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

Theorem sxbrsigalem2 28418
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 28416 . . 3  |-  U. ran  R  =  ( RR  X.  RR )
5 sxbrsigalem0 28403 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( RR  X.  RR )
64, 5eqtr4i 2489 . 2  |-  U. ran  R  =  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )
7 vex 3112 . . . . . 6  |-  u  e. 
_V
8 vex 3112 . . . . . 6  |-  v  e. 
_V
97, 8xpex 6603 . . . . 5  |-  ( u  X.  v )  e. 
_V
103, 9elrnmpt2 6414 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
11 simpr 461 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
121, 2dya2icobrsiga 28408 . . . . . . . . . . . . 13  |-  ran  I  C_ 𝔅
13 brsigasspwrn 28317 . . . . . . . . . . . . 13  |- 𝔅 
C_  ~P RR
1412, 13sstri 3508 . . . . . . . . . . . 12  |-  ran  I  C_ 
~P RR
1514sseli 3495 . . . . . . . . . . 11  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
1615elpwid 4025 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  C_  RR )
1714sseli 3495 . . . . . . . . . . 11  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
1817elpwid 4025 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  C_  RR )
19 xpinpreima2 28042 . . . . . . . . . 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 477 . . . . . . . . 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 9600 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2221mptex 6144 . . . . . . . . . . . . . . . 16  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e. 
_V
2322rnex 6733 . . . . . . . . . . . . . . 15  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e.  _V
2421mptex 6144 . . . . . . . . . . . . . . . 16  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  e. 
_V
2524rnex 6733 . . . . . . . . . . . . . . 15  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  e.  _V
2623, 25unex 6597 . . . . . . . . . . . . . 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 28303 . . . . . . . . . . . 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 1404 . . . . . . . . . . 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 27673 . . . . . . . . . . . . 13  |-  ( u 
C_  RR  ->  ( `' ( 1st  |`  ( RR  X.  RR ) )
" u )  =  ( u  X.  RR ) )
3216, 31syl 16 . . . . . . . . . . . 12  |-  ( u  e.  ran  I  -> 
( `' ( 1st  |`  ( RR  X.  RR ) ) " u
)  =  ( u  X.  RR ) )
33 ovex 6324 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
342, 33elrnmpt2 6414 . . . . . . . . . . . . 13  |-  ( u  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
35 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  u  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3635xpeq1d 5031 . . . . . . . . . . . . . . . 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 5439 . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  ZZ )
3938zred 10990 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
40 2rp 11250 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  2  e.  RR+
4140a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  2  e.  RR+ )
42 simpr 461 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  n  e.  ZZ )
4341, 42rpexpcld 12335 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
4439, 43rerpdivcld 11308 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
4544rexrd 9660 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
46 1red 9628 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
4739, 46readdcld 9640 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
4847, 43rerpdivcld 11308 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
4948rexrd 9660 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
50 pnfxr 11346 . . . . . . . . . . . . . . . . . . . . . 22  |- +oo  e.  RR*
5150a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> +oo  e.  RR* )
5239lep1d 10497 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  <_  ( x  +  1 ) )
5339, 47, 43, 52lediv1dd 11335 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  <_  ( (
x  +  1 )  /  ( 2 ^ n ) ) )
54 pnfge 11364 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  +  1 )  /  ( 2 ^ n ) )  e.  RR*  ->  ( ( x  +  1 )  /  ( 2 ^ n ) )  <_ +oo )
5549, 54syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  <_ +oo )
56 difico 27746 . . . . . . . . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  \  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) ) )
5857xpeq1d 5031 . . . . . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . . . . . 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 3663 . . . . . . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
63 oveq1 6303 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
6463xpeq1d 5031 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
6564eqeq2d 2471 . . . . . . . . . . . . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
68 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
69 ovex 6324 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e [,) +oo )  e. 
_V
7069, 21xpex 6603 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
7168, 70elrnmpti 5263 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
7361, 72sseldi 3497 . . . . . . . . . . . . . . . . . . . 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 28308 . . . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
77 oveq1 6303 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
7877xpeq1d 5031 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
7978eqeq2d 2471 . . . . . . . . . . . . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
8268, 70elrnmpti 5263 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 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 3497 . . . . . . . . . . . . . . . . . . . 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 28308 . . . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . . . 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 28294 . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . 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 2954 . . . . . . . . . . . . 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 195 . . . . . . . . . . . 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 2545 . . . . . . . . . . 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 465 . . . . . . . . . 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 27674 . . . . . . . . . . . . 13  |-  ( v 
C_  RR  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" v )  =  ( RR  X.  v
) )
9818, 97syl 16 . . . . . . . . . . . 12  |-  ( v  e.  ran  I  -> 
( `' ( 2nd  |`  ( RR  X.  RR ) ) " v
)  =  ( RR 
X.  v ) )
992, 33elrnmpt2 6414 . . . . . . . . . . . . 13  |-  ( v  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
100 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  v  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
101100xpeq2d 5032 . . . . . . . . . . . . . . . 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 5440 . . . . . . . . . . . . . . . . . . 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 5032 . . . . . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . . . . . 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 3664 . . . . . . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
107 oveq1 6303 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
108107xpeq2d 5032 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) )
109108eqeq2d 2471 . . . . . . . . . . . . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
112 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
113 ovex 6324 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f [,) +oo )  e. 
_V
11421, 113xpex 6603 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
115112, 114elrnmpti 5263 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  e.  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
117105, 116sseldi 3497 . . . . . . . . . . . . . . . . . . . 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 28308 . . . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
121 oveq1 6303 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
122121xpeq2d 5032 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )
123122eqeq2d 2471 . . . . . . . . . . . . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
126112, 114elrnmpti 5263 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 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 3497 . . . . . . . . . . . . . . . . . . . 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 28308 . . . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . . . 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 28294 . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . 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 2954 . . . . . . . . . . . . 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 195 . . . . . . . . . . . 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 2545 . . . . . . . . . . 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 466 . . . . . . . . . 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 28296 . . . . . . . . . 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 1228 . . . . . . . . 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 2545 . . . . . . . 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 465 . . . . . . 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 2545 . . . . . 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 434 . . . . 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 2954 . . . 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 195 . . 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 3503 . 2  |-  ran  R  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
150 sigagenss2 28311 . 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 1324 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 369    = wceq 1395   T. wtru 1396    e. wcel 1819   E.wrex 2808   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009    |` cres 5010   "cima 5011   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   RRcr 9508   1c1 9510    + caddc 9512   +oocpnf 9642   RR*cxr 9644    <_ cle 9646    / cdiv 10227   2c2 10606   ZZcz 10885   RR+crp 11245   (,)cioo 11554   [,)cico 11556   ^cexp 12168   topGenctg 14854  sigAlgebracsiga 28268  sigaGencsigagen 28299  𝔅cbrsiga 28313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-ac2 8860  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-ac 8514  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-sin 13816  df-cos 13817  df-pi 13819  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-refld 18767  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-fcls 20567  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-cfil 21819  df-cmet 21821  df-cms 21899  df-limc 22395  df-dv 22396  df-log 23069  df-cxp 23070  df-logb 28160  df-siga 28269  df-sigagen 28300  df-brsiga 28314
This theorem is referenced by:  sxbrsigalem4  28419
  Copyright terms: Public domain W3C validator