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Theorem dya2iocnrect 24631
Description: For any point of an opened rectangle in  ( RR  X.  RR ), there is a closed below opened 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 . . . . . . 7  |-  B  =  ran  ( e  e. 
ran  (,) ,  f  e. 
ran  (,)  |->  ( e  X.  f ) )
21eleq2i 2500 . . . . . 6  |-  ( A  e.  B  <->  A  e.  ran  ( e  e.  ran  (,)
,  f  e.  ran  (,)  |->  ( e  X.  f
) ) )
3 eqid 2436 . . . . . . 7  |-  ( e  e.  ran  (,) , 
f  e.  ran  (,)  |->  ( e  X.  f
) )  =  ( e  e.  ran  (,) ,  f  e.  ran  (,)  |->  ( e  X.  f
) )
4 vex 2959 . . . . . . . 8  |-  e  e. 
_V
5 vex 2959 . . . . . . . 8  |-  f  e. 
_V
64, 5xpex 4990 . . . . . . 7  |-  ( e  X.  f )  e. 
_V
73, 6elrnmpt2 6183 . . . . . 6  |-  ( 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, 7bitri 241 . . . . 5  |-  ( A  e.  B  <->  E. e  e.  ran  (,) E. f  e.  ran  (,) A  =  ( e  X.  f
) )
98biimpi 187 . . . 4  |-  ( A  e.  B  ->  E. e  e.  ran  (,) E. f  e.  ran  (,) A  =  ( e  X.  f
) )
1093ad2ant2 979 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  E. e  e.  ran  (,)
E. f  e.  ran  (,) A  =  ( e  X.  f ) )
11 simp1 957 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
12 simp3 959 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  A )
1310, 11, 12jca32 522 . 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
) ) )
14 r19.41vv 23970 . . 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 ) ) )
1514biimpri 198 . 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 ) ) )
16 simprl 733 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( RR  X.  RR ) )
17 simpl 444 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  A  =  ( e  X.  f
) )
18 simprr 734 . . . . . . 7  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  A )
1918, 17eleqtrd 2512 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( e  X.  f
) )
2016, 17, 193jca 1134 . . . . 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 ) ) )
21 simpr 448 . . . . . 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 ) ) )
22 xp1st 6376 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 1st `  X )  e.  RR )
23223ad2ant1 978 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 1st `  X )  e.  RR )
2423adantl 453 . . . . . . . 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 )
25 simpll 731 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
e  e.  ran  (,) )
26 xp1st 6376 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 1st `  X )  e.  e )
27263ad2ant3 980 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 1st `  X )  e.  e )
2827adantl 453 . . . . . . . 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 )
29 sxbrsiga.0 . . . . . . . . 9  |-  J  =  ( topGen `  ran  (,) )
30 dya2ioc.1 . . . . . . . . 9  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3129, 30dya2icoseg2 24628 . . . . . . . 8  |-  ( ( ( 1st `  X
)  e.  RR  /\  e  e.  ran  (,)  /\  ( 1st `  X )  e.  e )  ->  E. s  e.  ran  I ( ( 1st `  X )  e.  s  /\  s  C_  e
) )
3224, 25, 28, 31syl3anc 1184 . . . . . . 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
) )
33 xp2nd 6377 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 2nd `  X )  e.  RR )
34333ad2ant1 978 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 2nd `  X )  e.  RR )
3534adantl 453 . . . . . . . 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 )
36 simplr 732 . . . . . . . 8  |-  ( ( ( e  e.  ran  (,) 
/\  f  e.  ran  (,) )  /\  ( X  e.  ( RR  X.  RR )  /\  A  =  ( e  X.  f
)  /\  X  e.  ( e  X.  f
) ) )  -> 
f  e.  ran  (,) )
37 xp2nd 6377 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 2nd `  X )  e.  f )
38373ad2ant3 980 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 2nd `  X )  e.  f )
3938adantl 453 . . . . . . . 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 )
4029, 30dya2icoseg2 24628 . . . . . . . 8  |-  ( ( ( 2nd `  X
)  e.  RR  /\  f  e.  ran  (,)  /\  ( 2nd `  X )  e.  f )  ->  E. t  e.  ran  I ( ( 2nd `  X )  e.  t  /\  t  C_  f
) )
4135, 36, 39, 40syl3anc 1184 . . . . . . 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
) )
42 reeanv 2875 . . . . . . 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 ) ) )
4332, 41, 42sylanbrc 646 . . . . . 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 ) ) )
44 eqid 2436 . . . . . . . . . . . 12  |-  ( s  X.  t )  =  ( s  X.  t
)
45 xpeq1 4892 . . . . . . . . . . . . . 14  |-  ( u  =  s  ->  (
u  X.  v )  =  ( s  X.  v ) )
4645eqeq2d 2447 . . . . . . . . . . . . 13  |-  ( u  =  s  ->  (
( s  X.  t
)  =  ( u  X.  v )  <->  ( s  X.  t )  =  ( s  X.  v ) ) )
47 xpeq2 4893 . . . . . . . . . . . . . 14  |-  ( v  =  t  ->  (
s  X.  v )  =  ( s  X.  t ) )
4847eqeq2d 2447 . . . . . . . . . . . . 13  |-  ( v  =  t  ->  (
( s  X.  t
)  =  ( s  X.  v )  <->  ( s  X.  t )  =  ( s  X.  t ) ) )
4946, 48rspc2ev 3060 . . . . . . . . . . . 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
) )
5044, 49mp3an3 1268 . . . . . . . . . . 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
) )
51 dya2ioc.2 . . . . . . . . . . . 12  |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I  |->  ( u  X.  v ) )
52 vex 2959 . . . . . . . . . . . . 13  |-  u  e. 
