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Theorem dya2iocnrect 28077
Description: For any point of an open rectangle in  ( RR  X.  RR ), there is a closed-below open-above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-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 ) )
dya2iocnrect.1  |-  B  =  ran  ( e  e. 
ran  (,) ,  f  e. 
ran  (,)  |->  ( e  X.  f ) )
Assertion
Ref Expression
dya2iocnrect  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) )
Distinct variable groups:    x, n    x, I    v, u, I, x    e, b, f, A    R, b, e, f   
x, b, X, e, f
Allowed substitution hints:    A( x, v, u, n)    B( x, v, u, e, f, n, b)    R( x, v, u, n)    I( e, f, n, b)    J( x, v, u, e, f, n, b)    X( v, u, n)

Proof of Theorem dya2iocnrect
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dya2iocnrect.1 . . . . . 6  |-  B  =  ran  ( e  e. 
ran  (,) ,  f  e. 
ran  (,)  |->  ( e  X.  f ) )
21eleq2i 2545 . . . . 5  |-  ( A  e.  B  <->  A  e.  ran  ( e  e.  ran  (,)
,  f  e.  ran  (,)  |->  ( e  X.  f
) ) )
3 eqid 2467 . . . . . 6  |-  ( e  e.  ran  (,) , 
f  e.  ran  (,)  |->  ( e  X.  f
) )  =  ( e  e.  ran  (,) ,  f  e.  ran  (,)  |->  ( e  X.  f
) )
4 vex 3121 . . . . . . 7  |-  e  e. 
_V
5 vex 3121 . . . . . . 7  |-  f  e. 
_V
64, 5xpex 6599 . . . . . 6  |-  ( e  X.  f )  e. 
_V
73, 6elrnmpt2 6410 . . . . 5  |-  ( A  e.  ran  ( e  e.  ran  (,) , 
f  e.  ran  (,)  |->  ( e  X.  f
) )  <->  E. e  e.  ran  (,) E. f  e.  ran  (,) A  =  ( e  X.  f
) )
82, 7sylbb 197 . . . 4  |-  ( A  e.  B  ->  E. e  e.  ran  (,) E. f  e.  ran  (,) A  =  ( e  X.  f
) )
983ad2ant2 1018 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  E. e  e.  ran  (,)
E. f  e.  ran  (,) A  =  ( e  X.  f ) )
10 simp1 996 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
11 simp3 998 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  A )
129, 10, 11jca32 535 . 2  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  ( E. e  e. 
ran  (,) E. f  e. 
ran  (,) A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A
) ) )
13 r19.41vv 3020 . . 3  |-  ( E. e  e.  ran  (,) E. f  e.  ran  (,) ( A  =  (
e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A
) )  <->  ( E. e  e.  ran  (,) E. f  e.  ran  (,) A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A ) ) )
1413biimpri 206 . 2  |-  ( ( E. e  e.  ran  (,)
E. f  e.  ran  (,) A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  E. e  e.  ran  (,) E. f  e.  ran  (,) ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A ) ) )
15 simprl 755 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( RR  X.  RR ) )
16 simpl 457 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  A  =  ( e  X.  f
) )
17 simprr 756 . . . . . . 7  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  A )
1817, 16eleqtrd 2557 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( e  X.  f
) )
1915, 16, 183jca 1176 . . . . 5  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) ) )
20 simpr 461 . . . . . 6  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) ) )
21 xp1st 6825 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 1st `  X )  e.  RR )
22213ad2ant1 1017 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 1st `  X )  e.  RR )
2322adantl 466 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
( 1st `  X
)  e.  RR )
24 simpll 753 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
e  e.  ran  (,) )
