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Metamath Proof Explorer

Theorem phlstr

Description: A constructed pre-Hilbert space is a structure. Starting from lmodstr (which has 4 members), we chain strleun once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis phlfn.h 
|- H = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
Assertion phlstr 
|- H Struct <. 1 , 8 >.

Proof

Step Hyp Ref Expression
1 phlfn.h 
 |-  H = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
2 df-pr 
 |-  { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } = ( { <. ( .s ` ndx ) , .x. >. } u. { <. ( .i ` ndx ) , ., >. } )
3 2 uneq2i 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. ( { <. ( .s ` ndx ) , .x. >. } u. { <. ( .i ` ndx ) , ., >. } ) )
4 unass 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. } ) u. { <. ( .i ` ndx ) , ., >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. ( { <. ( .s ` ndx ) , .x. >. } u. { <. ( .i ` ndx ) , ., >. } ) )
5 3 1 4 3eqtr4i 
 |-  H = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. } ) u. { <. ( .i ` ndx ) , ., >. } )
6 eqid 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. } )
7 6 lmodstr 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. } ) Struct <. 1 , 6 >.
8 8nn 
 |-  8 e. NN
9 ipndx 
 |-  ( .i ` ndx ) = 8
10 8 9 strle1 
 |-  { <. ( .i ` ndx ) , ., >. } Struct <. 8 , 8 >.
11 6lt8 
 |-  6 < 8
12 7 10 11 strleun 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. } ) u. { <. ( .i ` ndx ) , ., >. } ) Struct <. 1 , 8 >.
13 5 12 eqbrtri 
 |-  H Struct <. 1 , 8 >.