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Theorem tpr2rico 26486
Description: For any point of an open set of the usual topology on  ( RR  X.  RR ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the  (
l ^ +oo ) norm  X. (Contributed by Thierry Arnoux, 21-Sep-2017.)
Hypotheses
Ref Expression
tpr2rico.0  |-  J  =  ( topGen `  ran  (,) )
tpr2rico.1  |-  G  =  ( u  e.  RR ,  v  e.  RR  |->  ( u  +  (
_i  x.  v )
) )
tpr2rico.2  |-  B  =  ran  ( x  e. 
ran  (,) ,  y  e. 
ran  (,)  |->  ( x  X.  y ) )
Assertion
Ref Expression
tpr2rico  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
Distinct variable groups:    v, u, x, y    x, r, A    B, r    x, G    x, J    x, X    y, r, X
Allowed substitution hints:    A( y, v, u)    B( x, y, v, u)    G( y, v, u, r)    J( y, v, u, r)    X( v, u)

Proof of Theorem tpr2rico
Dummy variables  z  m  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 11414 . . . . . . . . . 10  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
21ixxf 11420 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5666 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR*  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3mp1i 12 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  (,)  Fn  ( RR*  X.  RR* ) )
5 elssuni 4228 . . . . . . . . . . . . . 14  |-  ( A  e.  ( J  tX  J )  ->  A  C_ 
U. ( J  tX  J ) )
6 tpr2rico.0 . . . . . . . . . . . . . . . 16  |-  J  =  ( topGen `  ran  (,) )
7 retop 20471 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  e.  Top
86, 7eqeltri 2538 . . . . . . . . . . . . . . 15  |-  J  e. 
Top
9 uniretop 20472 . . . . . . . . . . . . . . . 16  |-  RR  =  U. ( topGen `  ran  (,) )
106unieqi 4207 . . . . . . . . . . . . . . . 16  |-  U. J  =  U. ( topGen `  ran  (,) )
119, 10eqtr4i 2486 . . . . . . . . . . . . . . 15  |-  RR  =  U. J
128, 8, 11, 11txunii 19297 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
135, 12syl6sseqr 3510 . . . . . . . . . . . . 13  |-  ( A  e.  ( J  tX  J )  ->  A  C_  ( RR  X.  RR ) )
1413ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  A  C_  ( RR  X.  RR ) )
15 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  A )
1614, 15sseldd 3464 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
17 xp1st 6715 . . . . . . . . . . 11  |-  ( X  e.  ( RR  X.  RR )  ->  ( 1st `  X )  e.  RR )
1816, 17syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR )
19 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2019rpred 11137 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR )
2120rehalfcld 10681 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR )
2218, 21resubcld 9886 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR )
2322rexrd 9543 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR* )
2418, 21readdcld 9523 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR )
2524rexrd 9543 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR* )
26 fnovrn 6347 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
( 1st `  X
)  -  ( d  /  2 ) )  e.  RR*  /\  (
( 1st `  X
)  +  ( d  /  2 ) )  e.  RR* )  ->  (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
274, 23, 25, 26syl3anc 1219 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
28 xp2nd 6716 . . . . . . . . . . 11  |-  ( X  e.  ( RR  X.  RR )  ->  ( 2nd `  X )  e.  RR )
2916, 28syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR )
3029, 21resubcld 9886 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR )
3130rexrd 9543 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR* )
3229, 21readdcld 9523 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR )
3332rexrd 9543 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR* )
34 fnovrn 6347 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
( 2nd `  X
)  -  ( d  /  2 ) )  e.  RR*  /\  (
( 2nd `  X
)  +  ( d  /  2 ) )  e.  RR* )  ->  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
354, 31, 33, 34syl3anc 1219 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
36 eqidd 2455 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
37 xpeq1 4961 . . . . . . . . 9  |-  ( x  =  ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  ->  ( x  X.  y )  =  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  y ) )
3837eqeq2d 2468 . . . . . . . 8  |-  ( x  =  ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  ->  ( (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  =  ( x  X.  y )  <-> 
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  y ) ) )
39 xpeq2 4962 . . . . . . . . 9  |-  ( y  =  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  y
)  =  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
4039eqeq2d 2468 . . . . . . . 8  |-  ( y  =  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) )  ->  ( (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  =  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  y )  <->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) ) )
4138, 40rspc2ev 3186 . . . . . . 7  |-  ( ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  e.  ran  (,)  /\  ( ( ( 2nd `  X )  -  (
d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  / 
2 ) ) )  e.  ran  (,)  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  E. x  e.  ran  (,)
E. y  e.  ran  (,) ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( x  X.  y
) )
4227, 35, 36, 41syl3anc 1219 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  E. x  e.  ran  (,) E. y  e.  ran  (,) ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( x  X.  y
) )
43 eqid 2454 . . . . . . 7  |-  ( x  e.  ran  (,) , 
y  e.  ran  (,)  |->  ( x  X.  y
) )  =  ( x  e.  ran  (,) ,  y  e.  ran  (,)  |->  ( x  X.  y
) )
44 vex 3079 . . . . . . . 8  |-  x  e. 
