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Theorem tpr2rico 28718
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 11639 . . . . . . . . . 10  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
21ixxf 11645 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5728 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR*  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3mp1i 13 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  (,)  Fn  ( RR*  X.  RR* ) )
5 elssuni 4227 . . . . . . . . . . . . . 14  |-  ( A  e.  ( J  tX  J )  ->  A  C_ 
U. ( J  tX  J ) )
6 tpr2rico.0 . . . . . . . . . . . . . . . 16  |-  J  =  ( topGen `  ran  (,) )
7 retop 21782 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  e.  Top
86, 7eqeltri 2525 . . . . . . . . . . . . . . 15  |-  J  e. 
Top
9 uniretop 21783 . . . . . . . . . . . . . . . 16  |-  RR  =  U. ( topGen `  ran  (,) )
106unieqi 4207 . . . . . . . . . . . . . . . 16  |-  U. J  =  U. ( topGen `  ran  (,) )
119, 10eqtr4i 2476 . . . . . . . . . . . . . . 15  |-  RR  =  U. J
128, 8, 11, 11txunii 20608 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
135, 12syl6sseqr 3479 . . . . . . . . . . . . 13  |-  ( A  e.  ( J  tX  J )  ->  A  C_  ( RR  X.  RR ) )
1413ad2antrr 732 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  A  C_  ( RR  X.  RR ) )
15 simplr 762 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  A )
1614, 15sseldd 3433 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
17 xp1st 6823 . . . . . . . . . . 11  |-  ( X  e.  ( RR  X.  RR )  ->  ( 1st `  X )  e.  RR )
1816, 17syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR )
19 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2019rpred 11341 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR )
2120rehalfcld 10859 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR )
2218, 21resubcld 10047 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR )
2322rexrd 9690 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR* )
2418, 21readdcld 9670 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR )
2524rexrd 9690 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR* )
26 fnovrn 6444 . . . . . . . 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 1268 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
28 xp2nd 6824 . . . . . . . . . . 11  |-  ( X  e.  ( RR  X.  RR )  ->  ( 2nd `  X )  e.  RR )
2916, 28syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR )
3029, 21resubcld 10047 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR )
3130rexrd 9690 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR* )
3229, 21readdcld 9670 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR )
3332rexrd 9690 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR* )
34 fnovrn 6444 . . . . . . . 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 1268 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
36 eqidd 2452 . . . . . . 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 4848 . . . . . . . . 9  |-  ( x  =  ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  ->  ( x  X.  y )  =  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  y ) )
3837eqeq2d 2461 . . . . . . . 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 4849 . . . . . . . . 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 2461 . . . . . . . 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 3161 . . . . . . 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 1268 . . . . . 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 2451 . . . . . . 7  |-  ( x  e.  ran  (,) , 
y  e.  ran  (,)  |->  ( x  X.  y
) )  =  ( x  e.  ran  (,) ,  y  e.  ran  (,)  |->  ( x  X.  y
) )
44 vex 3048 . . . . . . . 8  |-  x  e. 
_V
45 vex 3048 . . . . . . . 8  |-  y  e. 
_V
4644, 45xpex 6595 . . . . . . 7  |-  ( x  X.  y )  e. 
