Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sxbrsigalem2 Structured version   Unicode version

Theorem sxbrsigalem2 29117
Description: The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of  ( RR  X.  RR ) is a subset of the sigma-algebra generated by the closed half-spaces of  ( RR  X.  RR ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0  |-  J  =  ( topGen `  ran  (,) )
dya2ioc.1  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
dya2ioc.2  |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I  |->  ( u  X.  v ) )
Assertion
Ref Expression
sxbrsigalem2  |-  (sigaGen `  ran  R )  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
Distinct variable groups:    x, n    x, I    v, u, I, x    u, n, v    R, n, x    x, J   
e, f, n, u, v, x
Allowed substitution hints:    R( v, u, e, f)    I( e, f, n)    J( v, u, e, f, n)

Proof of Theorem sxbrsigalem2
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 sxbrsiga.0 . . . 4  |-  J  =  ( topGen `  ran  (,) )
2 dya2ioc.1 . . . 4  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3 dya2ioc.2 . . . 4  |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I  |->  ( u  X.  v ) )
41, 2, 3dya2iocucvr 29115 . . 3  |-  U. ran  R  =  ( RR  X.  RR )
5 sxbrsigalem0 29102 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( RR  X.  RR )
64, 5eqtr4i 2454 . 2  |-  U. ran  R  =  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )
7 vex 3083 . . . . . 6  |-  u  e. 
_V
8 vex 3083 . . . . . 6  |-  v  e. 
_V
97, 8xpex 6610 . . . . 5  |-  ( u  X.  v )  e. 
_V
103, 9elrnmpt2 6424 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
11 simpr 462 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
121, 2dya2icobrsiga 29107 . . . . . . . . . . . . 13  |-  ran  I  C_ 𝔅
13 brsigasspwrn 29016 . . . . . . . . . . . . 13  |- 𝔅 
C_  ~P RR
1412, 13sstri 3473 . . . . . . . . . . . 12  |-  ran  I  C_ 
~P RR
1514sseli 3460 . . . . . . . . . . 11  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
1615elpwid 3991 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  C_  RR )
1714sseli 3460 . . . . . . . . . . 11  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
1817elpwid 3991 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  C_  RR )
19 xpinpreima2 28722 . . . . . . . . . 10  |-  ( ( u  C_  RR  /\  v  C_  RR )  ->  (
u  X.  v )  =  ( ( `' ( 1st  |`  ( RR  X.  RR ) )
" u )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " v
) ) )
2016, 18, 19syl2an 479 . . . . . . . . 9  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
u  X.  v )  =  ( ( `' ( 1st  |`  ( RR  X.  RR ) )
" u )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " v
) ) )
21 reex 9638 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2221mptex 6152 . . . . . . . . . . . . . . . 16  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e. 
_V
2322rnex 6742 . . . . . . . . . . . . . . 15  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e.  _V
2421mptex 6152 . . . . . . . . . . . . . . . 16  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  e. 
_V
2524rnex 6742 . . . . . . . . . . . . . . 15  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  e.  _V
2623, 25unex 6604 . . . . . . . . . . . . . 14  |-  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  e. 
_V
2726a1i 11 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )  e.  _V )
2827sgsiga 28973 . . . . . . . . . . . 12  |-  ( T. 
