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Theorem dya2iocnrect 26701
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 2507 . . . . 5  |-  ( A  e.  B  <->  A  e.  ran  ( e  e.  ran  (,)
,  f  e.  ran  (,)  |->  ( e  X.  f
) ) )
3 eqid 2443 . . . . . 6  |-  ( e  e.  ran  (,) , 
f  e.  ran  (,)  |->  ( e  X.  f
) )  =  ( e  e.  ran  (,) ,  f  e.  ran  (,)  |->  ( e  X.  f
) )
4 vex 2980 . . . . . . 7  |-  e  e. 
_V
5 vex 2980 . . . . . . 7  |-  f  e. 
_V
64, 5xpex 6513 . . . . . 6  |-  ( e  X.  f )  e. 
_V
73, 6elrnmpt2 6208 . . . . 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 1010 . . 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 988 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
11 simp3 990 . . 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 2879 . . 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 2519 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( e  X.  f
) )
1915, 16, 183jca 1168 . . . . 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 6611 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 1st `  X )  e.  RR )
22213ad2ant1 1009 . . . . . . . . 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 6611 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 1st `  X )  e.  e )
26253ad2ant3 1011 . . . . . . . . 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 26698 . . . . . . . 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 1218 . . . . . . 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 6612 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 2nd `  X )  e.  RR )
33323ad2ant1 1009 . . . . . . . . 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 6612 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 2nd `  X )  e.  f )
37363ad2ant3 1011 . . . . . . . . 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 26698 . . . . . . . 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 1218 . . . . . . 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 2893 . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( s  X.  t )  =  ( s  X.  t
)
44 xpeq1 4859 . . . . . . . . . . . . . 14  |-  ( u  =  s  ->  (
u  X.  v )  =  ( s  X.  v ) )
4544eqeq2d 2454 . . . . . . . . . . . . 13  |-  ( u  =  s  ->  (
( s  X.  t
)  =  ( u  X.  v )  <->  ( s  X.  t )  =  ( s  X.  v ) ) )
46 xpeq2 4860 . . . . . . . . . . . . . 14  |-  ( v  =  t  ->  (
s  X.  v )  =  ( s  X.  t ) )
4746eqeq2d 2454 . . . . . . . . . . . . 13  |-  ( v  =  t  ->  (
( s  X.  t
)  =  ( s  X.  v )  <->  ( s  X.  t )  =  ( s  X.  t ) ) )
4845, 47rspc2ev 3086 . . . . . . . . . . . 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 1303 . . . . . . . . . . 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 2980 . . . . . . . . . . . . 13  |-  u  e. 
_V
52 vex 2980 . . . . . . . . . . . . 13  |-  v  e. 
_V
5351, 52xpex 6513 . . . . . . . . . . . 12  |-  ( u  X.  v )  e. 
_V
5450, 53elrnmpt2 6208 . . . . . . . . . . 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 4951 . . . . . . . . . . 11  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
58 simpl1 991 . . . . . . . . . . 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 3359 . . . . . . . . . 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 6614 . . . . . . . . . . 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 1216 . . . . . . . . 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 4950 . . . . . . . . . . 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 992 . . . . . . . . . 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 3398 . . . . . . . . 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 2504 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  ( X  e.  b  <->  X  e.  ( s  X.  t
) ) )
74 sseq1 3382 . . . . . . . . . . 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 3078 . . . . . . . . 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 1216 . . . . . . . 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 2852 . . . . . 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 2851 . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2721   _Vcvv 2977    C_ wss 3333    X. cxp 4843   ran crn 4846   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581   RRcr 9286   1c1 9288    + caddc 9290    / cdiv 9998   2c2 10376   ZZcz 10651   (,)cioo 11305   [,)cico 11307   ^cexp 11870   topGenctg 14381
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-refld 18040  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-cmp 18995  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-fcls 19519  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-cfil 20771  df-cmet 20773  df-cms 20851  df-limc 21346  df-dv 21347  df-log 22013  df-cxp 22014  df-logb 26455
This theorem is referenced by:  dya2iocnei  26702
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