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Theorem tpr2rico 27558
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 11533 . . . . . . . . . 10  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
21ixxf 11539 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5731 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR*  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3mp1i 12 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  (,)  Fn  ( RR*  X.  RR* ) )
5 elssuni 4275 . . . . . . . . . . . . . 14  |-  ( A  e.  ( J  tX  J )  ->  A  C_ 
U. ( J  tX  J ) )
6 tpr2rico.0 . . . . . . . . . . . . . . . 16  |-  J  =  ( topGen `  ran  (,) )
7 retop 21031 . . . . . . . . . . . . . . . 16  |-  ( topGen ` 
ran  (,) )  e.  Top
86, 7eqeltri 2551 . . . . . . . . . . . . . . 15  |-  J  e. 
Top
9 uniretop 21032 . . . . . . . . . . . . . . . 16  |-  RR  =  U. ( topGen `  ran  (,) )
106unieqi 4254 . . . . . . . . . . . . . . . 16  |-  U. J  =  U. ( topGen `  ran  (,) )
119, 10eqtr4i 2499 . . . . . . . . . . . . . . 15  |-  RR  =  U. J
128, 8, 11, 11txunii 19857 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
135, 12syl6sseqr 3551 . . . . . . . . . . . . 13  |-  ( A  e.  ( J  tX  J )  ->  A  C_  ( RR  X.  RR ) )
1413ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  A  C_  ( RR  X.  RR ) )
15 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  A )
1614, 15sseldd 3505 . . . . . . . . . . 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 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR )
19 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2019rpred 11256 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  d  e.  RR )
2120rehalfcld 10785 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR )
2218, 21resubcld 9987 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR )
2322rexrd 9643 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  e.  RR* )
2418, 21readdcld 9623 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR )
2524rexrd 9643 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  e.  RR* )
26 fnovrn 6434 . . . . . . . 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 1228 . . . . . . 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 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR )
3029, 21resubcld 9987 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR )
3130rexrd 9643 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  e.  RR* )
3229, 21readdcld 9623 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR )
3332rexrd 9643 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  e.  RR* )
34 fnovrn 6434 . . . . . . . 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 1228 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  e.  ran  (,) )
36 eqidd 2468 . . . . . . 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 5013 . . . . . . . . 9  |-  ( x  =  ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  ->  ( x  X.  y )  =  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  y ) )
3837eqeq2d 2481 . . . . . . . 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 5014 . . . . . . . . 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 2481 . . . . . . . 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 3225 . . . . . . 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 1228 . . . . . 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 2467 . . . . . . 7  |-  ( x  e.  ran  (,) , 
y  e.  ran  (,)  |->  ( x  X.  y
) )  =  ( x  e.  ran  (,) ,  y  e.  ran  (,)  |->  ( x  X.  y
) )
44 vex 3116 . . . . . . . 8  |-  x  e. 
_V
45 vex 3116 . . . . . . . 8  |-  y  e. 
_V
4644, 45xpex 6588 . . . . . . 7  |-  ( x  X.  y )  e. 
_V
4743, 46elrnmpt2 6399 . . . . . 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 2566 . . . 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 2878 . . 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 5109 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
5352, 16sseldi 3502 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( _V  X.  _V )
)
5418rexrd 9643 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  RR* )
5519rphalfcld 11268 . . . . . . . . 9  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( d  /  2 )  e.  RR+ )
5618, 55ltsubrpd 11284 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X ) )
5718, 55ltaddrpd 11285 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) )
58 elioo1 11569 . . . . . . . . 9  |-  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) )  e.  RR*  /\  (
( 1st `  X
)  +  ( d  /  2 ) )  e.  RR* )  ->  (
( 1st `  X
)  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  <->  ( ( 1st `  X )  e. 
RR*  /\  ( ( 1st `  X )  -  ( d  /  2
) )  <  ( 1st `  X )  /\  ( 1st `  X )  <  ( ( 1st `  X )  +  ( d  /  2 ) ) ) ) )
5923, 25, 58syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  e.  ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  <-> 
( ( 1st `  X
)  e.  RR*  /\  (
( 1st `  X
)  -  ( d  /  2 ) )  <  ( 1st `  X
)  /\  ( 1st `  X )  <  (
( 1st `  X
)  +  ( d  /  2 ) ) ) ) )
6054, 56, 57, 59mpbir3and 1179 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 1st `  X )  e.  ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) ) )
6129rexrd 9643 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  RR* )
6229, 55ltsubrpd 11284 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X ) )
6329, 55ltaddrpd 11285 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) )
64 elioo1 11569 . . . . . . . . 9  |-  ( ( ( ( 2nd `  X
)  -  ( d  /  2 ) )  e.  RR*  /\  (
( 2nd `  X
)  +  ( d  /  2 ) )  e.  RR* )  ->  (
( 2nd `  X
)  e.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  <->  ( ( 2nd `  X )  e. 