->  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  e.  U. ran sigAlgebra )
2928trud 1446 . . . . . . . . . . 11  |-  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  e.  U. ran sigAlgebra
3029a1i 11 . . . . . . . . . 10  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )  e.  U. ran sigAlgebra )
31 1stpreima 28290 . . . . . . . . . . . . 13  |-  ( u 
C_  RR  ->  ( `' ( 1st  |`  ( RR  X.  RR ) )
" u )  =  ( u  X.  RR ) )
3216, 31syl 17 . . . . . . . . . . . 12  |-  ( u  e.  ran  I  -> 
( `' ( 1st  |`  ( RR  X.  RR ) ) " u
)  =  ( u  X.  RR ) )
33 ovex 6334 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
342, 33elrnmpt2 6424 . . . . . . . . . . . . 13  |-  ( u  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
35 simpr 462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  u  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3635xpeq1d 4876 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( u  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  X.  RR ) )
37 difxp1 5281 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  \  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  X.  RR )  =  ( (
( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  \ 
( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
38 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  ZZ )
3938zred 11048 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
40 2rp 11315 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  2  e.  RR+
4140a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  2  e.  RR+ )
42 simpr 462 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  n  e.  ZZ )
4341, 42rpexpcld 12446 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
4439, 43rerpdivcld 11377 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
4544rexrd 9698 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
46 1red 9666 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
4739, 46readdcld 9678 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
4847, 43rerpdivcld 11377 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
4948rexrd 9698 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
50 pnfxr 11420 . . . . . . . . . . . . . . . . . . . . . 22  |- +oo  e.  RR*
5150a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> +oo  e.  RR* )
5239lep1d 10546 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  <_  ( x  +  1 ) )
5339, 47, 43, 52lediv1dd 11404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  <_  ( (
x  +  1 )  /  ( 2 ^ n ) ) )
54 pnfge 11440 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  +  1 )  /  ( 2 ^ n ) )  e.  RR*  ->  ( ( x  +  1 )  /  ( 2 ^ n ) )  <_ +oo )
5549, 54syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  <_ +oo )
56 difico 28372 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( x  / 
( 2 ^ n
) )  e.  RR*  /\  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR*  /\ +oo  e.  RR* )  /\  (
( x  /  (
2 ^ n ) )  <_  ( (
x  +  1 )  /  ( 2 ^ n ) )  /\  ( ( x  + 
1 )  /  (
2 ^ n ) )  <_ +oo )
)  ->  ( (
( x  /  (
2 ^ n ) ) [,) +oo )  \  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
5745, 49, 51, 53, 55, 56syl32anc 1272 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  \  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) ) )
5857xpeq1d 4876 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  \ 
( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) )  X.  RR )  =  ( (
( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  X.  RR ) )
5937, 58syl5reqr 2478 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  X.  RR )  =  ( (
( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  \ 
( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) ) )
6029a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  e.  U. ran sigAlgebra )
61 ssun1 3629 . . . . . . . . . . . . . . . . . . . . 21  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ( ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) ) )
62 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
63 oveq1 6313 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
6463xpeq1d 4876 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
6564eqeq2d 2436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR )  <->  ( (
( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR ) ) )
6665rspcev 3182 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) )  e.  RR  /\  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR ) )  ->  E. e  e.  RR  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
6744, 62, 66sylancl 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
68 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
69 ovex 6334 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e [,) +oo )  e. 
_V
7069, 21xpex 6610 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
7168, 70elrnmpti 5104 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  e. 
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  <->  E. e  e.  RR  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( e [,) +oo )  X.  RR ) )
7267, 71sylibr 215 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
7361, 72sseldi 3462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
74 elsigagen 28978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )  e.  _V  /\  (
( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )  ->  ( (
( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
7526, 73, 74sylancr 667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
76 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
77 oveq1 6313 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
7877xpeq1d 4876 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
7978eqeq2d 2436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR )  <->  ( (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR ) ) )
8079rspcev 3182 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR  /\  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR ) )  ->  E. e  e.  RR  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
8148, 76, 80sylancl 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
8268, 70elrnmpti 5104 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  e. 
