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Theorem tpr2rico 26196
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 11292 . . . . . . . . . 10  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
21ixxf 11298 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5547 . . . . . . . . 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 4109 . . . . . . . . . . . . . 14  |-  ( A  e.  ( J  tX  J )  ->  A  C_ 
U. ( J  tX  J ) )
6 tpr2rico.0 . . . . . . . . . . . . . . . 16  |-  J  =  ( topGen `  ran  (,) )
7 retop 20182 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  e.  Top
86, 7eqeltri 2503 . . . . . . . . . . . . . . 15  |-  J  e. 
Top
9 uniretop 20183 . . . . . . . . . . . . . . . 16  |-  RR  =  U. ( topGen `  ran  (,) )
106unieqi 4088 . . . . . . . . . . . . . . . 16  |-  U. J  =  U. ( topGen `  ran  (,) )
119, 10eqtr4i 2456 . . . . . . . . . . . . . . 15  |-  RR  =  U. J
128, 8, 11, 11txunii 19008 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
135, 12syl6sseqr 3391 . . . . . . . . . . . . 13  |-  ( A  e.  ( J  tX  J )  ->  A  C_  ( RR  X.  RR ) )
1413ad2antrr 718 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  A  C_  ( RR  X.  RR ) )
15 simplr 747 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  A )
1614, 15sseldd 3345 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
17 xp1st 6595 . . . . . . . . . . 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 458 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2019rpred 11015 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR )
2120rehalfcld 10559 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR )
2218, 21resubcld 9764 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR )
2322rexrd 9421 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR* )
2418, 21readdcld 9401 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR )
2524rexrd 9421 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR* )
26 fnovrn 6227 . . . . . . . 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 1211 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
28 xp2nd 6596 . . . . . . . . . . 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 9764 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR )
3130rexrd 9421 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR* )
3229, 21readdcld 9401 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR )
3332rexrd 9421 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR* )
34 fnovrn 6227 . . . . . . . 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 1211 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
36 eqidd 2434 . . . . . . 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 4841 . . . . . . . . 9  |-  ( x  =  ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  ->  ( x  X.  y )  =  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  y ) )
3837eqeq2d 2444 . . . . . . . 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 4842 . . . . . . . . 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 2444 . . . . . . . 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 3070 . . . . . . 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 1211 . . . . . 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 2433 . . . . . . 7  |-  ( x  e.  ran  (,) , 
y  e.  ran  (,)  |->  ( x  X.  y
) )  =  ( x  e.  ran  (,) ,  y  e.  ran  (,)  |->  ( x  X.  y
) )
44 vex 2965 . . . . . . . 8  |-  x  e. 
_V
45 vex 2965 . . . . . . . 8  |-  y  e. 
_V
4644, 45xpex 6497 . . . . . . 7  |-  ( x  X.  y )  e. 
_V
4743, 46elrnmpt2 6192 . . . . . 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 2524 . . . 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 2789 . . 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 4933 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
5352, 16sseldi 3342 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( _V  X.  _V )
)
5418rexrd 9421 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR* )
5519rphalfcld 11027 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR+ )
5618, 55ltsubrpd 11043 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X ) )
5718, 55ltaddrpd 11044 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) )
58 elioo1 11328 . . . . . . . . 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 654 . . . . . . . 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 1164 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) ) )
6129rexrd 9421 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR* )
6229, 55ltsubrpd 11043 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X ) )
6329, 55ltaddrpd 11044 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) )
64 elioo1 11328 . . . . . . . . 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 654 . . . . . . . 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 1164 . . . . . . 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 529 . . . . . 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 6598 . . . . . 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 657 . . . . 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 2789 . . . 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 11101 . . . . . . . . . . . . . . . . . 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 11098 . . . . . . . . . . . . . . . . . 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 11082 . . . . . . . . . . . . . . . . . 18  |- -oo  e.  RR*
76 pnfxr 11080 . . . . . . . . . . . . . . . . . 18  |- +oo  e.  RR*
77 ioossioo 25886 . . . . . . . . . . . . . . . . . 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 675 . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . 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 11358 . . . . . . . . . . . . . . . 16  |-  ( -oo (,) +oo )  =  RR
8179, 80syl6sseq 3390 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  RR )
82 mnfle 11101 . . . . . . . . . . . . . . . . . 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 11098 . . . . . . . . . . . . . . . . . 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 25886 . . . . . . . . . . . . . . . . . 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 675 . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . 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 3390 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )
90 xpss12 4932 . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . 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 3344 . . . . . . . . . . . . 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 548 . . . . . . . . . . 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 684 . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  A  C_  ( RR  X.  RR ) )
97 simpr1 987 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  A )
9896, 97sseldd 3345 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  ( RR  X.  RR ) )
99983anassrs 1202 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  X  e.  ( RR  X.  RR ) )
100 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  x  e.  ( RR  X.  RR ) )
101 simplr 747 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR+ )
102101rphalfcld 11027 . . . . . . . . . . . . . 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 26195 . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . 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 2433 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107103, 6, 106cnrehmeo 20367 . . . . . . . . . . . . . . . . . . . 20  |-  G  e.  ( ( J  tX  J ) Homeo ( TopOpen ` fld )