RR*  /\  ( ( 2nd `  X )  -  ( d  /  2
) )  <  ( 2nd `  X )  /\  ( 2nd `  X )  <  ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
6531, 33, 64syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  e.  ( ( ( 2nd `  X )  -  (
d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  / 
2 ) ) )  <-> 
( ( 2nd `  X
)  e.  RR*  /\  (
( 2nd `  X
)  -  ( d  /  2 ) )  <  ( 2nd `  X
)  /\  ( 2nd `  X )  <  (
( 2nd `  X
)  +  ( d  /  2 ) ) ) ) )
6661, 62, 63, 65mpbir3and 1179 . . . . . . 7  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( 2nd `  X )  e.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )
6760, 66jca 532 . . . . . 6  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  e.  ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  /\  ( 2nd `  X
)  e.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
68 elxp7 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 664 . . . . 5  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )
7069ralrimiva 2878 . . . 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 11342 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  X
)  -  ( d  /  2 ) )  e.  RR*  -> -oo  <_  ( ( 1st `  X
)  -  ( d  /  2 ) ) )
7223, 71syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  -> -oo  <_  (
( 1st `  X
)  -  ( d  /  2 ) ) )
73 pnfge 11339 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  X
)  +  ( d  /  2 ) )  e.  RR*  ->  ( ( 1st `  X )  +  ( d  / 
2 ) )  <_ +oo )
7425, 73syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 1st `  X )  +  ( d  /  2
) )  <_ +oo )
75 mnfxr 11323 . . . . . . . . . . . . . . . . . 18  |- -oo  e.  RR*
76 pnfxr 11321 . . . . . . . . . . . . . . . . . 18  |- +oo  e.  RR*
77 ioossioo 11616 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <_  ( ( 1st `  X )  -  (
d  /  2 ) )  /\  ( ( 1st `  X )  +  ( d  / 
2 ) )  <_ +oo ) )  ->  (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
7875, 76, 77mpanl12 682 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  <_  ( ( 1st `  X )  -  ( d  /  2
) )  /\  (
( 1st `  X
)  +  ( d  /  2 ) )  <_ +oo )  ->  (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
7972, 74, 78syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
80 ioomax 11599 . . . . . . . . . . . . . . . 16  |-  ( -oo (,) +oo )  =  RR
8179, 80syl6sseq 3550 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  C_  RR )
82 mnfle 11342 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  X
)  -  ( d  /  2 ) )  e.  RR*  -> -oo  <_  ( ( 2nd `  X
)  -  ( d  /  2 ) ) )
8331, 82syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  -> -oo  <_  (
( 2nd `  X
)  -  ( d  /  2 ) ) )
84 pnfge 11339 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  X
)  +  ( d  /  2 ) )  e.  RR*  ->  ( ( 2nd `  X )  +  ( d  / 
2 ) )  <_ +oo )
8533, 84syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( ( 2nd `  X )  +  ( d  /  2
) )  <_ +oo )
86 ioossioo 11616 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <_  ( ( 2nd `  X )  -  (
d  /  2 ) )  /\  ( ( 2nd `  X )  +  ( d  / 
2 ) )  <_ +oo ) )  ->  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
8775, 76, 86mpanl12 682 . . . . . . . . . . . . . . . . 17  |-  ( ( -oo  <_  ( ( 2nd `  X )  -  ( d  /  2
) )  /\  (
( 2nd `  X
)  +  ( d  /  2 ) )  <_ +oo )  ->  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
8883, 85, 87syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  ( -oo (,) +oo ) )
8988, 80syl6sseq 3550 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )
90 xpss12 5108 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) ) 
C_  RR  /\  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) )  C_  RR )  ->  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( RR  X.  RR ) )
9181, 89, 90syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( (
( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( RR  X.  RR ) )
9291sselda 3504 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  x  e.  ( RR  X.  RR ) )
9392expcom 435 . . . . . . . . . . . 12  |-  ( x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  ->  x  e.  ( RR  X.  RR ) ) )
9493ancld 553 . . . . . . . . . . 11  |-  ( x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  ->  (
( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) ) ) )
9594imdistanri 691 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) ) )
9613adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  A  C_  ( RR  X.  RR ) )
97 simpr1 1002 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  A )
9896, 97sseldd 3505 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( J 
tX  J )  /\  ( X  e.  A  /\  d  e.  RR+  /\  x  e.  ( RR  X.  RR ) ) )  ->  X  e.  ( RR  X.  RR ) )