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  <->  E. e  e.  RR  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( e [,) +oo )  X.  RR ) )
8381, 82sylibr 215 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
8461, 83sseldi 3462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
85 elsigagen 28978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )  e.  _V  /\  (
( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )  ->  ( (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
8626, 84, 85sylancr 667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
87 difelsiga 28964 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  e.  U. ran sigAlgebra  /\  (
( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  /\  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )  ->  (
( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  \  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
8860, 75, 86, 87syl3anc 1264 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  \  (
( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
8959, 88eqeltrd 2507 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
9089adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
9136, 90eqeltrd 2507 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( u  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
9291ex 435 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  ( u  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) ) )
9392rexlimivv 2919 . . . . . . . . . . . . 13  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  u  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  (
u  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
9434, 93sylbi 198 . . . . . . . . . . . 12  |-  ( u  e.  ran  I  -> 
( u  X.  RR )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
9532, 94eqeltrd 2507 . . . . . . . . . . 11  |-  ( u  e.  ran  I  -> 
( `' ( 1st  |`  ( RR  X.  RR ) ) " u
)  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
9695adantr 466 . . . . . . . . . 10  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  ( `' ( 1st  |`  ( RR  X.  RR ) )
" u )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
97 2ndpreima 28291 . . . . . . . . . . . . 13  |-  ( v 
C_  RR  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" v )  =  ( RR  X.  v
) )
9818, 97syl 17 . . . . . . . . . . . 12  |-  ( v  e.  ran  I  -> 
( `' ( 2nd  |`  ( RR  X.  RR ) ) " v
)  =  ( RR 
X.  v ) )
992, 33elrnmpt2 6424 . . . . . . . . . . . . 13  |-  ( v  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
100 simpr 462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  v  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
101100xpeq2d 4877 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( RR  X.  v )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) ) ) )
102 difxp2 5282 . . . . . . . . . . . . . . . . . . 19  |-  ( RR 
X.  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  \ 
( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) ) )  =  ( ( RR  X.  ( ( x  / 
( 2 ^ n
) ) [,) +oo ) )  \  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )
10357xpeq2d 4877 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( ( x  / 
( 2 ^ n
) ) [,) +oo )  \  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )  =  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) ) )
104102, 103syl5reqr 2478 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  =  ( ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) 
\  ( RR  X.  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) ) ) )
105 ssun2 3630 . . . . . . . . . . . . . . . . . . . . 21  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )
106 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
107 oveq1 6313 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
108107xpeq2d 4877 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) )
109108eqeq2d 2436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  (
( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) )  <->  ( RR  X.  ( ( x  / 
( 2 ^ n
) ) [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) ) )
110109rspcev 3182 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) )  e.  RR  /\  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) )  ->  E. f  e.  RR  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
11144, 106, 110sylancl 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
112 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
113 ovex 6334 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f [,) +oo )  e. 
_V
11421, 113xpex 6610 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
115112, 114elrnmpti 5104 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  <->  E. f  e.  RR  ( RR  X.  ( ( x  / 
( 2 ^ n
) ) [,) +oo ) )  =  ( RR  X.  ( f [,) +oo ) ) )
116111, 115sylibr 215 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  e.  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
117105, 116sseldi 3462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
118 elsigagen 28978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )  e.  _V  /\  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )  ->  ( RR  X.  ( ( x  / 
( 2 ^ n
) ) [,) +oo ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
11926, 117, 118sylancr 667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
120 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
121 oveq1 6313 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
122121xpeq2d 4877 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )
123122eqeq2d 2436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) )  <->  ( RR  X.  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) ) )
124123rspcev 3182 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR  /\  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )  ->  E. f  e.  RR  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
12548, 120, 124sylancl 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
126112, 114elrnmpti 5104 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  <->  E. f  e.  RR  ( RR  X.  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) )  =  ( RR  X.  ( f [,) +oo ) ) )
127125, 126sylibr 215 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  e.  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
