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

Theorem sxbrsigalem2 24589
Description: The sigma-algebra generated by the dyadic closed below, opened 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 24587 . . 3  |-  U. ran  R  =  ( RR  X.  RR )
5 sxbrsigalem0 24574 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )  =  ( RR  X.  RR )
64, 5eqtr4i 2427 . 2  |-  U. ran  R  =  U. ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )
7 vex 2919 . . . . . 6  |-  u  e. 
_V
8 vex 2919 . . . . . 6  |-  v  e. 
_V
97, 8xpex 4949 . . . . 5  |-  ( u  X.  v )  e. 
_V
103, 9elrnmpt2 6142 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
11 simpr 448 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
121, 2dya2icobrsiga 24579 . . . . . . . . . . . . 13  |-  ran  I  C_ 𝔅
13 brsigasspwrn 24492 . . . . . . . . . . . . 13  |- 𝔅 
C_  ~P RR
1412, 13sstri 3317 . . . . . . . . . . . 12  |-  ran  I  C_ 
~P RR
1514sseli 3304 . . . . . . . . . . 11  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
1615elpwid 3768 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  C_  RR )
1714sseli 3304 . . . . . . . . . . 11  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
1817elpwid 3768 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  C_  RR )
19 xpinpreima2 24258 . . . . . . . . . 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 464 . . . . . . . . 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 9037 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2221mptex 5925 . . . . . . . . . . . . . . . 16  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  e. 
_V
2322rnex 5092 . . . . . . . . . . . . . . 15  |-  ran  (
e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  e.  _V
2421mptex 5925 . . . . . . . . . . . . . . . 16  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  e.  _V
2524rnex 5092 . . . . . . . . . . . . . . 15  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) )  e.  _V
2623, 25unex 4666 . . . . . . . . . . . . . 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 24478 . . . . . . . . . . . 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 1329 . . . . . . . . . . 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 24048 . . . . . . . . . . . . 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 6065 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
342, 33elrnmpt2 6142 . . . . . . . . . . . . 13  |-  ( u  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
35 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  u  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  u  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3635xpeq1d 4860 . . . . . . . . . . . . . . . 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 6340 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  ZZ )
3938zred 10331 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
40 2rp 10573 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  2  e.  RR+
4140a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  2  e.  RR+ )
42 simpr 448 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  n  e.  ZZ )
4341, 42rpexpcld 11501 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
4439, 43rerpdivcld 10631 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
4544rexrd 9090 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
46 1re 9046 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  RR
4746a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
4839, 47readdcld 9071 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
4948, 43rerpdivcld 10631 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
5049rexrd 9090 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
51 pnfxr 10669 . . . . . . . . . . . . . . . . . . . . . 22  |-  +oo  e.  RR*
5251a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  +oo  e.  RR* )
5339lep1d 9898 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  <_  ( x  +  1 ) )
5439, 48, 43, 53lediv1dd 10658 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  <_  ( (
x  +  1 )  /  ( 2 ^ n ) ) )
55 pnfge 10683 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  +  1 )  /  ( 2 ^ n ) )  e.  RR*  ->  ( ( x  +  1 )  /  ( 2 ^ n ) )  <_  +oo )
5650, 55syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  <_  +oo )
57 difico 24099 . . . . . . . . . . . . . . . . . . . . 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
) ) ) )
5845, 50, 52, 54, 56, 57syl32anc 1192 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) 
+oo )  \  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,)  +oo )
)  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) ) )
5958xpeq1d 4860 . . . . . . . . . . . . . . . . . . 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 ) )
6037, 59syl5reqr 2451 . . . . . . . . . . . . . . . . . 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 ) ) )
6129a1i 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 )
62 ssun1 3470 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
63 eqid 2404 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) ) [,)  +oo )  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,)  +oo )  X.  RR )
64 oveq1 6047 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
e [,)  +oo )  =  ( ( x  / 
( 2 ^ n
) ) [,)  +oo ) )
6564xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
( e [,)  +oo )  X.  RR )  =  ( ( ( x  /  ( 2 ^ n ) ) [,) 
+oo )  X.  RR ) )
6665eqeq2d 2415 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
6766rspcev 3012 . . . . . . . . . . . . . . . . . . . . . . 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 ) )
6844, 63, 67sylancl 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( x  /  ( 2 ^ n ) ) [,) 
+oo )  X.  RR )  =  ( (
e [,)  +oo )  X.  RR ) )
69 eqid 2404 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )
70 ovex 6065 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e [,)  +oo )  e.  _V
7170, 21xpex 4949 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( e [,)  +oo )  X.  RR )  e.  _V
7269, 71elrnmpti 5080 . . . . . . . . . . . . . . . . . . . . . 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 ) )
7368, 72sylibr 204 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) 
+oo )  X.  RR )  e.  ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) )
7462, 73sseldi 3306 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
75 elsigagen 24483 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
7626, 74, 75sylancr 645 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
77 eqid 2404 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) ) [,)  +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,)  +oo )  X.  RR )
78 oveq1 6047 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
e [,)  +oo )  =  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,)  +oo ) )
7978xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( e [,)  +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) 
+oo )  X.  RR ) )
8079eqeq2d 2415 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
8180rspcev 3012 . . . . . . . . . . . . . . . . . . . . . . 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 ) )
8249, 77, 81sylancl 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) 
+oo )  X.  RR )  =  ( (
e [,)  +oo )  X.  RR ) )
8369, 71elrnmpti 5080 . . . . . . . . . . . . . . . . . . . . . 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 ) )
8482, 83sylibr 204 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) 