25 xp1st 6825 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 1st `  X )  e.  e )
26253ad2ant3 1019 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 1st `  X )  e.  e )
2726adantl 466 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
( 1st `  X
)  e.  e )
28 sxbrsiga.0 . . . . . . . . 9  |-  J  =  ( topGen `  ran  (,) )
29 dya2ioc.1 . . . . . . . . 9  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3028, 29dya2icoseg2 28074 . . . . . . . 8  |-  ( ( ( 1st `  X
)  e.  RR  /\  e  e.  ran  (,)  /\  ( 1st `  X )  e.  e )  ->  E. s  e.  ran  I ( ( 1st `  X )  e.  s  /\  s  C_  e
) )
3123, 24, 27, 30syl3anc 1228 . . . . . . 7  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  ->  E. s  e.  ran  I ( ( 1st `  X )  e.  s  /\  s  C_  e
) )
32 xp2nd 6826 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 2nd `  X )  e.  RR )
33323ad2ant1 1017 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 2nd `  X )  e.  RR )
3433adantl 466 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
( 2nd `  X
)  e.  RR )
35 simplr 754 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
f  e.  ran  (,) )
36 xp2nd 6826 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 2nd `  X )  e.  f )
37363ad2ant3 1019 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 2nd `  X )  e.  f )
3837adantl 466 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
( 2nd `  X
)  e.  f )
3928, 29dya2icoseg2 28074 . . . . . . . 8  |-  ( ( ( 2nd `  X
)  e.  RR  /\  f  e.  ran  (,)  /\  ( 2nd `  X )  e.  f )  ->  E. t  e.  ran  I ( ( 2nd `  X )  e.  t  /\  t  C_  f
) )
4034, 35, 38, 39syl3anc 1228 . . . . . . 7  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  ->  E. t  e.  ran  I ( ( 2nd `  X )  e.  t  /\  t  C_  f
) )
41 reeanv 3034 . . . . . . 7  |-  ( E. s  e.  ran  I E. t  e.  ran  I ( ( ( 1st `  X )  e.  s  /\  s  C_  e )  /\  (
( 2nd `  X
)  e.  t  /\  t  C_  f ) )  <-> 
( E. s  e. 
ran  I ( ( 1st `  X )  e.  s  /\  s  C_  e )  /\  E. t  e.  ran  I ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) )
4231, 40, 41sylanbrc 664 . . . . . 6  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  ->  E. s  e.  ran  I E. t  e.  ran  I ( ( ( 1st `  X )  e.  s  /\  s  C_  e )  /\  (
( 2nd `  X
)  e.  t  /\  t  C_  f ) ) )
43 eqid 2467 . . . . . . . . . . . 12  |-  ( s  X.  t )  =  ( s  X.  t
)
44 xpeq1 5019 . . . . . . . . . . . . . 14  |-  ( u  =  s  ->  (
u  X.  v )  =  ( s  X.  v ) )
4544eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( u  =  s  ->  (
( s  X.  t
)  =  ( u  X.  v )  <->  ( s  X.  t )  =  ( s  X.  v ) ) )
46 xpeq2 5020 . . . . . . . . . . . . . 14  |-  ( v  =  t  ->  (
s  X.  v )  =  ( s  X.  t ) )
4746eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( v  =  t  ->  (
( s  X.  t
)  =  ( s  X.  v )  <->  ( s  X.  t )  =  ( s  X.  t ) ) )
4845, 47rspc2ev 3230 . . . . . . . . . . . 12  |-  ( ( s  e.  ran  I  /\  t  e.  ran  I  /\  ( s  X.  t )  =  ( s  X.  t ) )  ->  E. u  e.  ran  I E. v  e.  ran  I ( s  X.  t )  =  ( u  X.  v
) )
4943, 48mp3an3 1313 . . . . . . . . . . 11  |-  ( ( s  e.  ran  I  /\  t  e.  ran  I )  ->  E. u  e.  ran  I E. v  e.  ran  I ( s  X.  t )  =  ( u  X.  v
) )
50 dya2ioc.2 . . . . . . . . . . . 12  |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I  |->  ( u  X.  v ) )
51 vex 3121 . . . . . . . . . . . . 13  |-  u  e. 
_V
52 vex 3121 . . . . . . . . . . . . 13  |-  v  e. 
_V
5351, 52xpex 6599 . . . . . . . . . . . 12  |-  ( u  X.  v )  e. 
