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Theorem tpr2rico 28347
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 11586 . . . . . . . . . 10  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
21ixxf 11592 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5714 . . . . . . . . 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 4220 . . . . . . . . . . . . . 14  |-  ( A  e.  ( J  tX  J )  ->  A  C_ 
U. ( J  tX  J ) )
6 tpr2rico.0 . . . . . . . . . . . . . . . 16  |-  J  =  ( topGen `  ran  (,) )
7 retop 21560 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  e.  Top
86, 7eqeltri 2486 . . . . . . . . . . . . . . 15  |-  J  e. 
Top
9 uniretop 21561 . . . . . . . . . . . . . . . 16  |-  RR  =  U. ( topGen `  ran  (,) )
106unieqi 4200 . . . . . . . . . . . . . . . 16  |-  U. J  =  U. ( topGen `  ran  (,) )
119, 10eqtr4i 2434 . . . . . . . . . . . . . . 15  |-  RR  =  U. J
128, 8, 11, 11txunii 20386 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
135, 12syl6sseqr 3489 . . . . . . . . . . . . 13  |-  ( A  e.  ( J  tX  J )  ->  A  C_  ( RR  X.  RR ) )
1413ad2antrr 724 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  A  C_  ( RR  X.  RR ) )
15 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  A )
1614, 15sseldd 3443 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
17 xp1st 6814 . . . . . . . . . . 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 459 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2019rpred 11304 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR )
2120rehalfcld 10826 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR )
2218, 21resubcld 10028 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR )
2322rexrd 9673 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR* )
2418, 21readdcld 9653 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR )
2524rexrd 9673 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR* )
26 fnovrn 6431 . . . . . . . 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 1230 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
28 xp2nd 6815 . . . . . . . . . . 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 10028 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR )
3130rexrd 9673 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR* )
3229, 21readdcld 9653 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR )
3332rexrd 9673 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR* )
34 fnovrn 6431 . . . . . . . 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 1230 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
36 eqidd 2403 . . . . . . 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 4837 . . . . . . . . 9  |-  ( x  =  ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  ->  ( x  X.  y )  =  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  y ) )
3837eqeq2d 2416 . . . . . . . 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 4838 . . . . . . . . 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 2416 . . . . . . . 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 3171 . . . . . . 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 1230 . . . . . 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 2402 . . . . . . 7  |-  ( x  e.  ran  (,) , 
y  e.  ran  (,)  |->  ( x  X.  y
) )  =  ( x  e.  ran  (,) ,  y  e.  ran  (,)  |->  ( x  X.  y
) )
44 vex 3062 . . . . . . . 8  |-  x  e. 
_V
45 vex 3062 . . . . . . . 8  |-  y  e. 
_V
4644, 45xpex 6586 . . . . . . 7  |-  ( x  X.  y )  e. 
_V
4743, 46elrnmpt2 6396 . . . . . 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 2501 . . . 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 2818 . . 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 4930 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
5352, 16sseldi 3440 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( _V  X.  _V )
)
5418rexrd 9673 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR* )
5519rphalfcld 11316 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR+ )
5618, 55ltsubrpd 11332 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X ) )
5718, 55ltaddrpd 11333 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) )
58 elioo1 11622 . . . . . . . . 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 659 . . . . . . . 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 1180 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) ) )
6129rexrd 9673 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR* )
6229, 55ltsubrpd 11332 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X ) )
6329, 55ltaddrpd 11333 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) )
64 elioo1 11622 . . . . . . . . 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 659 . . . . . . . 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 1180 . . . . . . 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 530 . . . . . 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 6817 . . . . . 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 662 . . . . 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 2818 . . . 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 11395 . . . . . . . . . . . . . . . . . 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 11392 . . . . . . . . . . . . . . . . . 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 11376 . . . . . . . . . . . . . . . . . 18  |- -oo  e.  RR*
76 pnfxr 11374 . . . . . . . . . . . . . . . . . 18  |- +oo  e.  RR*
77 ioossioo 11670 . . . . . . . . . . . . . . . . . 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 680 . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . 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 11653 . . . . . . . . . . . . . . . 16  |-  ( -oo (,) +oo )  =  RR
8179, 80syl6sseq 3488 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  RR )
82 mnfle 11395 . . . . . . . . . . . . . . . . . 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 11392 . . . . . . . . . . . . . . . . . 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 11670 . . . . . . . . . . . . . . . . . 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 680 . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . 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 3488 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )
90 xpss12 4929 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . 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 3442 . . . . . . . . . . . . 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 433 . . . . . . . . . . . 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 551 . . . . . . . . . . 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 689 . . . . . . . . . 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 463 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  A  C_  ( RR  X.  RR ) )
97 simpr1 1003 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  A )
9896, 97sseldd 3443 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  ( RR  X.  RR ) )
99983anassrs 1220 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  X  e.  ( RR  X.  RR ) )
100 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  x  e.  ( RR  X.  RR ) )
101 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR+ )
102101rphalfcld 11316 . . . . . . . . . . . . . 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 28346 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 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 2402 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107103, 6, 106cnrehmeo 21745 . . . . . . . . . . . . . . . . . . . 20  |-  G  e.  ( ( J  tX  J ) Homeo ( TopOpen ` fld )
)
