Metamath Proof Explorer

Theorem ipsvsca

Description: The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

Ref Expression
Hypothesis ipspart.a 
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
Assertion ipsvsca 
|- ( .x. e. V -> .x. = ( .s ` A ) )

Proof

Step Hyp Ref Expression
1 ipspart.a 
 |-  A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
2 1 ipsstr 
 |-  A Struct <. 1 , 8 >.
3 vscaid 
 |-  .s = Slot ( .s ` ndx )
4 snsstp2 
 |-  { <. ( .s ` ndx ) , .x. >. } C_ { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. }
5 ssun2 
 |-  { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
6 5 1 sseqtrri 
 |-  { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } C_ A
7 4 6 sstri 
 |-  { <. ( .s ` ndx ) , .x. >. } C_ A
8 2 3 7 strfv 
 |-  ( .x. e. V -> .x. = ( .s ` A ) )