_V
45 vex 3079 . . . . . . . 8  |-  y  e. 
_V
4644, 45xpex 6617 . . . . . . 7  |-  ( x  X.  y )  e. 
_V
4743, 46elrnmpt2 6312 . . . . . 6  |-  ( ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  e.  ran  ( x  e.  ran  (,)
,  y  e.  ran  (,)  |->  ( x  X.  y
) )  <->  E. x  e.  ran  (,) E. y  e.  ran  (,) ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  =  ( x  X.  y
) )
4842, 47sylibr 212 . . . . 5  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e. 
ran  ( x  e. 
ran  (,) ,  y  e. 
ran  (,)  |->  ( x  X.  y ) ) )
49 tpr2rico.2 . . . . 5  |-  B  =  ran  ( x  e. 
ran  (,) ,  y  e. 
ran  (,)  |->  ( x  X.  y ) )
5048, 49syl6eleqr 2553 . . . 4  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B )
5150ralrimiva 2829 . . 3  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  A. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B )
52 xpss 5053 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
5352, 16sseldi 3461 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( _V  X.  _V )
)
5418rexrd 9543 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR* )
5519rphalfcld 11149 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR+ )
5618, 55ltsubrpd 11165 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X ) )
5718, 55ltaddrpd 11166 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) )
58 elioo1 11450 . . . . . . . . 9  |-  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) )  e.  RR*  /\  (
( 1st `  X
)  +  ( d  /  2 ) )  e.  RR* )  ->  (
( 1st `  X
)  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  <->  ( ( 1st `  X )  e. 
RR*  /\  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X )  /\  ( 1st `  X )  <  ( ( 1st `  X )  +  ( d  /  2 ) ) ) ) )
5923, 25, 58syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  e.  ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  <-> 
( ( 1st `  X
)  e.  RR*  /\  (
( 1st `  X
)  -  ( d  /  2 ) )  <  ( 1st `  X
)  /\  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) ) ) )
6054, 56, 57, 59mpbir3and 1171 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) ) )
6129rexrd 9543 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR* )
6229, 55ltsubrpd 11165 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X ) )
6329, 55ltaddrpd 11166 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) )
64 elioo1 11450 . . . . . . . . 9  |-  ( ( ( ( 2nd `  X
)  -  ( d  /  2 ) )  e.  RR*  /\  (
( 2nd `  X
)  +  ( d  /  2 ) )  e.  RR* )  ->  (
( 2nd `  X
)  e.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  <->  ( ( 2nd `  X )  e. 
