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Theorem bcthlem1 18694
Description: Lemma for bcth 18699. Substitutions for the function  F. (Contributed by Mario Carneiro, 9-Jan-2014.)
Hypotheses
Ref Expression
bcth.2  |-  J  =  ( MetOpen `  D )
bcthlem.4  |-  ( ph  ->  D  e.  ( CMet `  X ) )
bcthlem.5  |-  F  =  ( k  e.  NN ,  z  e.  ( X  X.  RR+ )  |->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) } )
Assertion
Ref Expression
bcthlem1  |-  ( (
ph  /\  ( A  e.  NN  /\  B  e.  ( X  X.  RR+ ) ) )  -> 
( C  e.  ( A F B )  <-> 
( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
Distinct variable groups:    k, r, x, z, A    B, k,
r, x, z    C, r, x    D, k, r, x, z    k, F, r, x, z    k, J, r, x, z    k, M, r, x, z    ph, k,
r, x, z    k, X, r, x, z
Allowed substitution hints:    C( z, k)

Proof of Theorem bcthlem1
StepHypRef Expression
1 opabssxp 4736 . . . . . . 7  |-  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) } 
C_  ( X  X.  RR+ )
2 bcthlem.4 . . . . . . . . 9  |-  ( ph  ->  D  e.  ( CMet `  X ) )
3 elfvdm 5474 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  dom  CMet )
42, 3syl 17 . . . . . . . 8  |-  ( ph  ->  X  e.  dom  CMet )
5 reex 8782 . . . . . . . . 9  |-  RR  e.  _V
6 rpssre 10317 . . . . . . . . 9  |-  RR+  C_  RR
75, 6ssexi 4119 . . . . . . . 8  |-  RR+  e.  _V
8 xpexg 4774 . . . . . . . 8  |-  ( ( X  e.  dom  CMet  /\  RR+  e.  _V )  -> 
( X  X.  RR+ )  e.  _V )
94, 7, 8sylancl 646 . . . . . . 7  |-  ( ph  ->  ( X  X.  RR+ )  e.  _V )
10 ssexg 4120 . . . . . . 7  |-  ( ( { <. x ,  r
>.  |  ( (
x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) } 
C_  ( X  X.  RR+ )  /\  ( X  X.  RR+ )  e.  _V )  ->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  e.  _V )
111, 9, 10sylancr 647 . . . . . 6  |-  ( ph  ->  { <. x ,  r
>.  |  ( (
x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  e.  _V )
12 oveq2 5786 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
1  /  k )  =  ( 1  /  A ) )
1312breq2d 3995 . . . . . . . . . 10  |-  ( k  =  A  ->  (
r  <  ( 1  /  k )  <->  r  <  ( 1  /  A ) ) )
14 fveq2 5444 . . . . . . . . . . . 12  |-  ( k  =  A  ->  ( M `  k )  =  ( M `  A ) )
1514difeq2d 3255 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
( ( ball `  D
) `  z )  \  ( M `  k ) )  =  ( ( ( ball `  D ) `  z
)  \  ( M `  A ) ) )
1615sseq2d 3167 . . . . . . . . . 10  |-  ( k  =  A  ->  (
( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  k ) )  <->  ( ( cls `  J ) `  ( x ( ball `  D ) r ) )  C_  ( (
( ball `  D ) `  z )  \  ( M `  A )
) ) )
1713, 16anbi12d 694 . . . . . . . . 9  |-  ( k  =  A  ->  (
( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) )  <->  ( r  <  ( 1  /  A
)  /\  ( ( cls `  J ) `  ( x ( ball `  D ) r ) )  C_  ( (
( ball `  D ) `  z )  \  ( M `  A )
) ) ) )
1817anbi2d 687 . . . . . . . 8  |-  ( k  =  A  ->  (
( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  k )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  k ) ) ) )  <->  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  A ) ) ) ) ) )
1918opabbidv 4042 . . . . . . 7  |-  ( k  =  A  ->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) }  =  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  A )
) ) ) } )
20 fveq2 5444 . . . . . . . . . . . 12  |-  ( z  =  B  ->  (
( ball `  D ) `  z )  =  ( ( ball `  D
) `  B )
)
2120difeq1d 3254 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
( ( ball `  D
) `  z )  \  ( M `  A ) )  =  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) )
2221sseq2d 3167 . . . . . . . . . 10  |-  ( z  =  B  ->  (
( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  A ) )  <->  ( ( cls `  J ) `  ( x ( ball `  D ) r ) )  C_  ( (
( ball `  D ) `  B )  \  ( M `  A )
) ) )
2322anbi2d 687 . . . . . . . . 9  |-  ( z  =  B  ->  (
( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  A )
) )  <->  ( r  <  ( 1  /  A
)  /\  ( ( cls `  J ) `  ( x ( ball `  D ) r ) )  C_  ( (