99983anassrs 1218 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  X  e.  ( RR  X.  RR ) )
100 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  x  e.  ( RR  X.  RR ) )
101 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR+ )
102101rphalfcld 11268 . . . . . . . . . . . . . 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 27557 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107103, 6, 106cnrehmeo 21216 . . . . . . . . . . . . . . . . . . . 20  |-  G  e.  ( ( J  tX  J ) Homeo ( TopOpen ` fld )
)
108106cnfldtopon 21053 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
109108toponunii 19228 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  =  U. ( TopOpen ` fld )
11012, 109hmeof1o 20028 . . . . . . . . . . . . . . . . . . . 20  |-  ( G  e.  ( ( J 
tX  J ) Homeo (
TopOpen ` fld ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
111 f1of 5816 . . . . . . . . . . . . . . . . . . . 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 6022 . . . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  CC )
117 sqsscirc2 27555 . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( ( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) )  ->  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
) )
119118imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  (
( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) ) )  -> 
( abs `  (
( G `  x
)  -  ( G `
 X ) ) )  <  d )
120101rpxrd 11257 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  d  e.  RR* )
121120adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  d  e.  RR* )
122 cnxmet 21043 . . . . . . . . . . . . . . . . 17  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123121, 122jctil 537 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  d  e.  RR* ) )
124114adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( G `  X )  e.  CC )
125116adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( G `  x )  e.  CC )
126124, 125jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( G `  X )  e.  CC  /\  ( G `
 x )  e.  CC ) )
127 eqid 2467 . . . . . . . . . . . . . . . . . . 19  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
128127cnmetdval 21041 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( G `  x
)  e.  CC  /\  ( G `  X )  e.  CC )  -> 
( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  =  ( abs `  ( ( G `  x )  -  ( G `  X )
) ) )
129125, 124, 128syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( ( G `  x )
( abs  o.  -  )
( G `  X
) )  =  ( abs `  ( ( G `  x )  -  ( G `  X ) ) ) )
130 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( abs `  ( ( G `  x )  -  ( G `  X )
) )  <  d
)
131129, 130eqbrtrd 4467 . . . . . . . . . . . . . . . 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 20658 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  d  e.  RR* )  /\  ( ( G `
 X )  e.  CC  /\  ( G `
 x )  e.  CC ) )  -> 
( ( G `  x )  e.  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  <->  ( ( G `  x )
( abs  o.  -  )
( G `  X
) )  <  d
) )
133132biimpar 485 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( abs 
o.  -  )  e.  ( *Met `  CC )  /\  d  e.  RR* )  /\  ( ( G `
 X )  e.  CC  /\  ( G `
 x )  e.  CC ) )  /\  ( ( G `  x ) ( abs 
o.  -  ) ( G `  X )
)  <  d )  ->  ( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )
134123, 126, 131, 133syl21anc 1227 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  ( abs `  ( ( G `
 x )  -  ( G `  X ) ) )  <  d
)  ->  ( G `  x )  e.  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
135119, 134syldan 470 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  (
( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) ) )  -> 
( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )
136135ex 434 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( ( abs `  (
Re `  ( ( G `  x )  -  ( G `  X ) ) ) )  <  ( d  /  2 )  /\  ( abs `  ( Im
`  ( ( G `
 x )  -  ( G `  X ) ) ) )  < 
( d  /  2
) )  ->  ( G `  x )  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
137105, 136syld 44 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
138 f1ocnv 5828 . . . . . . . . . . . . . . 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 5818 . . . . . . . . . . . . . 14  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  Fun  `' G
)
141139, 140ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  `' G
142 f1odm 5820 . . . . . . . . . . . . . . 15  |-  ( `' G : CC -1-1-onto-> ( RR  X.  RR )  ->  dom  `' G  =  CC )
143139, 142ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  `' G  =  CC
144116, 143syl6eleqr 2566 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( G `  x )  e.  dom  `' G )