108106cnfldtopon 21582 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
109108toponunii 19725 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  =  U. ( TopOpen ` fld )
11012, 109hmeof1o 20557 . . . . . . . . . . . . . . . . . . . 20  |-  ( G  e.  ( ( J 
tX  J ) Homeo (
TopOpen ` fld ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
111 f1of 5799 . . . . . . . . . . . . . . . . . . . 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 6010 . . . . . . . . . . . . . . . . 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 6009 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  CC )
117 sqsscirc2 28344 . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . 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 11305 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR* )
121120adantr 463 . . . . . . . . . . . . . . . . 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 21572 . . . . . . . . . . . . . . . . 17  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123121, 122jctil 535 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 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 530 . . . . . . . . . . . . . . . 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 2402 . . . . . . . . . . . . . . . . . . 19  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
128127cnmetdval 21570 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( G `  x
)  e.  CC  /\  ( G `  X )  e.  CC )  -> 
( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  =  ( abs `  ( ( G `  x )  -  ( G `  X )
) ) )
129125, 124, 128syl2anc 659 . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . 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 4415 . . . . . . . . . . . . . . . 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 21187 . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . 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 42 . . . . . . . . . . . 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 5811 . . . . . . . . . . . . . . 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 5801 . . . . . . . . . . . . . 14  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  Fun  `' G
)
141139, 140ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  `' G
142 f1odm 5803 . . . . . . . . . . . . . . 15  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  dom  `' G  =  CC )
143139, 142ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  `' G  =  CC
144116, 143syl6eleqr 2501 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  dom  `' G )
145 funfvima 6128 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 6163 . . . . . . . . . . . . . . 15  |-  ( ( G : ( RR 
X.  RR ) -1-1-onto-> CC  /\  x  e.  ( RR  X.  RR ) )  -> 
( `' G `  ( G `  x ) )  =  x )
149147, 100, 148syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( `' G `  ( G `
 x ) )  =  x )
150149eleq1d 2471 . . . . . . . . . . . . 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 54 . . . . . . . . . . 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 427 . . . . . . . . . 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 432 . . . . . . . 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 3448 . . . . . . 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 2818 . . . . . 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 6385 . . . . . . . . . 10  |-  Fun  G
159158a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  Fun  G )
16013sselda 3442 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
161 f1odm 5803 . . . . . . . . . . 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 2501 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  dom  G
)
164 simpr 459 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  A )
165 funfvima 6128 . . . . . . . . . 10  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( X  e.  A  ->  ( G `  X
)  e.  ( G
" A ) ) )
166165imp 427 . . . . . . . . 9  |-  ( ( ( Fun  G  /\  X  e.  dom  G )  /\  X  e.  A
)  ->  ( G `  X )  e.  ( G " A ) )
167159, 163, 164, 166syl21anc 1229 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( G `  X
)  e.  ( G
" A ) )
168 hmeoima 20558 . . . . . . . . . . 11  |-  ( ( G  e.  ( ( J  tX  J )
Homeo ( TopOpen ` fld ) )  /\  A  e.  ( J  tX  J
) )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
169107, 168mpan 668 . . . . . . . . . 10  |-  ( A  e.  ( J  tX  J )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
170106cnfldtopn 21581 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
171170elmopn2 21240 . . . . . . . . . . . 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 462 . . . . . . . . . 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 463 . . . . . . . 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 6285 . . . . . . . . . . 11  |-  ( m  =  ( G `  X )  ->  (
m ( ball `  ( abs  o.  -  ) ) d )  =  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
177176sseq1d 3469 . . . . . . . . . 10  |-  ( m  =  ( G `  X )  ->  (
( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  <->  ( ( G `  X )
( ball `  ( abs  o. 
-  ) ) d )  C_  ( G " A ) ) )
178177rexbidv 2918 . . . . . . . . 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 3158 . . . . . . . 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 659 . . . . . . 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 5798 . . . . . . . . . . . . 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 5815 . . . . . . . . . . . 12  |-  ( ( G : ( RR 
X.  RR ) -1-1-> CC  /\  A  C_  ( RR  X.  RR ) )  -> 
( `' G "
( G " A
) )  =  A )
185183, 13, 184sylancr 661 . . . . . . . . . . 11  |-  ( A  e.  ( J  tX  J )  ->  ( `' G " ( G
" A ) )  =  A )
186185sseq2d 3470 . . . . . . . . . 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 2878 . . . . . . . 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 463 . . . . . . 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 2942 . . . . . 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 659 . . . . 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 3450 . . . . . 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 2872 . . . . 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 2942 . . . 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 659 . . 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 2942 . . 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 659 . 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 2475 . . . . 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 3463 . . . . 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 709 . . . 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 3160 . . 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 2893 . 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 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059    C_ wss 3414   ~Pcpw 3955   U.cuni 4191   class class class wbr 4395    X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826    o. ccom 4827   Fun wfun 5563    Fn wfn 5564   -->wf 5565   -1-1->wf1 5566   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783   CCcc 9520   RRcr 9521   _ici 9524    + caddc 9525    x. cmul 9527   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658    <_ cle 9659    - cmin 9841    / cdiv 10247   2c2 10626   RR+crp 11265   (,)cioo 11582   Recre 13079   Imcim 13080   abscabs 13216   TopOpenctopn 15036   topGenctg 15052   *Metcxmt 18723   ballcbl 18725  ℂfldccnfld 18740   Topctop 19686    tX ctx 20353   Homeochmeo 20546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cn 20021  df-cnp 20022  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674
This theorem is referenced by:  dya2iocnei  28730
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