( ball `  D ) `  B )  \  ( M `  A )
) ) ) )
2423anbi2d 687 . . . . . . . 8  |-  ( z  =  B  ->  (
( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  A ) ) ) )  <->  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) ) )
2524opabbidv 4042 . . . . . . 7  |-  ( z  =  B  ->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  A )
) ) ) }  =  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) } )
26 bcthlem.5 . . . . . . 7  |-  F  =  ( k  e.  NN ,  z  e.  ( X  X.  RR+ )  |->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) } )
2719, 25, 26ovmpt2g 5902 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( X  X.  RR+ )  /\  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) }  e.  _V )  ->  ( A F B )  =  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) } )
2811, 27syl3an3 1222 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( X  X.  RR+ )  /\  ph )  ->  ( A F B )  =  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) } )
29283expa 1156 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  ( X  X.  RR+ ) )  /\  ph )  ->  ( A F B )  =  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) } )
3029ancoms 441 . . 3  |-  ( (
ph  /\  ( A  e.  NN  /\  B  e.  ( X  X.  RR+ ) ) )  -> 
( A F B )  =  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) } )
3130eleq2d 2323 . 2  |-  ( (
ph  /\  ( A  e.  NN  /\  B  e.  ( X  X.  RR+ ) ) )  -> 
( C  e.  ( A F B )  <-> 
C  e.  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) } ) )
321sseli 3137 . . 3  |-  ( C  e.  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  ->  C  e.  ( X  X.  RR+ )
)
33 simp1 960 . . 3  |-  ( ( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C )  < 
( 1  /  A
)  /\  ( ( cls `  J ) `  ( ( ball `  D
) `  C )
)  C_  ( (
( ball `  D ) `  B )  \  ( M `  A )
) )  ->  C  e.  ( X  X.  RR+ ) )
34 1st2nd2 6079 . . . . . 6  |-  ( C  e.  ( X  X.  RR+ )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
3534eleq1d 2322 . . . . 5  |-  ( C  e.  ( X  X.  RR+ )  ->  ( C  e.  { <. x ,  r
>.  |  ( (
x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  <->  <. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) } ) )
36 fvex 5458 . . . . . 6  |-  ( 1st `  C )  e.  _V
37 fvex 5458 . . . . . 6  |-  ( 2nd `  C )  e.  _V
38 eleq1 2316 . . . . . . . 8  |-  ( x  =  ( 1st `  C
)  ->  ( x  e.  X  <->  ( 1st `  C
)  e.  X ) )
39 eleq1 2316 . . . . . . . 8  |-  ( r  =  ( 2nd `  C
)  ->  ( r  e.  RR+  <->  ( 2nd `  C
)  e.  RR+ )
)
4038, 39bi2anan9 848 . . . . . . 7  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
( x  e.  X  /\  r  e.  RR+ )  <->  ( ( 1st `  C
)  e.  X  /\  ( 2nd `  C )  e.  RR+ ) ) )
41 simpr 449 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  r  =  ( 2nd `  C
) )
4241breq1d 3993 . . . . . . . 8  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
r  <  ( 1  /  A )  <->  ( 2nd `  C )  <  (
1  /  A ) ) )
43 oveq12 5787 . . . . . . . . . 10  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
x ( ball `  D
) r )  =  ( ( 1st `  C
) ( ball `  D
) ( 2nd `  C
) ) )
4443fveq2d 5448 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
( cls `  J
) `  ( x
( ball `  D )
r ) )  =  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) ) )
4544sseq1d 3166 . . . . . . . 8  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) )  <->  ( ( cls `  J ) `  ( ( 1st `  C
) ( ball `  D
) ( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) )
4642, 45anbi12d 694 . . . . . . 7  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) )  <->  ( ( 2nd `  C )  < 
( 1  /  A
)  /\  ( ( cls `  J ) `  ( ( 1st `  C
) ( ball `  D
) ( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
4740, 46anbi12d 694 . . . . . 6  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) )  <->  ( ( ( 1st `  C )  e.  X  /\  ( 2nd `  C )  e.  RR+ )  /\  (
( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) ) )
4836, 37, 47opelopaba 4239 . . . . 5  |-  ( <.