128105, 127sseldi 3462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
129 elsigagen 28978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )  e.  _V  /\  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )  ->  ( RR  X.  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
13026, 128, 129sylancr 667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
131 difelsiga 28964 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  e.  U. ran sigAlgebra  /\  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )  /\  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )  ->  (
( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  \  ( RR  X.  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo ) ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
13260, 119, 130, 131syl3anc 1264 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( RR  X.  ( ( x  / 
( 2 ^ n
) ) [,) +oo ) )  \  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
133104, 132eqeltrd 2507 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
134133adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( RR  X.  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
135101, 134eqeltrd 2507 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( RR  X.  v )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
136135ex 435 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  ( RR  X.  v )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) ) )
137136rexlimivv 2919 . . . . . . . . . . . . 13  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  v  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  ( RR  X.  v )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
13899, 137sylbi 198 . . . . . . . . . . . 12  |-  ( v  e.  ran  I  -> 
( RR  X.  v
)  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
13998, 138eqeltrd 2507 . . . . . . . . . . 11  |-  ( v  e.  ran  I  -> 
( `' ( 2nd  |`  ( RR  X.  RR ) ) " v
)  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
140139adantl 467 . . . . . . . . . 10  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" v )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
141 inelsiga 28966 . . . . . . . . . 10  |-  ( ( (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  e.  U. ran sigAlgebra  /\  ( `' ( 1st  |`  ( RR  X.  RR ) )
" u )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )  /\  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
v )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )  ->  (
( `' ( 1st  |`  ( RR  X.  RR ) ) " u
)  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
v ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
14230, 96, 140, 141syl3anc 1264 . . . . . . . . 9  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
( `' ( 1st  |`  ( RR  X.  RR ) ) " u
)  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
v ) )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
14320, 142eqeltrd 2507 . . . . . . . 8  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
u  X.  v )  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
144143adantr 466 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
( u  X.  v
)  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
14511, 144eqeltrd 2507 . . . . . 6  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
146145ex 435 . . . . 5  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
d  =  ( u  X.  v )  -> 
d  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) ) )
147146rexlimivv 2919 . . . 4  |-  ( E. u  e.  ran  I E. v  e.  ran  I  d  =  (
u  X.  v )  ->  d  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) ) )
14810, 147sylbi 198 . . 3  |-  ( d  e.  ran  R  -> 
d  e.  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) ) )
149148ssriv 3468 . 2  |-  ran  R  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
150 sigagenss2 28981 . 2  |-  ( ( U. ran  R  = 
U. ( ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) ) )  /\  ran  R  C_  (sigaGen `  ( ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) ) ) )  /\  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  e. 
_V )  ->  (sigaGen ` 
ran  R )  C_  (sigaGen `  ( ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) ) ) ) )
1516, 149, 26, 150mp3an 1360 1  |-  (sigaGen `  ran  R )  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1872   E.wrex 2772   _Vcvv 3080    \ cdif 3433    u. cun 3434    i^i cin 3435    C_ wss 3436   ~Pcpw 3981   U.cuni 4219   class class class wbr 4423    |-> cmpt 4482    X. cxp 4851   `'ccnv 4852   ran crn 4854    |` cres 4855   "cima 4856   ` cfv 5601  (class class class)co 6306    |-> cmpt2 6308   1stc1st 6806   2ndc2nd 6807   RRcr 9546   1c1 9548    + caddc 9550   +oocpnf 9680   RR*cxr 9682    <_ cle 9684    / cdiv 10277   2c2 10667   ZZcz 10945   RR+crp 11310   (,)cioo 11643   [,)cico 11645   ^cexp 12279   topGenctg 15336  sigAlgebracsiga 28938  sigaGencsigagen 28969  𝔅cbrsiga 29012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-ac2 8901  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625  ax-addf 9626  ax-mulf 9627
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-ixp 7535  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-fi 7935  df-sup 7966  df-inf 7967  df-oi 8035  df-card 8382  df-acn 8385  df-ac 8555  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-q 11273  df-rp 11311  df-xneg 11417  df-xadd 11418  df-xmul 11419  df-ioo 11647  df-ioc 11648  df-ico 11649  df-icc 11650  df-fz 11793  df-fzo 11924  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-shft 13131  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-refld 19172  df-top 19920  df-bases 19921  df-topon 19922  df-topsp 19923  df-cld 20033  df-ntr 20034  df-cls 20035  df-nei 20113  df-lp 20151  df-perf 20152  df-cn 20242  df-cnp 20243  df-haus 20330  df-cmp 20401  df-tx 20576  df-hmeo 20769  df-fil 20860  df-fm 20952  df-flim 20953  df-flf 20954  df-fcls 20955  df-xms 21334  df-ms 21335  df-tms 21336  df-cncf 21909  df-cfil 22224  df-cmet 22226  df-cms 22302  df-limc 22820  df-dv 22821  df-log 23505  df-cxp 23506  df-logb 23701  df-siga 28939  df-sigagen 28970  df-brsiga 29013
This theorem is referenced by:  sxbrsigalem4  29118
  Copyright terms: Public domain W3C validator