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Theorem sxbrsigalem2 26653
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 26651 . . 3  |-  U. ran  R  =  ( RR  X.  RR )
5 sxbrsigalem0 26638 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( RR  X.  RR )
64, 5eqtr4i 2461 . 2  |-  U. ran  R  =  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )
7 vex 2970 . . . . . 6  |-  u  e. 
_V
8 vex 2970 . . . . . 6  |-  v  e. 
_V
97, 8xpex 6503 . . . . 5  |-  ( u  X.  v )  e. 
_V
103, 9elrnmpt2 6198 . . . 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 26643 . . . . . . . . . . . . 13  |-  ran  I  C_ 𝔅
13 brsigasspwrn 26551 . . . . . . . . . . . . 13  |- 𝔅 
C_  ~P RR
1412, 13sstri 3360 . . . . . . . . . . . 12  |-  ran  I  C_ 
~P RR
1514sseli 3347 . . . . . . . . . . 11  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
1615elpwid 3865 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  C_  RR )
1714sseli 3347 . . . . . . . . . . 11  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
1817elpwid 3865 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  C_  RR )
19 xpinpreima2 26289 . . . . . . . . . 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 9365 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2221mptex 5943 . . . . . . . . . . . . . . . 16  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e. 
_V
2322rnex 6507 . . . . . . . . . . . . . . 15  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  e.  _V
2421mptex 5943 . . . . . . . . . . . . . . . 16  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  e. 
_V
2524rnex 6507 . . . . . . . . . . . . . . 15  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  e.  _V
2623, 25unex 6373 . . . . . . . . . . . . . 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 26537 . . . . . . . . . . . 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 1378 . . . . . . . . . . 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 25952 . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
342, 33elrnmpt2 6198 . . . . . . . . . . . . 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 4858 . . . . . . . . . . . . . . . 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 5258 . . . . . . . . . . . . . . . . . . 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 10739 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
40 2rp 10988 . . . . . . . . . . . . . . . . . . . . . . . . 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 12023 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
4439, 43rerpdivcld 11046 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
4544rexrd 9425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
46 1re 9377 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  RR
4746a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
4839, 47readdcld 9405 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
4948, 43rerpdivcld 11046 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
5049rexrd 9425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
51 pnfxr 11084 . . . . . . . . . . . . . . . . . . . . . 22  |- +oo  e.  RR*
5251a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> +oo  e.  RR* )
5339lep1d 10256 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  <_  ( x  +  1 ) )
5439, 48, 43, 53lediv1dd 11073 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  <_  ( (
x  +  1 )  /  ( 2 ^ n ) ) )
55 pnfge 11102 . . . . . . . . . . . . . . . . . . . . . 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 26024 . . . . . . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  \  (
( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )
)  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) ) )
5958xpeq1d 4858 . . . . . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . . . . . . . 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 3514 . . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( x  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
64 oveq1 6093 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
6564xpeq1d 4858 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( x  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( x  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
6665eqeq2d 2449 . . . . . . . . . . . . . . . . . . . . . . . 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 3068 . . . . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
70 ovex 6111 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( e [,) +oo )  e. 
_V
7170, 21xpex 6503 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
7269, 71elrnmpti 5085 . . . . . . . . . . . . . . . . . . . . . 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 3349 . . . . . . . . . . . . . . . . . . . 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 26542 . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( x  + 
1 )  /  (
2 ^ n ) ) [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  / 
( 2 ^ n
) ) [,) +oo )  X.  RR )
78 oveq1 6093 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
e [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
7978xpeq1d 4858 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( e  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
( e [,) +oo )  X.  RR )  =  ( ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo )  X.  RR ) )
8079eqeq2d 2449 . . . . . . . . . . . . . . . . . . . . . . . 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 3068 . . . . . . . . . . . . . . . . . . . . . . 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 5085 . . . . . . . . . . . . . . . . . . . . . 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 3349 . . . . . . . . . . . . . . . . . . . 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 26542 . . . . . . . . . . . . . . . . . . . 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 26528 . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 2512 . . . . . . . . . . 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 25953 . . . . . . . . . . . . 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 6198 . . . . . . . . . . . . 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 4859 . . . . . . . . . . . . . . . 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 5259 . . . . . . . . . . . . . . . . . . 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 4859 . . . . . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . . . . . . . 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 3515 . . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
108 oveq1 6093 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) )
109108xpeq2d 4859 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( x  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( x  /  ( 2 ^ n ) ) [,) +oo ) ) )
110109eqeq2d 2449 . . . . . . . . . . . . . . . . . . . . . . . 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 3068 . . . . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
114 ovex 6111 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f [,) +oo )  e. 
_V
11521, 114xpex 6503 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
116113, 115elrnmpti 5085 . . . . . . . . . . . . . . . . . . . . . 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 3349 . . . . . . . . . . . . . . . . . . . 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 26542 . . . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( RR 
X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
122 oveq1 6093 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  (
f [,) +oo )  =  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) )
123122xpeq2d 4859 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  =  ( ( x  +  1 )  / 
( 2 ^ n
) )  ->  ( RR  X.  ( f [,) +oo ) )  =  ( RR  X.  ( ( ( x  +  1 )  /  ( 2 ^ n ) ) [,) +oo ) ) )
124123eqeq2d 2449 . . . . . . . . . . . . . . . . . . . . . . . 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 3068 . . . . . . . . . . . . . . . . . . . . . . 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 5085 . . . . . . . . . . . . . . . . . . . . . 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 3349 . . . . . . . . . . . . . . . . . . . 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 26542 . . . . . . . . . . . . . . . . . . . 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 26528 . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 2512 . . . . . . . . . . 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 26530 . . . . . . . . . 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 1218 . . . . . . . . 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 2512 . . . . . . . 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 2512 . . . . . 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 2841 . . . 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 3355 . 2  |-  ran  R  C_  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
151 sigagenss2 26545 . 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 1314 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 1369   T. wtru 1370    e. wcel 1756   E.wrex 2711   _Vcvv 2967    \ cdif 3320    u. cun 3321    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   class class class wbr 4287    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   ran crn 4836    |` cres 4837   "cima 4838   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   RRcr 9273   1c1 9275    + caddc 9277   +oocpnf 9407   RR*cxr 9409    <_ cle 9411    / cdiv 9985   2c2 10363   ZZcz 10638   RR+crp 10983   (,)cioo 11292   [,)cico 11294   ^cexp 11857   topGenctg 14368  sigAlgebracsiga 26502  sigaGencsigagen 26533  𝔅cbrsiga 26547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-ac2 8624  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-ac 8278  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-refld 18010  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-fcls 19489  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-cfil 20741  df-cmet 20743  df-cms 20821  df-limc 21316  df-dv 21317  df-log 21983  df-cxp 21984  df-logb 26402  df-siga 26503  df-sigagen 26534  df-brsiga 26548
This theorem is referenced by:  sxbrsigalem4  26654
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