RR*  /\  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X )  /\  ( 2nd `  X )  <  ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
6531, 33, 64syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  e.  ( ( ( 2nd `  X )  -  (
d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  / 
2 ) ) )  <-> 
( ( 2nd `  X
)  e.  RR*  /\  (
( 2nd `  X
)  -  ( d  /  2 ) )  <  ( 2nd `  X
)  /\  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) ) ) )
6661, 62, 63, 65mpbir3and 1171 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )
6760, 66jca 532 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  e.  ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  /\  ( 2nd `  X
)  e.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
68 elxp7 6718 . . . . . 6  |-  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  <->  ( X  e.  ( _V  X.  _V )  /\  ( ( 1st `  X )  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  /\  ( 2nd `  X )  e.  ( ( ( 2nd `  X )  -  (
d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  / 
2 ) ) ) ) ) )
6953, 67, 68sylanbrc 664 . . . . 5  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
7069ralrimiva 2829 . . . 4  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  A. d  e.  RR+  X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
71 mnfle 11223 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  X
)  -  ( d  /  2 ) )  e.  RR*  -> -oo  <_  ( ( 1st `  X
)  -  ( d  /  2 ) ) )
7223, 71syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  -> -oo  <_  (
( 1st `  X
)  -  ( d  /  2 ) ) )
73 pnfge 11220 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  X
)  +  ( d  /  2 ) )  e.  RR*  ->  ( ( 1st `  X )  +  ( d  / 
2 ) )  <_ +oo )
7425, 73syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  <_ +oo )
75 mnfxr 11204 . . . . . . . . . . . . . . . . . 18  |- -oo  e.  RR*
76 pnfxr 11202 . . . . . . . . . . . . . . . . . 18  |- +oo  e.  RR*
77 ioossioo 11497 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <_  ( ( 1st `  X )  -  (
d  /  2 ) )  /\  ( ( 1st `  X )  +  ( d  / 
2 ) )  <_ +oo ) )  ->  (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
7875, 76, 77mpanl12 682 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  <_  ( ( 1st `  X )  -  ( d  /  2
) )  /\  (
( 1st `  X
)  +  ( d  /  2 ) )  <_ +oo )  ->  (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
7972, 74, 78syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
80 ioomax 11480 . . . . . . . . . . . . . . . 16  |-  ( -oo (,) +oo )  =  RR
8179, 80syl6sseq 3509 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  RR )
82 mnfle 11223 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  X
)  -  ( d  /  2 ) )  e.  RR*  -> -oo  <_  ( ( 2nd `  X
)  -  ( d  /  2 ) ) )
8331, 82syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  -> -oo  <_  (
( 2nd `  X
)  -  ( d  /  2 ) ) )
84 pnfge 11220 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  X
)  +  ( d  /  2 ) )  e.  RR*  ->  ( ( 2nd `  X )  +  ( d  / 
2 ) )  <_ +oo )
8533, 84syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  <_ +oo )
86 ioossioo 11497 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <_  ( ( 2nd `  X )  -  (
d  /  2 ) )  /\  ( ( 2nd `  X )  +  ( d  / 
2 ) )  <_ +oo ) )  ->  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
8775, 76, 86mpanl12 682 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  <_  ( ( 2nd `  X )  -  ( d  /  2
) )  /\  (
( 2nd `  X
)  +  ( d  /  2 ) )  <_ +oo )  ->  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
8883, 85, 87syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
8988, 80syl6sseq 3509 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )
90 xpss12 5052 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) ) 
C_  RR  /\  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )  ->  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( RR  X.  RR ) )
9181, 89, 90syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( RR  X.  RR ) )
9291sselda 3463 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  x  e.  ( RR  X.  RR ) )
9392expcom 435 . . . . . . . . . . . 12  |-  ( x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  ->  x  e.  ( RR  X.  RR ) ) )
9493ancld 553 . . . . . . . . . . 11  |-  ( x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  ->  (
( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) ) ) )
9594imdistanri 691 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) ) )
9613adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  A  C_  ( RR  X.  RR ) )
97 simpr1 994 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  A )
9896, 97sseldd 3464 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  ( RR  X.  RR ) )
99983anassrs 1210 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  X  e.  ( RR  X.  RR ) )
100 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  x  e.  ( RR  X.  RR ) )
101 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR+ )
102101rphalfcld 11149 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
d  /  2 )  e.  RR+ )
103 tpr2rico.1 . . . . . . . . . . . . . . 15  |-  G  =  ( u  e.  RR ,  v  e.  RR  |->  ( u  +  (
_i  x.  v )
) )
104103cnre2csqima 26485 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( RR 
X.  RR )  /\  x  e.  ( RR  X.  RR )  /\  (
d  /  2 )  e.  RR+ )  ->  (
x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) ) ) )
10599, 100, 102, 104syl3anc 1219 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) ) ) )
106 eqid 2454 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107103, 6, 106cnrehmeo 20656 . . . . . . . . . . . . . . . . . . . 20  |-  G  e.  ( ( J  tX  J ) Homeo ( TopOpen ` fld )
)
108106cnfldtopon 20493 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
109108toponunii 18668 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  =  U. ( TopOpen ` fld )
11012, 109hmeof1o 19468 . . . . . . . . . . . . . . . . . . . 20  |-  ( G  e.  ( ( J 
tX  J ) Homeo (
TopOpen ` fld ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
111 f1of 5748 . . . . . . . . . . . . . . . . . . . 20  |-  ( G : ( RR  X.  RR ) -1-1-onto-> CC  ->  G :
( RR  X.  RR )
--> CC )
112107, 110, 111mp2b 10 . . . . . . . . . . . . . . . . . . 19  |-  G :
( RR  X.  RR )
--> CC
113112a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  G : ( RR  X.  RR ) --> CC )
114113, 99ffvelrnd 5952 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  X )  e.  CC )
115112a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  G :
( RR  X.  RR )
--> CC )
116115ffvelrnda 5951 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  CC )
117 sqsscirc2 26483 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G `  X )  e.  CC  /\  ( G `  x
)  e.  CC )  /\  d  e.  RR+ )  ->  ( ( ( abs `  ( Re
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
)  /\  ( abs `  ( Im `  (
( G `  x
)  -  ( G `
 X ) ) ) )  <  (
d  /  2 ) )  ->  ( abs `  ( ( G `  x )  -  ( G `  X )
) )  <  d
) )
118114, 116, 101, 117syl21anc 1218 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( ( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) )  ->  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
) )
119118imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  (
( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) ) )  -> 
( abs `  (
( G `  x
)  -  ( G `
 X ) ) )  <  d )
120101rpxrd 11138 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR* )
121120adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  d  e.  RR* )
122 cnxmet 20483 . . . . . . . . . . . . . . . . 17  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123121, 122jctil 537 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  d  e.  RR* ) )
124114adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( G `  X )  e.  CC )
125116adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( G `  x )  e.  CC )
126124, 125jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( G `  X )  e.  CC  /\  ( G `
 x )  e.  CC ) )
127 eqid 2454 . . . . . . . . . . . . . . . . . . 19  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
128127cnmetdval 20481 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( G `  x
)  e.  CC  /\  ( G `  X )  e.  CC )  -> 
( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  =  ( abs `  ( ( G `  x )  -  ( G `  X )
) ) )
129125, 124, 128syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( G `  x )
( abs  o.  -  )
( G `  X
) )  =  ( abs `  ( ( G `  x )  -  ( G `  X ) ) ) )
130 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( abs `  ( ( G `  x )  -  ( G `  X )
) )  <  d
)
131129, 130eqbrtrd 4419 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( G `  x )
( abs  o.  -  )
( G `  X
) )  <  d
)
132 elbl3 20098 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  d  e.  RR* )  /\  ( ( G `
 X )  e.  CC  /\  ( G `
 x )  e.  CC ) )  -> 
( ( G `  x )  e.  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  <->  ( ( G `  x )
( abs  o.  -  )
( G `  X
) )  <  d
) )
133132biimpar 485 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( abs 
o.  -  )  e.  ( *Met `  CC )  /\  d  e.  RR* )  /\  ( ( G `
 X )  e.  CC  /\  ( G `
 x )  e.  CC ) )  /\  ( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  <  d )  ->  ( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )
134123, 126, 131, 133syl21anc 1218 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( G `  x )  e.  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
135119, 134syldan 470 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  (
( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) ) )  -> 
( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )
136135ex 434 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( ( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) )  ->  ( G `  x )  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
137105, 136syld 44 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
138 f1ocnv 5760 . . . . . . . . . . . . . . 15  |-  ( G : ( RR  X.  RR ) -1-1-onto-> CC  ->  `' G : CC -1-1-onto-> ( RR  X.  RR ) )
