34	9, 17, 10, 33	grpsubval 13801	. . . 4 |- ( ( A e. K /\ B e. K ) -> ( A S B ) = ( A ( +g ` F ) ( ( invg ` F ) ` B ) ) )
35	2, 8, 34	syl2anc 411	. . 3 |- ( ph -> ( A S B ) = ( A ( +g ` F ) ( ( invg ` F ) ` B ) ) )
36	35	oveq1d 6073	. 2 |- ( ph -> ( ( A S B ) .x. X ) = ( ( A ( +g ` F ) ( ( invg ` F ) ` B ) ) .x. X ) )
37	14, 3, 16, 9	lmodvscl 14579	. . . 4 |- ( ( W e. LMod /\ A e. K /\ X e. V ) -> ( A .x. X ) e. V )
38	1, 2, 13, 37	syl3anc 1274	. . 3 |- ( ph -> ( A .x. X ) e. V )
39	14, 3, 16, 9	lmodvscl 14579	. . . 4 |- ( ( W e. LMod /\ B e. K /\ X e. V ) -> ( B .x. X ) e. V )
40	1, 8, 13, 39	syl3anc 1274	. . 3 |- ( ph -> ( B .x. X ) e. V )
41		lmodsubdir.m	. . . 4 |- .- = ( -g ` W )
42	14, 15, 41, 3, 16, 10, 21	lmodvsubval2 14616	. . 3 |- ( ( W e. LMod /\ ( A .x. X ) e. V /\ ( B .x. X ) e. V ) -> ( ( A .x. X ) .- ( B .x. X ) ) = ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) ) )
43	1, 38, 40, 42	syl3anc 1274	. 2 |- ( ph -> ( ( A .x. X ) .- ( B .x. X ) ) = ( ( A .x. X ) ( +g ` W ) ( ( ( invg ` F ) ` ( 1r ` F ) ) .x. ( B .x. X ) ) ) )
44	32, 36, 43	3eqtr4d 2277	1 |- ( ph -> ( ( A S B ) .x. X ) = ( ( A .x. X ) .- ( B .x. X ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375  Scalarcsca 13377   .scvsca 13378   Grpcgrp 13755   invgcminusg 13756   -gcsg 13757   1rcur 14202   Ringcrg 14239   LModclmod 14561

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-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

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-nel 2510  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-nul 3513  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-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-mgp 14160  df-ur 14203  df-ring 14241  df-lmod 14563

This theorem is referenced by: (None)