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Metamath Proof Explorer

Theorem imasip

Description: The inner product of an image structure. (Contributed by Thierry Arnoux, 16-Jun-2019)

Ref Expression
Hypotheses imasbas.u 
|- ( ph -> U = ( F "s R ) )
imasbas.v 
|- ( ph -> V = ( Base ` R ) )
imasbas.f 
|- ( ph -> F : V -onto-> B )
imasbas.r 
|- ( ph -> R e. Z )
imasip.i 
|- ., = ( .i ` R )
imasip.w 
|- I = ( .i ` U )
Assertion imasip 
|- ( ph -> I = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )

Proof

Step Hyp Ref Expression
1 imasbas.u 
 |-  ( ph -> U = ( F "s R ) )
2 imasbas.v 
 |-  ( ph -> V = ( Base ` R ) )
3 imasbas.f 
 |-  ( ph -> F : V -onto-> B )
4 imasbas.r 
 |-  ( ph -> R e. Z )
5 imasip.i 
 |-  ., = ( .i ` R )
6 imasip.w 
 |-  I = ( .i ` U )
7 eqid 
 |-  ( +g ` R ) = ( +g ` R )
8 eqid 
 |-  ( .r ` R ) = ( .r ` R )
9 eqid 
 |-  ( Scalar ` R ) = ( Scalar ` R )
10 eqid 
 |-  ( Base ` ( Scalar ` R ) ) = ( Base ` ( Scalar ` R ) )
11 eqid 
 |-  ( .s ` R ) = ( .s ` R )
12 eqid 
 |-  ( TopOpen ` R ) = ( TopOpen ` R )
13 eqid 
 |-  ( dist ` R ) = ( dist ` R )
14 eqid 
 |-  ( le ` R ) = ( le ` R )
15 eqid 
 |-  ( +g ` U ) = ( +g ` U )
16 1 2 3 4 7 15 imasplusg 
 |-  ( ph -> ( +g ` U ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
17 eqid 
 |-  ( .r ` U ) = ( .r ` U )
18 1 2 3 4 8 17 imasmulr 
 |-  ( ph -> ( .r ` U ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
19 eqid 
 |-  ( .s ` U ) = ( .s ` U )
20 1 2 3 4 9 10 11 19 imasvsca 
 |-  ( ph -> ( .s ` U ) = U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) q ) ) ) )
21 eqidd 
 |-  ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
22 eqidd 
 |-  ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( TopOpen ` R ) qTop F ) )
23 eqid 
 |-  ( dist ` U ) = ( dist ` U )
24 1 2 3 4 13 23 imasds 
 |-  ( ph -> ( dist ` U ) = ( x e. B , y e. B |-> inf ( U_ u e. NN ran ( z e. { w e. ( ( V X. V ) ^m ( 1 ... u ) ) | ( ( F ` ( 1st ` ( w ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( w ` u ) ) ) = y /\ A. v e. ( 1 ... ( u - 1 ) ) ( F ` ( 2nd ` ( w ` v ) ) ) = ( F ` ( 1st ` ( w ` ( v + 1 ) ) ) ) ) } |-> ( RR*s gsum ( ( dist ` R ) o. z ) ) ) , RR* , < ) ) )
25 eqidd 
 |-  ( ph -> ( ( F o. ( le ` R ) ) o. `' F ) = ( ( F o. ( le ` R ) ) o. `' F ) )
26 1 2 7 8 9 10 11 5 12 13 14 16 18 20 21 22 24 25 3 4 imasval 
 |-  ( ph -> U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) )
27 eqid 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
28 27 imasvalstr 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) Struct <. 1 , ; 1 2 >.
29 ipid 
 |-  .i = Slot ( .i ` ndx )
30 snsstp3 
 |-  { <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } C_ { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. }
31 ssun2 
 |-  { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } )
32 30 31 sstri 
 |-  { <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } )
33 ssun1 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
34 32 33 sstri 
 |-  { <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
35 fvex 
 |-  ( Base ` R ) e. _V
36 2 35 eqeltrdi 
 |-  ( ph -> V e. _V )
37 snex 
 |-  { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V
38 37 rgenw 
 |-  A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V
39 iunexg 
 |-  ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V )
40 36 38 39 sylancl 
 |-  ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V )
41 40 ralrimivw 
 |-  ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V )
42 iunexg 
 |-  ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V )
43 36 41 42 syl2anc 
 |-  ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } e. _V )
44 26 28 29 34 43 6 strfv3 
 |-  ( ph -> I = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )