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Metamath Proof Explorer

Theorem slmdvscl

Description: Closure of scalar product for a semiring left module. ( hvmulcl analog.) (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step Hyp Ref Expression
1 slmdvscl.v 
 |-  V = ( Base ` W )
2 slmdvscl.f 
 |-  F = ( Scalar ` W )
3 slmdvscl.s 
 |-  .x. = ( .s ` W )
4 slmdvscl.k 
 |-  K = ( Base ` F )
5 biid 
 |-  ( W e. SLMod <-> W e. SLMod )
6 pm4.24 
 |-  ( R e. K <-> ( R e. K /\ R e. K ) )
7 pm4.24 
 |-  ( X e. V <-> ( X e. V /\ X e. V ) )
8 eqid 
 |-  ( +g ` W ) = ( +g ` W )
9 eqid 
 |-  ( 0g ` W ) = ( 0g ` W )
10 eqid 
 |-  ( +g ` F ) = ( +g ` F )
11 eqid 
 |-  ( .r ` F ) = ( .r ` F )
12 eqid 
 |-  ( 1r ` F ) = ( 1r ` F )
13 eqid 
 |-  ( 0g ` F ) = ( 0g ` F )
14 1 8 3 9 2 4 10 11 12 13 slmdlema 
 |-  ( ( W e. SLMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( R ( +g ` F ) R ) .x. X ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) ) /\ ( ( ( R ( .r ` F ) R ) .x. X ) = ( R .x. ( R .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( ( 0g ` F ) .x. X ) = ( 0g ` W ) ) ) )
15 14 simpld 
 |-  ( ( W e. SLMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( R ( +g ` F ) R ) .x. X ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) ) )
16 15 simp1d 
 |-  ( ( W e. SLMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( R .x. X ) e. V )
17 5 6 7 16 syl3anb 
 |-  ( ( W e. SLMod /\ R e. K /\ X e. V ) -> ( R .x. X ) e. V )