( 1st `  C
) ,  ( 2nd `  C ) >.  e.  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) }  <->  ( (
( 1st `  C
)  e.  X  /\  ( 2nd `  C )  e.  RR+ )  /\  (
( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
4935, 48syl6bb 254 . . . 4  |-  ( C  e.  ( X  X.  RR+ )  ->  ( C  e.  { <. x ,  r
>.  |  ( (
x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  <-> 
( ( ( 1st `  C )  e.  X  /\  ( 2nd `  C
)  e.  RR+ )  /\  ( ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) ) )
5034eleq1d 2322 . . . . . . 7  |-  ( C  e.  ( X  X.  RR+ )  ->  ( C  e.  ( X  X.  RR+ ) 
<-> 
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  ( X  X.  RR+ )
) )
51 opelxp 4693 . . . . . . 7  |-  ( <.
( 1st `  C
) ,  ( 2nd `  C ) >.  e.  ( X  X.  RR+ )  <->  ( ( 1st `  C
)  e.  X  /\  ( 2nd `  C )  e.  RR+ ) )
5250, 51syl6rbb 255 . . . . . 6  |-  ( C  e.  ( X  X.  RR+ )  ->  ( (
( 1st `  C
)  e.  X  /\  ( 2nd `  C )  e.  RR+ )  <->  C  e.  ( X  X.  RR+ )
) )
5334fveq2d 5448 . . . . . . . . . 10  |-  ( C  e.  ( X  X.  RR+ )  ->  ( ( ball `  D ) `  C )  =  ( ( ball `  D
) `  <. ( 1st `  C ) ,  ( 2nd `  C )
>. ) )
54 df-ov 5781 . . . . . . . . . 10  |-  ( ( 1st `  C ) ( ball `  D
) ( 2nd `  C
) )  =  ( ( ball `  D
) `  <. ( 1st `  C ) ,  ( 2nd `  C )
>. )
5553, 54syl6reqr 2307 . . . . . . . . 9  |-  ( C  e.  ( X  X.  RR+ )  ->  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) )  =  ( ( ball `  D
) `  C )
)
5655fveq2d 5448 . . . . . . . 8  |-  ( C  e.  ( X  X.  RR+ )  ->  ( ( cls `  J ) `  ( ( 1st `  C
) ( ball `  D
) ( 2nd `  C
) ) )  =  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) ) )
5756sseq1d 3166 . . . . . . 7  |-  ( C  e.  ( X  X.  RR+ )  ->  ( (
( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) )  <->  ( ( cls `  J ) `  ( ( ball `  D
) `  C )
)  C_  ( (
( ball `  D ) `  B )  \  ( M `  A )
) ) )
5857anbi2d 687 . . . . . 6  |-  ( C  e.  ( X  X.  RR+ )  ->  ( (
( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) )  <-> 
( ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
5952, 58anbi12d 694 . . . . 5  |-  ( C  e.  ( X  X.  RR+ )  ->  ( (
( ( 1st `  C
)  e.  X  /\  ( 2nd `  C )  e.  RR+ )  /\  (
( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) )  <->  ( C  e.  ( X  X.  RR+ )  /\  ( ( 2nd `  C )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) ) )
60 3anass 943 . . . . 5  |-  ( ( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C )  < 
( 1  /  A
)  /\  ( ( cls `  J ) `  ( ( ball `  D
) `  C )
)  C_  ( (
( ball `  D ) `  B )  \  ( M `  A )
) )  <->  ( C  e.  ( X  X.  RR+ )  /\  ( ( 2nd `  C )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
6159, 60syl6bbr 256 . . . 4  |-  ( C  e.  ( X  X.  RR+ )  ->  ( (
( ( 1st `  C
)  e.  X  /\  ( 2nd `  C )  e.  RR+ )  /\  (
( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) )  <->  ( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
6249, 61bitrd 246 . . 3  |-  ( C  e.  ( X  X.  RR+ )  ->  ( C  e.  { <. x ,  r
>.  |  ( (
x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  <-> 
( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
6332, 33, 62pm5.21nii 344 . 2  |-  ( C  e.  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  B )  \  ( M `  A )
) ) ) }  <-> 
( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) )
6431, 63syl6bb 254 1  |-  ( (
ph  /\  ( A  e.  NN  /\  B  e.  ( X  X.  RR+ ) ) )  -> 
( C  e.  ( A F B )  <-> 
( C  e.  ( X  X.  RR+ )  /\  ( 2nd `  C
)  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) )  C_  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2757    \ cdif 3110    C_ wss 3113   <.cop 3603   class class class wbr 3983   {copab 4036    X. cxp 4645   dom cdm 4647   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   1stc1st 6040   2ndc2nd 6041   RRcr 8690   1c1 8692    < clt 8821    / cdiv 9377   NNcn 9700   RR+crp 10307   ballcbl 16319   MetOpencmopn 16320   clsccl 16703   CMetcms 18628
This theorem is referenced by:  bcthlem2  18695  bcthlem3  18696  bcthlem4  18697
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-rp 10308
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