_V
4743, 46elrnmpt2 6409 . . . . . 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 216 . . . . 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 2540 . . . 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 2802 . . 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 4941 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
5352, 16sseldi 3430 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( _V  X.  _V )
)
5418rexrd 9690 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR* )
5519rphalfcld 11353 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR+ )
5618, 55ltsubrpd 11370 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X ) )
5718, 55ltaddrpd 11371 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) )
58 elioo1 11676 . . . . . . . . 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 667 . . . . . . . 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 1191 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) ) )
6129rexrd 9690 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR* )
6229, 55ltsubrpd 11370 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X ) )
6329, 55ltaddrpd 11371 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) )
64 elioo1 11676 . . . . . . . . 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 667 . . . . . . . 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 1191 . . . . . . 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 535 . . . . . 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 6826 . . . . . 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 670 . . . . 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 2802 . . . 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 11435 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  X
)  -  ( d  /  2 ) )  e.  RR*  -> -oo  <_  ( ( 1st `  X
)  -  ( d  /  2 ) ) )
7223, 71syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  -> -oo  <_  (
( 1st `  X
)  -  ( d  /  2 ) ) )
73 pnfge 11432 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  X
)  +  ( d  /  2 ) )  e.  RR*  ->  ( ( 1st `  X )  +  ( d  / 
2 ) )  <_ +oo )
7425, 73syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  <_ +oo )
75 mnfxr 11414 . . . . . . . . . . . . . . . . . 18  |- -oo  e.  RR*
76 pnfxr 11412 . . . . . . . . . . . . . . . . . 18  |- +oo  e.  RR*
77 ioossioo 11726 . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . 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 11709 . . . . . . . . . . . . . . . 16  |-  ( -oo (,) +oo )  =  RR
8179, 80syl6sseq 3478 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  RR )
82 mnfle 11435 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  X
)  -  ( d  /  2 ) )  e.  RR*  -> -oo  <_  ( ( 2nd `  X
)  -  ( d  /  2 ) ) )
8331, 82syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  -> -oo  <_  (
( 2nd `  X
)  -  ( d  /  2 ) ) )
84 pnfge 11432 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  X
)  +  ( d  /  2 ) )  e.  RR*  ->  ( ( 2nd `  X )  +  ( d  / 
2 ) )  <_ +oo )
8533, 84syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  <_ +oo )
86 ioossioo 11726 . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . 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 3478 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )
90 xpss12 4940 . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . 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 3432 . . . . . . . . . . . . 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 437 . . . . . . . . . . . 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 556 . . . . . . . . . . 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 697 . . . . . . . . . 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 467 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  A  C_  ( RR  X.  RR ) )
97 simpr1 1014 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  A )
9896, 97sseldd 3433 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  ( RR  X.  RR ) )
99983anassrs 1232 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  X  e.  ( RR  X.  RR ) )
100 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  x  e.  ( RR  X.  RR ) )
101 simplr 762 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR+ )
102101rphalfcld 11353 . . . . . . . . . . . . . 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 28717 . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107103, 6, 106cnrehmeo 21981 . . . . . . . . . . . . . . . . . . . 20  |-  G  e.  ( ( J  tX  J ) Homeo ( TopOpen ` fld )
)
108106cnfldtopon 21803 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
109108toponunii 19947 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  =  U. ( TopOpen ` fld )
11012, 109hmeof1o 20779 . . . . . . . . . . . . . . . . . . . 20  |-  ( G  e.  ( ( J 
tX  J ) Homeo (
TopOpen ` fld ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
111 f1of 5814 . . . . . . . . . . . . . . . . . . . 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 6023 . . . . . . . . . . . . . . . . 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 6022 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  CC )
117 sqsscirc2 28715 . . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . . 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 11342 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR* )
121120adantr 467 . . . . . . . . . . . . . . . . 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 21793 . . . . . . . . . . . . . . . . 17  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123121, 122jctil 540 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
128127cnmetdval 21791 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( G `  x
)  e.  CC  /\  ( G `  X )  e.  CC )  -> 
( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  =  ( abs `  ( ( G `  x )  -  ( G `  X )
) ) )
129125, 124, 128syl2anc 667 . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 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 4423 . . . . . . . . . . . . . . . 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 21407 . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . 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 45 . . . . . . . . . . . 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 5826 . . . . . . . . . . . . . . 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 5816 . . . . . . . . . . . . . 14  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  Fun  `' G
