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Theorem dya2iocnrect 24584
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 2468 . . . . . 6  |-  ( A  e.  B  <->  A  e.  ran  ( e  e.  ran  (,)
,  f  e.  ran  (,)  |->  ( e  X.  f
) ) )
3 eqid 2404 . . . . . . 7  |-  ( e  e.  ran  (,) , 
f  e.  ran  (,)  |->  ( e  X.  f
) )  =  ( e  e.  ran  (,) ,  f  e.  ran  (,)  |->  ( e  X.  f
) )
4 vex 2919 . . . . . . . 8  |-  e  e. 
_V
5 vex 2919 . . . . . . . 8  |-  f  e. 
_V
64, 5xpex 4949 . . . . . . 7  |-  ( e  X.  f )  e. 
_V
73, 6elrnmpt2 6142 . . . . . 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 23923 . . 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 2480 . . . . . 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 6335 . . . . . . . . . 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 6335 . . . . . . . . . 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 24581 . . . . . . . 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 6336 . . . . . . . . . 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 6336 . . . . . . . . . 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 24581 . . . . . . . 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 2835 . . . . . . 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 2404 . . . . . . . . . . . 12  |-  ( s  X.  t )  =  ( s  X.  t
)
45 xpeq1 4851 . . . . . . . . . . . . . 14  |-  ( u  =  s  ->  (
u  X.  v )  =  ( s  X.  v ) )
4645eqeq2d 2415 . . . . . . . . . . . . 13  |-  ( u  =  s  ->  (
( s  X.  t
)  =  ( u  X.  v )  <->  ( s  X.  t )  =  ( s  X.  v ) ) )
47 xpeq2 4852 . . . . . . . . . . . . . 14  |-  ( v  =  t  ->  (
s  X.  v )  =  ( s  X.  t ) )
4847eqeq2d 2415 . . . . . . . . . . . . 13  |-  ( v  =  t  ->  (
( s  X.  t
)  =  ( s  X.  v )  <->  ( s  X.  t )  =  ( s  X.  t ) ) )
4946, 48rspc2ev 3020 . . . . . . . . . . . 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 2919 . . . . . . . . . . . . 13  |-  u  e. 
_V
53 vex 2919 . . . . . . . . . . . . 13  |-  v  e. 
_V
5452, 53xpex 4949 . . . . . . . . . . . 12  |-  ( u  X.  v )  e. 
_V
5551, 54elrnmpt2 6142 . . . . . . . . . . 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 4941 . . . . . . . . . . 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 3306 . . . . . . . . . 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 6338 . . . . . . . . . . 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 4940 . . . . . . . . . . 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 3345 . . . . . . . . 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 2465 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  ( X  e.  b  <->  X  e.  ( s  X.  t
) ) )
75 sseq1 3329 . . . . . . . . . . 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 3012 . . . . . . . . 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 2796 . . . . . 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 2795 . 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 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916    C_ wss 3280    X. cxp 4835   ran crn 4838   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   RRcr 8945   1c1 8947    + caddc 8949    / cdiv 9633   2c2 10005   ZZcz 10238   (,)cioo 10872   [,)cico 10874   ^cexp 11337   topGenctg 13620
This theorem is referenced by:  dya2iocnei  24585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-logb 24342
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