)
108106cnfldtopon 20204 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
109108toponunii 18379 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  =  U. ( TopOpen ` fld )
11012, 109hmeof1o 19179 . . . . . . . . . . . . . . . . . . . 20  |-  ( G  e.  ( ( J 
tX  J ) Homeo (
TopOpen ` fld ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
111 f1of 5629 . . . . . . . . . . . . . . . . . . . 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 5832 . . . . . . . . . . . . . . . . 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 5831 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  CC )
117 sqsscirc2 26193 . . . . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . . . 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 11016 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR* )
121120adantr 462 . . . . . . . . . . . . . . . . 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 20194 . . . . . . . . . . . . . . . . 17  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123121, 122jctil 534 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . 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 2433 . . . . . . . . . . . . . . . . . . 19  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
128127cnmetdval 20192 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( G `  x
)  e.  CC  /\  ( G `  X )  e.  CC )  -> 
( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  =  ( abs `  ( ( G `  x )  -  ( G `  X )
) ) )
129125, 124, 128syl2anc 654 . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . 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 4300 . . . . . . . . . . . . . . . 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 19809 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 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 5641 . . . . . . . . . . . . . . 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 5631 . . . . . . . . . . . . . 14  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  Fun  `' G
)
141139, 140ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  `' G
142 f1odm 5633 . . . . . . . . . . . . . . 15  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  dom  `' G  =  CC )
143139, 142ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  `' G  =  CC
144116, 143syl6eleqr 2524 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  dom  `' G )
145 funfvima 5939 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 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 5970 . . . . . . . . . . . . . . 15  |-  ( ( G : ( RR 
X.  RR ) -1-1-onto-> CC  /\  x  e.  ( RR  X.  RR ) )  -> 
( `' G `  ( G `  x ) )  =  x )
149147, 100, 148syl2anc 654 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( `' G `  ( G `
 x ) )  =  x )
150149eleq1d 2499 . . . . . . . . . . . . 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 3350 . . . . . . 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 2789 . . . . . 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 6181 . . . . . . . . . 10  |-  Fun  G
159158a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  Fun  G )
16013sselda 3344 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
161 f1odm 5633 . . . . . . . . . . 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 2524 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  dom  G
)
164 simpr 458 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  A )
165 funfvima 5939 . . . . . . . . . 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 1210 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( G `  X
)  e.  ( G
" A ) )
168 hmeoima 19180 . . . . . . . . . . 11  |-  ( ( G  e.  ( ( J  tX  J )
Homeo ( TopOpen ` fld ) )  /\  A  e.  ( J  tX  J
) )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
169107, 168mpan 663 . . . . . . . . . 10  |-  ( A  e.  ( J  tX  J )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
170106cnfldtopn 20203 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
171170elmopn2 19862 . . . . . . . . . . . 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 461 . . . . . . . . . 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 462 . . . . . . . 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 6087 . . . . . . . . . . 11  |-  ( m  =  ( G `  X )  ->  (
m ( ball `  ( abs  o.  -  ) ) d )  =  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
177176sseq1d 3371 . . . . . . . . . 10  |-  ( m  =  ( G `  X )  ->  (
( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  <->  ( ( G `  X )
( ball `  ( abs  o. 
-  ) ) d )  C_  ( G " A ) ) )
178177rexbidv 2726 . . . . . . . . 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 3060 . . . . . . . 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 654 . . . . . . 7  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
181 imass2 5192 . . . . . . . . . 10  |-  ( ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  -> 
( `' G "
( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  ( `' G " ( G " A ) ) )
182 f1of1 5628 . . . . . . . . . . . . 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 5645 . . . . . . . . . . . 12  |-  ( ( G : ( RR 
X.  RR ) -1-1-> CC  /\  A  C_  ( RR  X.  RR ) )  -> 
( `' G "
( G " A
) )  =  A )
185183, 13, 184sylancr 656 . . . . . . . . . . 11  |-  ( A  e.  ( J  tX  J )  ->  ( `' G " ( G
" A ) )  =  A )
186185sseq2d 3372 . . . . . . . . . 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 2817 . . . . . . . 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 462 . . . . . . 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 2847 . . . . . 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 654 . . . . 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 3352 . . . . . 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 2813 . . . . 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 2847 . . . 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 654 . . 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 2847 . . 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 654 . 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 2494 . . . . 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 3365 . . . . 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 703 . . . 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 3062 . . 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 2827 . 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962    C_ wss 3316   ~Pcpw 3848   U.cuni 4079   class class class wbr 4280    X. cxp 4825   `'ccnv 4826   dom cdm 4827   ran crn 4828   "cima 4830    o. ccom 4831   Fun wfun 5400    Fn wfn 5401   -->wf 5402   -1-1->wf1 5403   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1stc1st 6564   2ndc2nd 6565   CCcc 9268   RRcr 9269   _ici 9272    + caddc 9273    x. cmul 9275   +oocpnf 9403   -oocmnf 9404   RR*cxr 9405    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   2c2 10359   RR+crp 10979   (,)cioo 11288   Recre 12570   Imcim 12571   abscabs 12707   TopOpenctopn 14343   topGenctg 14359   *Metcxmt 17645   ballcbl 17647  ℂfldccnfld 17662   Topctop 18340    tX ctx 18975   Homeochmeo 19168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cn 18673  df-cnp 18674  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296
This theorem is referenced by:  dya2iocnei  26551
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