_V
5450, 53elrnmpt2 6410 . . . . . . . . . . 11  |-  ( ( s  X.  t )  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I ( s  X.  t )  =  ( u  X.  v
) )
5549, 54sylibr 212 . . . . . . . . . 10  |-  ( ( s  e.  ran  I  /\  t  e.  ran  I )  ->  (
s  X.  t )  e.  ran  R )
5655ad2antrl 727 . . . . . . . . 9  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  (
s  X.  t )  e.  ran  R )
57 xpss 5115 . . . . . . . . . . 11  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
58 simpl1 999 . . . . . . . . . . 11  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  X  e.  ( RR  X.  RR ) )
5957, 58sseldi 3507 . . . . . . . . . 10  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  X  e.  ( _V  X.  _V ) )
60 simprrl 763 . . . . . . . . . . 11  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  (
( 1st `  X
)  e.  s  /\  s  C_  e ) )
6160simpld 459 . . . . . . . . . 10  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  ( 1st `  X )  e.  s )
62 simprrr 764 . . . . . . . . . . 11  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  (
( 2nd `  X
)  e.  t  /\  t  C_  f ) )
6362simpld 459 . . . . . . . . . 10  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  ( 2nd `  X )  e.  t )
64 elxp7 6828 . . . . . . . . . . 11  |-  ( X  e.  ( s  X.  t )  <->  ( X  e.  ( _V  X.  _V )  /\  ( ( 1st `  X )  e.  s  /\  ( 2nd `  X
)  e.  t ) ) )
6564biimpri 206 . . . . . . . . . 10  |-  ( ( X  e.  ( _V 
X.  _V )  /\  (
( 1st `  X
)  e.  s  /\  ( 2nd `  X )  e.  t ) )  ->  X  e.  ( s  X.  t ) )
6659, 61, 63, 65syl12anc 1226 . . . . . . . . 9  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  X  e.  ( s  X.  t
) )
6760simprd 463 . . . . . . . . . . 11  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  s  C_  e )
6862simprd 463 . . . . . . . . . . 11  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  t  C_  f )
69 xpss12 5114 . . . . . . . . . . 11  |-  ( ( s  C_  e  /\  t  C_  f )  -> 
( s  X.  t
)  C_  ( e  X.  f ) )
7067, 68, 69syl2anc 661 . . . . . . . . . 10  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  (
s  X.  t ) 
C_  ( e  X.  f ) )
71 simpl2 1000 . . . . . . . . . 10  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  A  =  ( e  X.  f ) )
7270, 71sseqtr4d 3546 . . . . . . . . 9  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  (
s  X.  t ) 
C_  A )
73 eleq2 2540 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  ( X  e.  b  <->  X  e.  ( s  X.  t
) ) )
74 sseq1 3530 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  (
b  C_  A  <->  ( s  X.  t )  C_  A
) )
7573, 74anbi12d 710 . . . . . . . . . 10  |-  ( b  =  ( s  X.  t )  ->  (
( X  e.  b  /\  b  C_  A
)  <->  ( X  e.  ( s  X.  t
)  /\  ( s  X.  t )  C_  A
) ) )
7675rspcev 3219 . . . . . . . . 9  |-  ( ( ( s  X.  t
)  e.  ran  R  /\  ( X  e.  ( s  X.  t )  /\  ( s  X.  t )  C_  A
) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
7756, 66, 72, 76syl12anc 1226 . . . . . . . 8  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f ) )  /\  ( ( s  e.  ran  I  /\  t  e.  ran  I )  /\  (
( ( 1st `  X
)  e.  s  /\  s  C_  e )  /\  ( ( 2nd `  X
)  e.  t  /\  t  C_  f ) ) ) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
7877exp32 605 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  (
( s  e.  ran  I  /\  t  e.  ran  I )  ->  (
( ( ( 1st `  X )  e.  s  /\  s  C_  e
)  /\  ( ( 2nd `  X )  e.  t  /\  t  C_  f ) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) ) ) )
7978rexlimdvv 2965 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( E. s  e.  ran  I E. t  e.  ran  I ( ( ( 1st `  X )  e.  s  /\  s  C_  e )  /\  (
( 2nd `  X
)  e.  t  /\  t  C_  f ) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) ) )
8020, 42, 79sylc 60 . . . . 5  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) )
8119, 80sylan2 474 . . . 4  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A ) ) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) )
8281ex 434 . . 3  |-  ( ( e  e.  ran  (,)  /\  f  e.  ran  (,) )  ->  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A ) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) ) )
8382rexlimivv 2964 . 2  |-  ( E. e  e.  ran  (,) E. f  e.  ran  (,) ( A  =  (
e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A
) )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A ) )
8412, 14, 833syl 20 1  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  E. b  e.  ran  R ( X  e.  b  /\  b  C_  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118    C_ wss 3481    X. cxp 5003   ran crn 5006   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   RRcr 9503   1c1 9505    + caddc 9507    / cdiv 10218   2c2 10597   ZZcz 10876   (,)cioo 11541   [,)cico 11543   ^cexp 12146   topGenctg 14710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-refld 18510  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-fcls 20310  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-cfil 21562  df-cmet 21564  df-cms 21642  df-limc 22138  df-dv 22139  df-log 22810  df-cxp 22811  df-logb 27832
This theorem is referenced by:  dya2iocnei  28078
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