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Theorem slmdvsdir 29769
Description: Distributive law for scalar product. (ax-hvdistr1 27865 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvsdir.v |- V = ( Base ` W )
slmdvsdir.a |- .+ = ( +g ` W )
slmdvsdir.f |- F = (Scalar ` W )
slmdvsdir.s |- .x. = ( .s ` W )
slmdvsdir.k |- K = ( Base ` F )
slmdvsdir.p |- .+^ = ( +g ` F )
Assertion
Ref Expression
slmdvsdir |- ( ( W e. SLMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
Proof of Theorem slmdvsdir
StepHypRef Expression
1 slmdvsdir.v . . . . . . . 8 |- V = ( Base ` W )
2 slmdvsdir.a . . . . . . . 8 |- .+ = ( +g ` W )
3 slmdvsdir.s . . . . . . . 8 |- .x. = ( .s ` W )
4 eqid 2622 . . . . . . . 8 |- ( 0g ` W ) = ( 0g ` W )
5 slmdvsdir.f . . . . . . . 8 |- F = (Scalar ` W )
6 slmdvsdir.k . . . . . . . 8 |- K = ( Base ` F )
7 slmdvsdir.p . . . . . . . 8 |- .+^ = ( +g ` F )
8 eqid 2622 . . . . . . . 8 |- ( .r ` F ) = ( .r ` F )
9 eqid 2622 . . . . . . . 8 |- ( 1r ` F ) = ( 1r ` F )
10 eqid 2622 . . . . . . . 8 |- ( 0g ` F ) = ( 0g ` F )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10slmdlema 29756 . . . . . . 7 |- ( ( W e. SLMod /\ ( 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 /\ ( ( 0g ` F ) .x. X ) = ( 0g ` W ) ) ) )
1211simpld 475 . . . . . 6 |- ( ( W e. SLMod /\ ( 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 ) ) ) )
1312simp3d 1075 . . . . 5 |- ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
14133expa 1265 . . . 4 |- ( ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) ) /\ ( X e. V /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
1514anabsan2 863 . . 3 |- ( ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) ) /\ X e. V ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
1615exp42 639 . 2 |- ( W e. SLMod -> ( Q e. K -> ( R e. K -> ( X e. V -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) ) ) ) )
17163imp2 1282 1 |- ( ( W e. SLMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   1rcur 18501  SLModcslmd 29753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-slmd 29754
This theorem is referenced by:  gsumvsca2  29783
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