Theorem lmodvsdir 14460

Description: Distributive law for scalar product (right-distributivity). (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
lmodvsdir.v	|- V = ( Base ` W )
lmodvsdir.a	|- .+ = ( +g ` W )
lmodvsdir.f	|- F = (Scalar ` W )
lmodvsdir.s	|- .x. = ( .s ` W )
lmodvsdir.k	|- K = ( Base ` F )
lmodvsdir.p	|- .+^ = ( +g ` F )

Assertion

Ref	Expression
lmodvsdir	|- ( ( W e. LMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )

Proof of Theorem lmodvsdir

Step	Hyp	Ref	Expression
1		lmodvsdir.v	. . . . . . . 8 |- V = ( Base ` W )
2		lmodvsdir.a	. . . . . . . 8 |- .+ = ( +g ` W )
3		lmodvsdir.s	. . . . . . . 8 |- .x. = ( .s ` W )
4		lmodvsdir.f	. . . . . . . 8 |- F = (Scalar ` W )
5		lmodvsdir.k	. . . . . . . 8 |- K = ( Base ` F )
6		lmodvsdir.p	. . . . . . . 8 |- .+^ = ( +g ` F )
7		eqid 2232	. . . . . . . 8 |- ( .r ` F ) = ( .r ` F )
8		eqid 2232	. . . . . . . 8 |- ( 1r ` F ) = ( 1r ` F )
9	1, 2, 3, 4, 5, 6, 7, 8	lmodlema 14440	. . . . . . 7 |- ( ( W e. LMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( R .x. X ) e. V /\ ( R .x. ( X .+ X ) ) = ( ( R .x. X ) .+ ( R .x. X ) ) /\ ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) ) /\ ( ( ( Q ( .r ` F ) R ) .x. X ) = ( Q .x. ( R .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X ) ) )
10	9	simpld 112	. . . . . 6 |- ( ( W e. LMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( R .x. X ) e. V /\ ( R .x. ( X .+ X ) ) = ( ( R .x. X ) .+ ( R .x. X ) ) /\ ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) ) )
11	10	simp3d 1038	. . . . 5 |- ( ( W e. LMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
12	11	3expa 1230	. . . 4 |- ( ( ( W e. LMod /\ ( Q e. K /\ R e. K ) ) /\ ( X e. V /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
13	12	anabsan2 586	. . 3 |- ( ( ( W e. LMod /\ ( Q e. K /\ R e. K ) ) /\ X e. V ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
14	13	exp42 371	. 2 |- ( W e. LMod -> ( Q e. K -> ( R e. K -> ( X e. V -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) ) ) ) )
15	14	3imp2 1249	1 |- ( ( W e. LMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291  Scalarcsca 13293   .scvsca 13294   1rcur 14103   LModclmod 14435

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-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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-lmod 14437

This theorem is referenced by:  lmod0vs  14469  lmodvsmmulgdi  14471  lmodvneg1  14478  lmodcom  14481  lmodsubdir  14493  islss3  14527  lss1d  14531