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Theorem dya2iocnrect 29176
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 2541 . . . . 5  |-  ( A  e.  B  <->  A  e.  ran  ( e  e.  ran  (,)
,  f  e.  ran  (,)  |->  ( e  X.  f
) ) )
3 eqid 2471 . . . . . 6  |-  ( e  e.  ran  (,) , 
f  e.  ran  (,)  |->  ( e  X.  f
) )  =  ( e  e.  ran  (,) ,  f  e.  ran  (,)  |->  ( e  X.  f
) )
4 vex 3034 . . . . . . 7  |-  e  e. 
_V
5 vex 3034 . . . . . . 7  |-  f  e. 
_V
64, 5xpex 6614 . . . . . 6  |-  ( e  X.  f )  e. 
_V
73, 6elrnmpt2 6428 . . . . 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 202 . . . 4  |-  ( A  e.  B  ->  E. e  e.  ran  (,) E. f  e.  ran  (,) A  =  ( e  X.  f
) )
983ad2ant2 1052 . . 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 1030 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
11 simp3 1032 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  e.  B  /\  X  e.  A )  ->  X  e.  A )
129, 10, 11jca32 544 . 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 2930 . . 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 211 . 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 772 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( RR  X.  RR ) )
16 simpl 464 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  A  =  ( e  X.  f
) )
17 simprr 774 . . . . . . 7  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  A )
1817, 16eleqtrd 2551 . . . . . 6  |-  ( ( A  =  ( e  X.  f )  /\  ( X  e.  ( RR  X.  RR )  /\  X  e.  A )
)  ->  X  e.  ( e  X.  f
) )
1915, 16, 183jca 1210 . . . . 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 468 . . . . . 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 6842 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 1st `  X )  e.  RR )
22213ad2ant1 1051 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 1st `  X )  e.  RR )
2322adantl 473 . . . . . . . 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 768 . . . . . . . 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 6842 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 1st `  X )  e.  e )
26253ad2ant3 1053 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 1st `  X )  e.  e )
2726adantl 473 . . . . . . . 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 29173 . . . . . . . 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 1292 . . . . . . 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 6843 . . . . . . . . . 10  |-  ( X  e.  ( RR  X.  RR )  ->  ( 2nd `  X )  e.  RR )
33323ad2ant1 1051 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 2nd `  X )  e.  RR )
3433adantl 473 . . . . . . . 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 770 . . . . . . . 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 6843 . . . . . . . . . 10  |-  ( X  e.  ( e  X.  f )  ->  ( 2nd `  X )  e.  f )
37363ad2ant3 1053 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  A  =  ( e  X.  f )  /\  X  e.  ( e  X.  f
) )  ->  ( 2nd `  X )  e.  f )
3837adantl 473 . . . . . . . 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 29173 . . . . . . . 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 1292 . . . . . . 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 2944 . . . . . . 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 677 . . . . . 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 2471 . . . . . . . . . . . 12  |-  ( s  X.  t )  =  ( s  X.  t
)
44 xpeq1 4853 . . . . . . . . . . . . . 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 4854 . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . 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 1379 . . . . . . . . . . 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 3034 . . . . . . . . . . . . 13  |-  u  e. 
_V
52 vex 3034 . . . . . . . . . . . . 13  |-  v  e. 
_V
5351, 52xpex 6614 . . . . . . . . . . . 12  |-  ( u  X.  v )  e. 
_V
5450, 53elrnmpt2 6428 . . . . . . . . . . 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 217 . . . . . . . . . 10  |-  ( ( s  e.  ran  I  /\  t  e.  ran  I )  ->  (
s  X.  t )  e.  ran  R )
5655ad2antrl 742 . . . . . . . . 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 4946 . . . . . . . . . . 11  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
58 simpl1 1033 . . . . . . . . . . 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 3416 . . . . . . . . . 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 782 . . . . . . . . . . 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 466 . . . . . . . . . 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 783 . . . . . . . . . . 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 466 . . . . . . . . . 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 6845 . . . . . . . . . . 11  |-  ( X  e.  ( s  X.  t )  <->  ( X  e.  ( _V  X.  _V )  /\  ( ( 1st `  X )  e.  s  /\  ( 2nd `  X
)  e.  t ) ) )
6564biimpri 211 . . . . . . . . . 10  |-  ( ( X  e.  ( _V 
X.  _V )  /\  (
( 1st `  X
)  e.  s  /\  ( 2nd `  X )  e.  t ) )  ->  X  e.  ( s  X.  t ) )
6659, 61, 63, 65syl12anc 1290 . . . . . . . . 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 470 . . . . . . . . . . 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 470 . . . . . . . . . . 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 4945 . . . . . . . . . . 11  |-  ( ( s  C_  e  /\  t  C_  f )  -> 
( s  X.  t
)  C_  ( e  X.  f ) )
7067, 68, 69syl2anc 673 . . . . . . . . . 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 1034 . . . . . . . . . 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 3455 . . . . . . . . 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 2538 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  ( X  e.  b  <->  X  e.  ( s  X.  t
) ) )
74 sseq1 3439 . . . . . . . . . . 11  |-  ( b  =  ( s  X.  t )  ->  (
b  C_  A  <->  ( s  X.  t )  C_  A
) )
7573, 74anbi12d 725 . . . . . . . . . 10  |-  ( b  =  ( s  X.  t )  ->  (
( X  e.  b  /\  b  C_  A
)  <->  ( X  e.  ( s  X.  t
)  /\  ( s  X.  t )  C_  A
) ) )
7675rspcev 3136 . . . . . . . . 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 1290 . . . . . . . 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 616 . . . . . . 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 2877 . . . . . 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 61 . . . . 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 482 . . . 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 441 . . 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 2876 . 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 18 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    C_ wss 3390    X. cxp 4837   ran crn 4840   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   RRcr 9556   1c1 9558    + caddc 9560    / cdiv 10291   2c2 10681   ZZcz 10961   (,)cioo 11660   [,)cico 11662   ^cexp 12310   topGenctg 15414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-refld 19250  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-fcls 21034  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-cfil 22303  df-cmet 22305  df-cms 22381  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-logb 23781
This theorem is referenced by:  dya2iocnei  29177
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