139107, 110, 138mp2b 10 . . . . . . . . . . . . . 14  |-  `' G : CC -1-1-onto-> ( RR  X.  RR )
140 f1ofun 5750 . . . . . . . . . . . . . 14  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  Fun  `' G
)
141139, 140ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  `' G
142 f1odm 5752 . . . . . . . . . . . . . . 15  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  dom  `' G  =  CC )
143139, 142ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  `' G  =  CC
144116, 143syl6eleqr 2553 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  dom  `' G )
145 funfvima 6060 . . . . . . . . . . . . 13  |-  ( ( Fun  `' G  /\  ( G `  x )  e.  dom  `' G
)  ->  ( ( G `  x )  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d )  ->  ( `' G `  ( G `  x
) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) ) )
146141, 144, 145sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d )  ->  ( `' G `  ( G `
 x ) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) ) )
147107, 110mp1i 12 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
148 f1ocnvfv1 6091 . . . . . . . . . . . . . . 15  |-  ( ( G : ( RR 
X.  RR ) -1-1-onto-> CC  /\  x  e.  ( RR  X.  RR ) )  -> 
( `' G `  ( G `  x ) )  =  x )
149147, 100, 148syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( `' G `  ( G `
 x ) )  =  x )
150149eleq1d 2523 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( `' G `  ( G `  x ) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  <->  x  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) ) )
151150biimpd 207 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( `' G `  ( G `  x ) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  ->  x  e.  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) ) ) )
152137, 146, 1513syld 55 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  ->  x  e.  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) ) ) )
153152imp 429 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  x  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
15495, 153syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  x  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
155154ex 434 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  ->  x  e.  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) ) ) )
156155ssrdv 3469 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
157156ralrimiva 2829 . . . . . 6  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  A. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
158103mpt2fun 6301 . . . . . . . . . 10  |-  Fun  G
159158a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  Fun  G )
16013sselda 3463 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
161 f1odm 5752 . . . . . . . . . . 11  |-  ( G : ( RR  X.  RR ) -1-1-onto-> CC  ->  dom  G  =  ( RR  X.  RR ) )
162107, 110, 161mp2b 10 . . . . . . . . . 10  |-  dom  G  =  ( RR  X.  RR )
163160, 162syl6eleqr 2553 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  dom  G
)
164 simpr 461 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  A )
165 funfvima 6060 . . . . . . . . . 10  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( X  e.  A  ->  ( G `  X
)  e.  ( G
" A ) ) )
166165imp 429 . . . . . . . . 9  |-  ( ( ( Fun  G  /\  X  e.  dom  G )  /\  X  e.  A
)  ->  ( G `  X )  e.  ( G " A ) )
167159, 163, 164, 166syl21anc 1218 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( G `  X
)  e.  ( G
" A ) )
168 hmeoima 19469 . . . . . . . . . . 11  |-  ( ( G  e.  ( ( J  tX  J )
Homeo ( TopOpen ` fld ) )  /\  A  e.  ( J  tX  J
) )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
169107, 168mpan 670 . . . . . . . . . 10  |-  ( A  e.  ( J  tX  J )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
170106cnfldtopn 20492 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
171170elmopn2 20151 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  e.  ( *Met `  CC )  ->  ( ( G " A )  e.  ( TopOpen ` fld )  <->  ( ( G
" A )  C_  CC  /\  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) ) ) )
172122, 171ax-mp 5 . . . . . . . . . . 11  |-  ( ( G " A )  e.  ( TopOpen ` fld )  <->  ( ( G
" A )  C_  CC  /\  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) ) )
173172simprbi 464 . . . . . . . . . 10  |-  ( ( G " A )  e.  ( TopOpen ` fld )  ->  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
174169, 173syl 16 . . . . . . . . 9  |-  ( A  e.  ( J  tX  J )  ->  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
175174adantr 465 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  A. m  e.  ( G " A ) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
176 oveq1 6206 . . . . . . . . . . 11  |-  ( m  =  ( G `  X )  ->  (
m ( ball `  ( abs  o.  -  ) ) d )  =  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
177176sseq1d 3490 . . . . . . . . . 10  |-  ( m  =  ( G `  X )  ->  (
( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  <->  ( ( G `  X )
( ball `  ( abs  o. 
