Metamath Proof Explorer

Theorem imasplusg

Description: The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015) (Revised by Mario Carneiro, 11-Jul-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 )
imasplusg.p 
|- .+ = ( +g ` R )
imasplusg.a 
|- .+b = ( +g ` U )
Assertion imasplusg 
|- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( 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 imasplusg.p 
 |-  .+ = ( +g ` R )
6 imasplusg.a 
 |-  .+b = ( +g ` U )
7 eqid 
 |-  ( .r ` R ) = ( .r ` R )
8 eqid 
 |-  ( Scalar ` R ) = ( Scalar ` R )
9 eqid 
 |-  ( Base ` ( Scalar ` R ) ) = ( Base ` ( Scalar ` R ) )
10 eqid 
 |-  ( .s ` R ) = ( .s ` R )
11 eqid 
 |-  ( .i ` R ) = ( .i ` R )
12 eqid 
 |-  ( TopOpen ` R ) = ( TopOpen ` R )
13 eqid 
 |-  ( dist ` R ) = ( dist ` R )
14 eqid 
 |-  ( le ` R ) = ( le ` R )
15 eqidd 
 |-  ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
16 eqidd 
 |-  ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
17 eqidd 
 |-  ( ph -> U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) q ) ) ) = U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) q ) ) ) )
18 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 ) >. } )
19 eqidd 
 |-  ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( TopOpen ` R ) qTop F ) )
20 eqid 
 |-  ( dist ` U ) = ( dist ` U )
21 1 2 3 4 13 20 imasds 
 |-  ( ph -> ( dist ` U ) = ( x e. B , y e. B |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsum ( ( dist ` R ) o. g ) ) ) , RR* , < ) ) )
22 eqidd 
 |-  ( ph -> ( ( F o. ( le ` R ) ) o. `' F ) = ( ( F o. ( le ` R ) ) o. `' F ) )
23 1 2 5 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 3 4 imasval 
 |-  ( ph -> U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 ) >. } ) )
24 eqid 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 ) >. } )
25 24 imasvalstr 
 |-  ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 >.
26 plusgid 
 |-  +g = Slot ( +g ` ndx )
27 snsstp2 
 |-  { <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. }
28 ssun1 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } )
29 27 28 sstri 
 |-  { <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } )
30 ssun1 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 ) >. } )
31 29 30 sstri 
 |-  { <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } |-> ( F ` ( p ( .s ` R ) 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 ) >. } )
32 fvex 
 |-  ( Base ` R ) e. _V
33 2 32 eqeltrdi 
 |-  ( ph -> V e. _V )
34 snex 
 |-  { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V
35 34 rgenw 
 |-  A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V
36 iunexg 
 |-  ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
37 33 35 36 sylancl 
 |-  ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
38 37 ralrimivw 
 |-  ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
39 iunexg 
 |-  ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
40 33 38 39 syl2anc 
 |-  ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
41 23 25 26 31 40 6 strfv3 
 |-  ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )