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Theorem sxbrsigalem2 26838
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 26836 . . 3  |-  U. ran  R  =  ( RR  X.  RR )
5 sxbrsigalem0 26823 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( RR  X.  RR )
64, 5eqtr4i 2483 . 2  |-  U. ran  R  =  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )
7 vex 3074 . . . . . 6  |-  u  e. 
_V
8 vex 3074 . . . . . 6  |-  v  e. 
_V
97, 8xpex 6611 . . . . 5  |-  ( u  X.  v )  e. 
_V
103, 9elrnmpt2 6306 . . . 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 26828 . . . . . . . . . . . . 13  |-  ran  I  C_ 𝔅
13 brsigasspwrn 26737 . . . . . . . . . . . . 13  |- 𝔅 
C_  ~P RR
1412, 13sstri 3466 . . . . . . . . . . . 12  |-  ran  I  C_ 
~P RR
1514sseli 3453 . . . . . . . . . . 11  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
1615elpwid 3971 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  C_  RR )
1714sseli 3453 . . . . . . . . . . 11  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
1817elpwid 3971 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  C_  RR )
19 xpinpreima2 26475 . . . . . . . . . 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 9477 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2221mptex 6050 . . . . . . . . . . . . . . . 16  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e. 
_V
2322rnex 6615 . . . . . . . . . . . . . . 15  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e.  _V
2421mptex 6050 . . . . . . . . . . . . . . . 16  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  e. 
_V
2524rnex 6615 . . . . . . . . . . . . . . 15  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  e.  _V
2623, 25unex 6481 . . . . . . . . . . . . . 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 26723 . . . . . . . . . . . 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 1379 . . . . . . . . . . 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 26145 . . . . . . . . . . . . 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 6218 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
342, 33elrnmpt2 6306 . . . . . . . . . . . . 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 4964 . . . . . . . . . . . . . . . 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 5364 . . . . . . . . . . . . . . . . . . 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 10851 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
40 2rp 11100 . . . . . . . . . . . . . . . . . . . . . . . . 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 12141 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
4439, 43rerpdivcld 11158 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
4544rexrd 9537 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
46 1re 9489 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  RR
4746a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
4839, 47readdcld 9517 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
4948, 43rerpdivcld 11158 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
5049rexrd 9537 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
51 pnfxr 11196 . . . . . . . . . . . . . . . . . . . . . 22  |- +oo  e.  RR*
5251a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> +oo  e.  RR* )
5339lep1d 10368 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  <_  ( x  +  1 ) )
5439, 48, 43, 53lediv1dd 11185 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  <_  ( (
x  +  1 )  /  ( 2 ^ n ) ) )
55 pnfge 11214 . . . . . . . . . . . . . . . . . . . . . 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 26211 . . . . . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  \  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) ) )
5958xpeq1d 4964 . . . . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . . . . 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 3620 . . . . . . . . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
64 oveq1 6200 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
6564xpeq1d 4964 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
6665eqeq2d 2465 . . . . . . . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
69 eqid 2451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
70 ovex 6218 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e [,) +oo )  e. 
_V
7170, 21xpex 6611 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
7269, 71elrnmpti 5191 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
7462, 73sseldi 3455 . . . . . . . . . . . . . . . . . . . 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 26728 . . . . . . . . . . . . . . . . . . . 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 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 )
) ) ) ) )
77 eqid 2451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
78 oveq1 6200 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
7978xpeq1d 4964 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
8079eqeq2d 2465 . . . . . . . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. e  e.  RR  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR )  =  ( (
e [,) +oo )  X.  RR ) )
8369, 71elrnmpti 5191 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 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 3455 . . . . . . . . . . . . . . . . . . . 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 26728 . . . . . . . . . . . . . . . . . . . 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 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 )
) ) ) ) )
88 difelsiga 26714 . . . . . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . . 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 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 )
) ) ) ) )
9236, 91eqeltrd 2539 . . . . . . . . . . . . . . 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 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 ) ) ) ) ) ) )
9493rexlimivv 2945 . . . . . . . . . . . . 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 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 )
) ) ) ) )
9632, 95eqeltrd 2539 . . . . . . . . . . 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 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 )
) ) ) ) )
98 2ndpreima 26146 . . . . . . . . . . . . 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 6306 . . . . . . . . . . . . 13  |-  ( v  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  v  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
101 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 ) ) ) )
102101xpeq2d 4965 . . . . . . . . . . . . . . . 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 5365 . . . . . . . . . . . . . . . . . . 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 4965 . . . . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
108 oveq1 6200 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
109108xpeq2d 4965 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) )
110109eqeq2d 2465 . . . . . . . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
113 eqid 2451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
114 ovex 6218 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f [,) +oo )  e. 
_V
11521, 114xpex 6611 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
116113, 115elrnmpti 5191 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( RR  X.  (
( x  /  (
2 ^ n ) ) [,) +oo )
)  e.  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
118106, 117sseldi 3455 . . . . . . . . . . . . . . . . . . . 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 26728 . . . . . . . . . . . . . . . . . . . 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 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 )
) ) ) ) )
121 eqid 2451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
122 oveq1 6200 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
123122xpeq2d 4965 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )
124123eqeq2d 2465 . . . . . . . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  E. f  e.  RR  ( RR  X.  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( RR 
X.  ( f [,) +oo ) ) )
127113, 115elrnmpti 5191 . . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . . . . 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 3455 . . . . . . . . . . . . . . . . . . . 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 26728 . . . . . . . . . . . . . . . . . . . 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 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 )
) ) ) ) )
132 difelsiga 26714 . . . . . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . . 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 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 ) ) ) ) ) )
136102, 135eqeltrd 2539 . . . . . . . . . . . . . . 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 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 ) ) ) ) ) ) )
138137rexlimivv 2945 . . . . . . . . . . . . 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 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 )
) ) ) ) )
14099, 139eqeltrd 2539 . . . . . . . . . . 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 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 )
) ) ) ) )
142 inelsiga 26716 . . . . . . . . . 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 1219 . . . . . . . . 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 2539 . . . . . . . 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 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 )
) ) ) ) )
14611, 145eqeltrd 2539 . . . . . 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 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 )
) ) ) ) ) )
148147rexlimivv 2945 . . . 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 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 )
) ) ) ) )
150149ssriv 3461 . 2  |-  ran  R  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
151 sigagenss2 26731 . 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 1315 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 1370   T. wtru 1371    e. wcel 1758   E.wrex 2796   _Vcvv 3071    \ cdif 3426    u. cun 3427    i^i cin 3428    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   class class class wbr 4393    |-> cmpt 4451    X. cxp 4939   `'ccnv 4940   ran crn 4942    |` cres 4943   "cima 4944   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   1stc1st 6678   2ndc2nd 6679   RRcr 9385   1c1 9387    + caddc 9389   +oocpnf 9519   RR*cxr 9521    <_ cle 9523    / cdiv 10097   2c2 10475   ZZcz 10750   RR+crp 11095   (,)cioo 11404   [,)cico 11406   ^cexp 11975   topGenctg 14487  sigAlgebracsiga 26688  sigaGencsigagen 26719  𝔅cbrsiga 26733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-ac2 8736  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-ac 8390  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467  df-pi 13469  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-refld 18153  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-fcls 19639  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-cfil 20891  df-cmet 20893  df-cms 20971  df-limc 21467  df-dv 21468  df-log 22134  df-cxp 22135  df-logb 26588  df-siga 26689  df-sigagen 26720  df-brsiga 26734
This theorem is referenced by:  sxbrsigalem4  26839
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