)
141139, 140ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  `' G
142 f1odm 5818 . . . . . . . . . . . . . . 15  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  dom  `' G  =  CC )
143139, 142ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  `' G  =  CC
144116, 143syl6eleqr 2540 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  dom  `' G )
145 funfvima 6140 . . . . . . . . . . . . 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 669 . . . . . . . . . . . 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 13 . . . . . . . . . . . . . . 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 6175 . . . . . . . . . . . . . . 15  |-  ( ( G : ( RR 
X.  RR ) -1-1-onto-> CC  /\  x  e.  ( RR  X.  RR ) )  -> 
( `' G `  ( G `  x ) )  =  x )
149147, 100, 148syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( `' G `  ( G `
 x ) )  =  x )
150149eleq1d 2513 . . . . . . . . . . . . 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 211 . . . . . . . . . . . 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 57 . . . . . . . . . . 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 431 . . . . . . . . . 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 17 . . . . . . . . 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 436 . . . . . . . 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 3438 . . . . . . 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 2802 . . . . . 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 6398 . . . . . . . . . 10  |-  Fun  G
159158a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  Fun  G )
16013sselda 3432 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
161 f1odm 5818 . . . . . . . . . . 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 2540 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  dom  G
)
164 simpr 463 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  A )
165 funfvima 6140 . . . . . . . . . 10  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( X  e.  A  ->  ( G `  X
)  e.  ( G
" A ) ) )
166165imp 431 . . . . . . . . 9  |-  ( ( ( Fun  G  /\  X  e.  dom  G )  /\  X  e.  A
)  ->  ( G `  X )  e.  ( G " A ) )
167159, 163, 164, 166syl21anc 1267 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( G `  X
)  e.  ( G
" A ) )
168 hmeoima 20780 . . . . . . . . . . 11  |-  ( ( G  e.  ( ( J  tX  J )
Homeo ( TopOpen ` fld ) )  /\  A  e.  ( J  tX  J
) )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
169107, 168mpan 676 . . . . . . . . . 10  |-  ( A  e.  ( J  tX  J )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
170106cnfldtopn 21802 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
171170elmopn2 21460 . . . . . . . . . . . 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 466 . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( A  e.  ( J  tX  J )  ->  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
175174adantr 467 . . . . . . . 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 6297 . . . . . . . . . . 11  |-  ( m  =  ( G `  X )  ->  (
m ( ball `  ( abs  o.  -  ) ) d )  =  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
177176sseq1d 3459 . . . . . . . . . 10  |-  ( m  =  ( G `  X )  ->  (
( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  <->  ( ( G `  X )
( ball `  ( abs  o. 
-  ) ) d )  C_  ( G " A ) ) )
178177rexbidv 2901 . . . . . . . . 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 3148 . . . . . . . 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 667 . . . . . . 7  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
181 imass2 5204 . . . . . . . . . 10  |-  ( ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  -> 
( `' G "
( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  ( `' G " ( G " A ) ) )
182 f1of1 5813 . . . . . . . . . . . . 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 5830 . . . . . . . . . . . 12  |-  ( ( G : ( RR 
X.  RR ) -1-1-> CC  /\  A  C_  ( RR  X.  RR ) )  -> 
( `' G "
( G " A
) )  =  A )
185183, 13, 184sylancr 669 . . . . . . . . . . 11  |-  ( A  e.  ( J  tX  J )  ->  ( `' G " ( G
" A ) )  =  A )
186185sseq2d 3460 . . . . . . . . . 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 223 . . . . . . . . 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 2861 . . . . . . . 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 467 . . . . . . 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 2925 . . . . . 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 667 . . . . 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 3440 . . . . . 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 2855 . . . . 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 17 . . . 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 2925 . . . 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 667 . . 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 2925 . . 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 667 . 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 2518 . . . . 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 3453 . . . . 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 717 . . . 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 3150 . . 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 2876 . 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 17 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837    o. ccom 4838   Fun wfun 5576    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792   CCcc 9537   RRcr 9538   _ici 9541    + caddc 9542    x. cmul 9544   +oocpnf 9672   -oocmnf 9673   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   2c2 10659   RR+crp 11302   (,)cioo 11635   Recre 13160   Imcim 13161   abscabs 13297   TopOpenctopn 15320   topGenctg 15336   *Metcxmt 18955   ballcbl 18957  ℂfldccnfld 18970   Topctop 19917    tX ctx 20575   Homeochmeo 20768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cn 20243  df-cnp 20244  df-tx 20577  df-hmeo 20770  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910
This theorem is referenced by:  dya2iocnei  29104
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