Metamath Proof Explorer

Theorem ipsbase

Description: The base set 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 ipsbase 
|- ( B e. V -> B = ( Base ` 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 baseid 
 |-  Base = Slot ( Base ` ndx )
4 snsstp1 
 |-  { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. }
5 ssun1 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
6 5 1 sseqtrri 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } C_ A
7 4 6 sstri 
 |-  { <. ( Base ` ndx ) , B >. } C_ A
8 2 3 7 strfv 
 |-  ( B e. V -> B = ( Base ` A ) )