145 funfvima 6135 . . . . . . . . . . . . 13  |-  ( ( Fun  `' G  /\  ( G `  x )  e.  dom  `' G
)  ->  ( ( G `  x )  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d )  ->  ( `' G `  ( G `  x
) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) ) )
146141, 144, 145sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( G `  x
)  e.  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d )  ->  ( `' G `  ( G `
 x ) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) ) )
147107, 110mp1i 12 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  G : ( RR  X.  RR ) -1-1-onto-> CC )
148 f1ocnvfv1 6170 . . . . . . . . . . . . . . 15  |-  ( ( G : ( RR 
X.  RR ) -1-1-onto-> CC  /\  x  e.  ( RR  X.  RR ) )  -> 
( `' G `  ( G `  x ) )  =  x )
149147, 100, 148syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  ( `' G `  ( G `
 x ) )  =  x )
150149eleq1d 2536 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( `' G `  ( G `  x ) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  <->  x  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) ) )
151150biimpd 207 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
( `' G `  ( G `  x ) )  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  ->  x  e.  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) ) ) )
152137, 146, 1513syld 55 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  ->  (
x  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  ->  x  e.  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) ) ) )
153152imp 429 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( RR  X.  RR ) )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  x  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
15495, 153syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( J  tX  J
)  /\  X  e.  A )  /\  d  e.  RR+ )  /\  x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) ) )  ->  x  e.  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) ) )
155154ex 434 . . . . . . . 8  |-  ( ( ( A  e.  ( J  tX  J )  /\  X  e.  A
)  /\  d  e.  RR+ )  ->  ( x  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  ->  x  e.  ( `' G " ( ( G `
 X ) (
ball `  ( abs  o. 
-  ) ) d ) ) ) )
156155ssrdv 3510 . . . . . . 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 2878 . . . . . 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 6388 . . . . . . . . . 10  |-  Fun  G
159158a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  Fun  G )
16013sselda 3504 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  ( RR 
X.  RR ) )
161 f1odm 5820 . . . . . . . . . . 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 2566 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  dom  G
)
164 simpr 461 . . . . . . . . 9  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  X  e.  A )
165 funfvima 6135 . . . . . . . . . 10  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( X  e.  A  ->  ( G `  X
)  e.  ( G
" A ) ) )
166165imp 429 . . . . . . . . 9  |-  ( ( ( Fun  G  /\  X  e.  dom  G )  /\  X  e.  A
)  ->  ( G `  X )  e.  ( G " A ) )
167159, 163, 164, 166syl21anc 1227 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( G `  X
)  e.  ( G
" A ) )
168 hmeoima 20029 . . . . . . . . . . 11  |-  ( ( G  e.  ( ( J  tX  J )
Homeo ( TopOpen ` fld ) )  /\  A  e.  ( J  tX  J
) )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
169107, 168mpan 670 . . . . . . . . . 10  |-  ( A  e.  ( J  tX  J )  ->  ( G " A )  e.  ( TopOpen ` fld ) )
170106cnfldtopn 21052 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
171170elmopn2 20711 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  e.  ( *Met `  CC )  ->  ( ( G " A )  e.  ( TopOpen ` fld )  <->  ( ( G
" A )  C_  CC  /\  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) ) ) )
172122, 171ax-mp 5 . . . . . . . . . . 11  |-  ( ( G " A )  e.  ( TopOpen ` fld )  <->  ( ( G
" A )  C_  CC  /\  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) ) )
173172simprbi 464 . . . . . . . . . 10  |-  ( ( G " A )  e.  ( TopOpen ` fld )  ->  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
174169, 173syl 16 . . . . . . . . 9  |-  ( A  e.  ( J  tX  J )  ->  A. m  e.  ( G " A
) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
175174adantr 465 . . . . . . . 8  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  A. m  e.  ( G " A ) E. d  e.  RR+  ( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
176 oveq1 6291 . . . . . . . . . . 11  |-  ( m  =  ( G `  X )  ->  (
m ( ball `  ( abs  o.  -  ) ) d )  =  ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d ) )
177176sseq1d 3531 . . . . . . . . . 10  |-  ( m  =  ( G `  X )  ->  (
( m ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  <->  ( ( G `  X )
( ball `  ( abs  o. 