-  ) ) d )  C_  ( G " A ) ) )
178177rexbidv 2864 . . . . . . . . 9  |-  ( m  =  ( G `  X )  ->  ( E. d  e.  RR+  (
m ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  <->  E. d  e.  RR+  ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d )  C_  ( G " A ) ) )
179178rspcva 3175 . . . . . . . 8  |-  ( ( ( G `  X
)  e.  ( G
" A )  /\  A. m  e.  ( G
" A ) E. d  e.  RR+  (
m ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A ) )  ->  E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
180167, 175, 179syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
181 imass2 5311 . . . . . . . . . 10  |-  ( ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  -> 
( `' G "
( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  ( `' G " ( G " A ) ) )
182 f1of1 5747 . . . . . . . . . . . . 13  |-  ( G : ( RR  X.  RR ) -1-1-onto-> CC  ->  G :
( RR  X.  RR ) -1-1-> CC )
183107, 110, 182mp2b 10 . . . . . . . . . . . 12  |-  G :
( RR  X.  RR ) -1-1-> CC
184 f1imacnv 5764 . . . . . . . . . . . 12  |-  ( ( G : ( RR 
X.  RR ) -1-1-> CC  /\  A  C_  ( RR  X.  RR ) )  -> 
( `' G "
( G " A
) )  =  A )
185183, 13, 184sylancr 663 . . . . . . . . . . 11  |-  ( A  e.  ( J  tX  J )  ->  ( `' G " ( G
" A ) )  =  A )
186185sseq2d 3491 . . . . . . . . . 10  |-  ( A  e.  ( J  tX  J )  ->  (
( `' G "
( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  ( `' G " ( G " A ) )  <->  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) )  C_  A
) )
187181, 186syl5ib 219 . . . . . . . . 9  |-  ( A  e.  ( J  tX  J )  ->  (
( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  ->  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
188187reximdv 2931 . . . . . . . 8  |-  ( A  e.  ( J  tX  J )  ->  ( E. d  e.  RR+  (
( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  ->  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
189188adantr 465 . . . . . . 7  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  ->  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )
)
190180, 189mpd 15 . . . . . 6  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )
191 r19.29 2961 . . . . . 6  |-  ( ( A. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )  ->  E. d  e.  RR+  ( ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
192157, 190, 191syl2anc 661 . . . . 5  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
193 sstr 3471 . . . . . 6  |-  ( ( ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )  ->  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
)
194193reximi 2927 . . . . 5  |-  ( E. d  e.  RR+  (
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )  ->  E. d  e.  RR+  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A )
195192, 194syl 16 . . . 4  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A )
196 r19.29 2961 . . . 4  |-  ( ( A. d  e.  RR+  X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  E. d  e.  RR+  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
)  ->  E. d  e.  RR+  ( X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )
19770, 195, 196syl2anc 661 . . 3  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( X  e.  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  /\  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
) )
198 r19.29 2961 . . 3  |-  ( ( A. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  E. d  e.  RR+  ( X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )  ->  E. d  e.  RR+  (
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) ) )
19951, 197, 198syl2anc 661 . 2  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) ) )
200 eleq2 2527 . . . . 5  |-  ( r  =  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( X  e.  r  <-> 
X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) ) )
201 sseq1 3484 . . . . 5  |-  ( r  =  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( r  C_  A  <->  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
) )
202200, 201anbi12d 710 . . . 4  |-  ( r  =  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( X  e.  r  /\  r  C_  A )  <->  ( X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) ) )
203202rspcev 3177 . . 3  |-  ( ( ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
204203rexlimivw 2941 . 2  |-  ( E. d  e.  RR+  (
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
205199, 204syl 16 1  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2798   E.wrex 2799   _Vcvv 3076    C_ wss 3435   ~Pcpw 3967   U.cuni 4198   class class class wbr 4399    X. cxp 4945   `'ccnv 4946   dom cdm 4947   ran crn 4948   "cima 4950    o. ccom 4951   Fun wfun 5519    Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   1stc1st 6684   2ndc2nd 6685   CCcc 9390   RRcr 9391   _ici 9394    + caddc 9395    x. cmul 9397   +oocpnf 9525   -oocmnf 9526   RR*cxr 9527    < clt 9528    <_ cle 9529    - cmin 9705    / cdiv 10103   2c2 10481   RR+crp 11101   (,)cioo 11410   Recre 12703   Imcim 12704   abscabs 12840   TopOpenctopn 14478   topGenctg 14494   *Metcxmt 17925   ballcbl 17927  ℂfldccnfld 17942   Topctop 18629    tX ctx 19264   Homeochmeo 19457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cn 18962  df-cnp 18963  df-tx 19266  df-hmeo 19459  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585
This theorem is referenced by:  dya2iocnei  26840
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