+oo )  X.  RR )  e.  ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) )
8562, 84sseldi 3306 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
86 elsigagen 24483 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
8726, 85, 86sylancr 645 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
88 difelsiga 24469 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
8961, 76, 87, 88syl3anc 1184 . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
9060, 89eqeltrd 2478 . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
9190adantr 452 . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
9236, 91eqeltrd 2478 . . . . . . . . . . . . . . 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 ) ) ) ) ) )
9392ex 424 . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
9493rexlimivv 2795 . . . . . . . . . . . . 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 ) ) ) ) ) )
9534, 94sylbi 188 . . . . . . . . . . . 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 ) ) ) ) ) )
9632, 95eqeltrd 2478 . . . . . . . . . . 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 ) ) ) ) ) )
9796adantr 452 . . . . . . . . . 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 ) ) ) ) ) )
98 2ndpreima 24049 . . . . . . . . . . . . 13  |-  ( v 
C_  RR  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) )
" v )  =  ( RR  X.  v
) )
9918, 98syl 16 . . . . . . . . . . . 12  |-  ( v  e.  ran  I  -> 
( `' ( 2nd  |`  ( RR  X.  RR ) ) " v
)  =  ( RR 
X.  v ) )
1002, 33elrnmpt2 6142 . . . . . . . . . . . . 13  |-  ( v  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
101 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  v  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
102101xpeq2d 4861 . . . . . . . . . . . . . . . 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
) ) ) ) )
103 difxp2 6341 . . . . . . . . . . . . . . . . . . 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 ) ) )
10458xpeq2d 4861 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
105103, 104syl5reqr 2451 . . . . . . . . . . . . . . . . . 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 )
) ) )
106 ssun2 3471 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
107 eqid 2404 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( x  /  ( 2 ^ n ) ) [,) 
+oo ) )  =  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,)  +oo )
)
108 oveq1 6047 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  (
f [,)  +oo )  =  ( ( x  / 
( 2 ^ n
) ) [,)  +oo ) )
109108xpeq2d 4861 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) 
+oo ) )  =  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,)  +oo )
) )
110109eqeq2d 2415 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
111110rspcev 3012 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
11244, 107, 111sylancl 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,)  +oo )
)  =  ( RR 
X.  ( f [,) 
+oo ) ) )
113 eqid 2404 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) )
114 ovex 6065 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f [,)  +oo )  e.  _V
11521, 114xpex 4949 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( f [,) 
+oo ) )  e. 
_V
116113, 115elrnmpti 5080 . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
117112, 116sylibr 204 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,)  +oo )
)  e.  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) )
118106, 117sseldi 3306 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
119 elsigagen 24483 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
12026, 118, 119sylancr 645 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
121 eqid 2404 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) 
+oo ) )  =  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,)  +oo )
)
122 oveq1 6047 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
f [,)  +oo )  =  ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,)  +oo ) )
123122xpeq2d 4861 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) 
+oo ) )  =  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,)  +oo )
) )
124123eqeq2d 2415 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
125124rspcev 3012 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
12649, 121, 125sylancl 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,)  +oo )
)  =  ( RR 
X.  ( f [,) 
+oo ) ) )
127113, 115elrnmpti 5080 . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
128126, 127sylibr 204 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,)  +oo )
)  e.  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) )
129106, 128sseldi 3306 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
130 elsigagen 24483 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
13126, 129, 130sylancr 645 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
132 difelsiga 24469 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
13361, 120, 131, 132syl3anc 1184 . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
134105, 133eqeltrd 2478 . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
135134adantr 452 . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
136102, 135eqeltrd 2478 . . . . . . . . . . . . . . 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 ) ) ) ) ) )
137136ex 424 . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
138137rexlimivv 2795 . . . . . . . . . . . . 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 ) ) ) ) ) )
139100, 138sylbi 188 . . . . . . . . . . . 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 ) ) ) ) ) )
14099, 139eqeltrd 2478 . . . . . . . . . . 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 ) ) ) ) ) )
141140adantl 453 . . . . . . . . . 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 ) ) ) ) ) )
142 inelsiga 24471 . . . . . . . . . 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 ) ) ) ) ) )
14330, 97, 141, 142syl3anc 1184 . . . . . . . . 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 ) ) ) ) ) )
14420, 143eqeltrd 2478 . . . . . . . 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 ) ) ) ) ) )
145144adantr 452 . . . . . . 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 ) ) ) ) ) )
14611, 145eqeltrd 2478 . . . . . 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 ) ) ) ) ) )
147146ex 424 . . . . 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 ) ) ) ) ) ) )
148147rexlimivv 2795 . . . 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 ) ) ) ) ) )
14910, 148sylbi 188 . . 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 ) ) ) ) ) )
150149ssriv 3312 . 2  |-  ran  R  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) ) )
151 sigagenss2 24486 . 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 ) ) ) ) ) )
1526, 150, 26, 151mp3an 1279 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 set class
Syntax hints:    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   ran crn 4838    |` cres 4839   "cima 4840   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   RRcr 8945   1c1 8947    + caddc 8949    +oocpnf 9073   RR*cxr 9075    <_ cle 9077    / cdiv 9633   2c2 10005   ZZcz 10238   RR+crp 10568   (,)cioo 10872   [,)cico 10874   ^cexp 11337   topGenctg 13620  sigAlgebracsiga 24443  sigaGencsigagen 24474  𝔅cbrsiga 24488
This theorem is referenced by:  sxbrsigalem4  24590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-ac 7953  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-logb 24342  df-siga 24444  df-sigagen 24475  df-brsiga 24489
  Copyright terms: Public domain W3C validator