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Metamath Proof Explorer

Theorem lmodscaf

Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Hypotheses scaffval.b 
|- B = ( Base ` W )
scaffval.f 
|- F = ( Scalar ` W )
scaffval.k 
|- K = ( Base ` F )
scaffval.a 
|- .xb = ( .sf ` W )
Assertion lmodscaf 
|- ( W e. LMod -> .xb : ( K X. B ) --> B )

Proof

Step Hyp Ref Expression
1 scaffval.b 
 |-  B = ( Base ` W )
2 scaffval.f 
 |-  F = ( Scalar ` W )
3 scaffval.k 
 |-  K = ( Base ` F )
4 scaffval.a 
 |-  .xb = ( .sf ` W )
5 eqid 
 |-  ( .s ` W ) = ( .s ` W )
6 1 2 5 3 lmodvscl 
 |-  ( ( W e. LMod /\ x e. K /\ y e. B ) -> ( x ( .s ` W ) y ) e. B )
7 6 3expb 
 |-  ( ( W e. LMod /\ ( x e. K /\ y e. B ) ) -> ( x ( .s ` W ) y ) e. B )
8 7 ralrimivva 
 |-  ( W e. LMod -> A. x e. K A. y e. B ( x ( .s ` W ) y ) e. B )
9 1 2 3 4 5 scaffval 
 |-  .xb = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) )
10 9 fmpo 
 |-  ( A. x e. K A. y e. B ( x ( .s ` W ) y ) e. B <-> .xb : ( K X. B ) --> B )
11 8 10 sylib 
 |-  ( W e. LMod -> .xb : ( K X. B ) --> B )