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Metamath Proof Explorer

Theorem slmdvsdir

Description: Distributive law for scalar product. ( ax-hvdistr1 analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 22-Sep-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step Hyp Ref Expression
1 slmdvsdir.v 
 |-  V = ( Base ` W )
2 slmdvsdir.a 
 |-  .+ = ( +g ` W )
3 slmdvsdir.f 
 |-  F = ( Scalar ` W )
4 slmdvsdir.s 
 |-  .x. = ( .s ` W )
5 slmdvsdir.k 
 |-  K = ( Base ` F )
6 slmdvsdir.p 
 |-  .+^ = ( +g ` F )
7 eqid 
 |-  ( 0g ` W ) = ( 0g ` W )
8 eqid 
 |-  ( .r ` F ) = ( .r ` F )
9 eqid 
 |-  ( 1r ` F ) = ( 1r ` F )
10 eqid 
 |-  ( 0g ` F ) = ( 0g ` F )
11 1 2 4 7 3 5 6 8 9 10 slmdlema 
 |-  ( ( 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 ) ) ) )
12 11 simpld 
 |-  ( ( 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 ) ) ) )
13 12 simp3d 
 |-  ( ( 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 ) ) )
14 13 3expa 
 |-  ( ( ( 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 ) ) )
15 14 anabsan2 
 |-  ( ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) ) /\ X e. V ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )
16 15 exp42 
 |-  ( W e. SLMod -> ( Q e. K -> ( R e. K -> ( X e. V -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) ) ) ) )
17 16 3imp2 
 |-  ( ( W e. SLMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )