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Metamath Proof Explorer

Theorem clmvsass

Description: Associative law for scalar product. Analogue of lmodvsass . (Contributed by Mario Carneiro, 16-Oct-2015)

Ref Expression
Hypotheses clmvscl.v 
|- V = ( Base ` W )
clmvscl.f 
|- F = ( Scalar ` W )
clmvscl.s 
|- .x. = ( .s ` W )
clmvscl.k 
|- K = ( Base ` F )
Assertion clmvsass 
|- ( ( W e. CMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q x. R ) .x. X ) = ( Q .x. ( R .x. X ) ) )

Proof

Step Hyp Ref Expression
1 clmvscl.v 
 |-  V = ( Base ` W )
2 clmvscl.f 
 |-  F = ( Scalar ` W )
3 clmvscl.s 
 |-  .x. = ( .s ` W )
4 clmvscl.k 
 |-  K = ( Base ` F )
5 2 clmmul 
 |-  ( W e. CMod -> x. = ( .r ` F ) )
6 5 adantr 
 |-  ( ( W e. CMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> x. = ( .r ` F ) )
7 6 oveqd 
 |-  ( ( W e. CMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( Q x. R ) = ( Q ( .r ` F ) R ) )
8 7 oveq1d 
 |-  ( ( W e. CMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q x. R ) .x. X ) = ( ( Q ( .r ` F ) R ) .x. X ) )
9 clmlmod 
 |-  ( W e. CMod -> W e. LMod )
10 eqid 
 |-  ( .r ` F ) = ( .r ` F )
11 1 2 3 4 10 lmodvsass 
 |-  ( ( W e. LMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q ( .r ` F ) R ) .x. X ) = ( Q .x. ( R .x. X ) ) )
12 9 11 sylan 
 |-  ( ( W e. CMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q ( .r ` F ) R ) .x. X ) = ( Q .x. ( R .x. X ) ) )
13 8 12 eqtrd 
 |-  ( ( W e. CMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q x. R ) .x. X ) = ( Q .x. ( R .x. X ) ) )