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Metamath Proof Explorer

Theorem lmodvsdir

Description: Distributive law for scalar product (right-distributivity). ( ax-hvdistr1 analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 22-Sep-2015)

Ref Expression
Hypotheses 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 lmodvsdir 
|- ( ( W e. LMod /\ ( 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 lmodvsdir.v 
 |-  V = ( Base ` W )
2 lmodvsdir.a 
 |-  .+ = ( +g ` W )
3 lmodvsdir.f 
 |-  F = ( Scalar ` W )
4 lmodvsdir.s 
 |-  .x. = ( .s ` W )
5 lmodvsdir.k 
 |-  K = ( Base ` F )
6 lmodvsdir.p 
 |-  .+^ = ( +g ` F )
7 eqid 
 |-  ( .r ` F ) = ( .r ` F )
8 eqid 
 |-  ( 1r ` F ) = ( 1r ` F )
9 1 2 4 3 5 6 7 8 lmodlema 
 |-  ( ( 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 
 |-  ( ( 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 
 |-  ( ( 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 
 |-  ( ( ( 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 
 |-  ( ( ( 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 
 |-  ( 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 
 |-  ( ( W e. LMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .+^ R ) .x. X ) = ( ( Q .x. X ) .+ ( R .x. X ) ) )