_V
53 vex 2959 . . . . . . . . . . . . 13  |-  v  e. 
_V
5452, 53xpex 4990 . . . . . . . . . . . 12  |-  ( u  X.  v )  e. 
_V
5551, 54elrnmpt2 6183 . . . . . . . . . . 11  |-  ( ( s  X.  t )  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I ( s  X.  t )  =  ( u  X.  v
) )
5650, 55sylibr 204 . . . . . . . . . 10  |-  ( ( s  e.  ran  I  /\  t  e.  ran  I )  ->  (
s  X.  t )  e.  ran  R )
5756ad2antrl 709 . . . . . . . . 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 )
58 xpss 4982 . . . . . . . . . . 11  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
59 simpl1 960 . . . . . . . . . . 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 ) )
6058, 59sseldi 3346 . . . . . . . . . 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 ) )
61 simprrl 741 . . . . . . . . . . 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 ) )
6261simpld 446 . . . . . . . . . 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 )
63 simprrr 742 . . . . . . . . . . 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 ) )
6463simpld 446 . . . . . . . . . 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 )
65 elxp7 6379 . . . . . . . . . . 11  |-  ( X  e.  ( s  X.  t )  <->  ( X  e.  ( _V  X.  _V )  /\  ( ( 1st `  X )  e.  s  /\  ( 2nd `  X
)  e.  t ) ) )
6665biimpri 198 . . . . . . . . . 10  |-  ( ( X  e.  ( _V 
X.  _V )  /\  (
( 1st `  X
)  e.  s  /\  ( 2nd `  X )  e.  t ) )  ->  X  e.  ( s  X.  t ) )
6760, 62, 64, 66syl12anc 1182 . . . . . . . . 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
) )
6861simprd 450 . . . . . . . . . . 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 )
6963simprd 450 . . . . . . . . . . 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 )
70 xpss12 4981 . . . . . . . . . . 11  |-  ( ( s  C_  e  /\  t  C_  f )  -> 
( s  X.  t
)  C_  ( e  X.  f ) )
7168, 69, 70syl2anc 643 . . . . . . . . . 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 ) )
72 simpl2 961 . . . . . . . . . 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 ) )
7371, 72sseqtr4d 3385 . . . . . . . . 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 )
74 eleq2 2497 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  ( X  e.  b  <->  X  e.  ( s  X.  t
) ) )
75 sseq1 3369 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  (
b  C_  A  <->  ( s  X.  t )  C_  A
) )
7674, 75anbi12d 692 . . . . . . . . . 10  |-  ( b  =  ( s  X.  t )  ->  (
( X  e.  b  /\  b  C_  A
)  <->  ( X  e.  ( s  X.  t
)  /\  ( s  X.  t )  C_  A
) ) )
7776rspcev 3052 . . . . . . . . 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 ) )
7857, 67, 73, 77syl12anc 1182 . . . . . . . 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 ) )
7978exp32 589 . . . . . . 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
) ) ) )
8079rexlimdvv 2836 . . . . . 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
) ) )
8121, 43, 80sylc 58 . . . . 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
) )
8220, 81sylan2 461 . . . 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
) )
8382ex 424 . . 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
) ) )
8483rexlimivv 2835 . 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 ) )
8513, 15, 843syl 19 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956    C_ wss 3320    X. cxp 4876   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   RRcr 8989   1c1 8991    + caddc 8993    / cdiv 9677   2c2 10049   ZZcz 10282   (,)cioo 10916   [,)cico 10918   ^cexp 11382   topGenctg 13665
This theorem is referenced by:  dya2iocnei  24632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454  df-cxp 20455  df-logb 24389
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