-  ) ) d )  C_  ( G " A ) ) )
178177rexbidv 2973 . . . . . . . . 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 3212 . . . . . . . 8  |-  ( ( ( G `  X
)  e.  ( G
" A )  /\  A. m  e.  ( G
" A ) E. d  e.  RR+  (
m ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A ) )  ->  E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
180167, 175, 179syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A ) )
181 imass2 5372 . . . . . . . . . 10  |-  ( ( ( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  -> 
( `' G "
( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  ( `' G " ( G " A ) ) )
182 f1of1 5815 . . . . . . . . . . . . 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 5832 . . . . . . . . . . . 12  |-  ( ( G : ( RR 
X.  RR ) -1-1-> CC  /\  A  C_  ( RR  X.  RR ) )  -> 
( `' G "
( G " A
) )  =  A )
185183, 13, 184sylancr 663 . . . . . . . . . . 11  |-  ( A  e.  ( J  tX  J )  ->  ( `' G " ( G
" A ) )  =  A )
186185sseq2d 3532 . . . . . . . . . 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 2937 . . . . . . . 8  |-  ( A  e.  ( J  tX  J )  ->  ( E. d  e.  RR+  (
( G `  X
) ( ball `  ( abs  o.  -  ) ) d )  C_  ( G " A )  ->  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
189188adantr 465 . . . . . . 7  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  ( E. d  e.  RR+  ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) 
C_  ( G " A )  ->  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )
)
190180, 189mpd 15 . . . . . 6  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )
191 r19.29 2997 . . . . . 6  |-  ( ( A. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  E. d  e.  RR+  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )  ->  E. d  e.  RR+  ( ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
192157, 190, 191syl2anc 661 . . . . 5  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A ) )
193 sstr 3512 . . . . . 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 2932 . . . . 5  |-  ( E. d  e.  RR+  (
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  /\  ( `' G " ( ( G `  X ) ( ball `  ( abs  o.  -  ) ) d ) )  C_  A )  ->  E. d  e.  RR+  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A )
195192, 194syl 16 . . . 4  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A )
196 r19.29 2997 . . . 4  |-  ( ( A. d  e.  RR+  X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  E. d  e.  RR+  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
)  ->  E. d  e.  RR+  ( X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )
19770, 195, 196syl2anc 661 . . 3  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( X  e.  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  /\  (
( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
) )
198 r19.29 2997 . . 3  |-  ( ( A. d  e.  RR+  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  E. d  e.  RR+  ( X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )  ->  E. d  e.  RR+  (
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) ) )
19951, 197, 198syl2anc 661 . 2  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. d  e.  RR+  ( ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) ) )
200 eleq2 2540 . . . . 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 3525 . . . . 5  |-  ( r  =  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( r  C_  A  <->  ( ( ( ( 1st `  X )  -  (
d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  / 
2 ) ) )  X.  ( ( ( 2nd `  X )  -  ( d  / 
2 ) ) (,) ( ( 2nd `  X
)  +  ( d  /  2 ) ) ) )  C_  A
) )
202200, 201anbi12d 710 . . . 4  |-  ( r  =  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  -> 
( ( X  e.  r  /\  r  C_  A )  <->  ( X  e.  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) ) )
203202rspcev 3214 . . 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 2952 . 2  |-  ( E. d  e.  RR+  (
( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  e.  B  /\  ( X  e.  ( ( ( ( 1st `  X
)  -  ( d  /  2 ) ) (,) ( ( 1st `  X )  +  ( d  /  2 ) ) )  X.  (
( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  /\  ( ( ( ( 1st `  X )  -  ( d  / 
2 ) ) (,) ( ( 1st `  X
)  +  ( d  /  2 ) ) )  X.  ( ( ( 2nd `  X
)  -  ( d  /  2 ) ) (,) ( ( 2nd `  X )  +  ( d  /  2 ) ) ) )  C_  A ) )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
205199, 204syl 16 1  |-  ( ( A  e.  ( J 
tX  J )  /\  X  e.  A )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    o. ccom 5003   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783   CCcc 9490   RRcr 9491   _ici 9494    + caddc 9495    x. cmul 9497   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   2c2 10585   RR+crp 11220   (,)cioo 11529   Recre 12893   Imcim 12894   abscabs 13030   TopOpenctopn 14677   topGenctg 14693   *Metcxmt 18202   ballcbl 18204  ℂfldccnfld 18219   Topctop 19189    tX ctx 19824   Homeochmeo 20017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-icc 11536  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cn 19522  df-cnp 19523  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145
This theorem is referenced by:  dya2iocnei  27921
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