Metamath Proof Explorer

Theorem imasvsca

Description: The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015) (Revised 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 )
imassca.g 
|- G = ( Scalar ` R )
imasvsca.k 
|- K = ( Base ` G )
imasvsca.q 
|- .x. = ( .s ` R )
imasvsca.s 
|- .xb = ( .s ` U )
Assertion imasvsca 
|- ( ph -> .xb = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. 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 imassca.g 
 |-  G = ( Scalar ` R )
6 imasvsca.k 
 |-  K = ( Base ` G )
7 imasvsca.q 
 |-  .x. = ( .s ` R )
8 imasvsca.s 
 |-  .xb = ( .s ` U )
9 eqid 
 |-  ( +g ` R ) = ( +g ` R )
10 eqid 
 |-  ( .r ` R ) = ( .r ` R )
11 eqid 
 |-  ( Scalar ` R ) = ( Scalar ` R )
12 5 fveq2i 
 |-  ( Base ` G ) = ( Base ` ( Scalar ` R ) )
13 6 12 eqtri 
 |-  K = ( Base ` ( Scalar ` R ) )
14 eqid 
 |-  ( .i ` R ) = ( .i ` R )
15 eqid 
 |-  ( TopOpen ` R ) = ( TopOpen ` R )
16 eqid 
 |-  ( dist ` R ) = ( dist ` R )
17 eqid 
 |-  ( le ` R ) = ( le ` R )
18 eqid 
 |-  ( +g ` U ) = ( +g ` U )
19 1 2 3 4 9 18 imasplusg 
 |-  ( ph -> ( +g ` U ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
20 eqid 
 |-  ( .r ` U ) = ( .r ` U )
21 1 2 3 4 10 20 imasmulr 
 |-  ( ph -> ( .r ` U ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
22 eqidd 
 |-  ( ph -> U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )
23 eqidd 
 |-  ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } )
24 eqidd 
 |-  ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( TopOpen ` R ) qTop F ) )
25 eqid 
 |-  ( dist ` U ) = ( dist ` U )
26 1 2 3 4 16 25 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* , < ) ) )
27 eqidd 
 |-  ( ph -> ( ( F o. ( le ` R ) ) o. `' F ) = ( ( F o. ( le ` R ) ) o. `' F ) )
28 1 2 9 10 11 13 7 14 15 16 17 19 21 22 23 24 26 27 3 4 imasval 
 |-  ( ph -> U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) )
29 eqid 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) 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 ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
30 29 imasvalstr 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) Struct <. 1 , ; 1 2 >.
31 vscaid 
 |-  .s = Slot ( .s ` ndx )
32 snsstp2 
 |-  { <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. } C_ { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. }
33 ssun2 
 |-  { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } )
34 ssun1 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
35 33 34 sstri 
 |-  { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
36 32 35 sstri 
 |-  { <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( ( TopOpen ` R ) qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
37 fvex 
 |-  ( Base ` R ) e. _V
38 2 37 eqeltrdi 
 |-  ( ph -> V e. _V )
39 6 fvexi 
 |-  K e. _V
40 snex 
 |-  { ( F ` q ) } e. _V
41 39 40 mpoex 
 |-  ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) e. _V
42 41 rgenw 
 |-  A. q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) e. _V
43 iunexg 
 |-  ( ( V e. _V /\ A. q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) e. _V ) -> U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) e. _V )
44 38 42 43 sylancl 
 |-  ( ph -> U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) e. _V )
45 28 30 31 36 44 8 strfv3 
 |-  ( ph -> .xb = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )