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Theorem bcthlem1 19279
Description: Lemma for bcth 19284. 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 4952 . . . . . . 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 5759 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  X  e.  dom  CMet )
42, 3syl 16 . . . . . . . 8  |-  ( ph  ->  X  e.  dom  CMet )
5 reex 9083 . . . . . . . . 9  |-  RR  e.  _V
6 rpssre 10624 . . . . . . . . 9  |-  RR+  C_  RR
75, 6ssexi 4350 . . . . . . . 8  |-  RR+  e.  _V
8 xpexg 4991 . . . . . . . 8  |-  ( ( X  e.  dom  CMet  /\  RR+  e.  _V )  -> 
( X  X.  RR+ )  e.  _V )
94, 7, 8sylancl 645 . . . . . . 7  |-  ( ph  ->  ( X  X.  RR+ )  e.  _V )
10 ssexg 4351 . . . . . . 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 646 . . . . . 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 6091 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
1  /  k )  =  ( 1  /  A ) )
1312breq2d 4226 . . . . . . . . . 10  |-  ( k  =  A  ->  (
r  <  ( 1  /  k )  <->  r  <  ( 1  /  A ) ) )
14 fveq2 5730 . . . . . . . . . . . 12  |-  ( k  =  A  ->  ( M `  k )  =  ( M `  A ) )
1514difeq2d 3467 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
( ( ball `  D
) `  z )  \  ( M `  k ) )  =  ( ( ( ball `  D ) `  z
)  \  ( M `  A ) ) )
1615sseq2d 3378 . . . . . . . . . 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 693 . . . . . . . . 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 686 . . . . . . . 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 4273 . . . . . . 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 5730 . . . . . . . . . . . 12  |-  ( z  =  B  ->  (
( ball `  D ) `  z )  =  ( ( ball `  D
) `  B )
)
2120difeq1d 3466 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
( ( ball `  D
) `  z )  \  ( M `  A ) )  =  ( ( ( ball `  D ) `  B
)  \  ( M `  A ) ) )
2221sseq2d 3378 . . . . . . . . . 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 686 . . . . . . . . 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 686 . . . . . . . 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 4273 . . . . . . 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 6210 . . . . . 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 1220 . . . . 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 1154 . . . 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 2505 . 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 3346 . . 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 958 . . 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 6388 . . . . . 6  |-  ( C  e.  ( X  X.  RR+ )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
3534eleq1d 2504 . . . . 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 5744 . . . . . 6  |-  ( 1st `  C )  e.  _V
37 fvex 5744 . . . . . 6  |-  ( 2nd `  C )  e.  _V
38 eleq1 2498 . . . . . . . 8  |-  ( x  =  ( 1st `  C
)  ->  ( x  e.  X  <->  ( 1st `  C
)  e.  X ) )
39 eleq1 2498 . . . . . . . 8  |-  ( r  =  ( 2nd `  C
)  ->  ( r  e.  RR+  <->  ( 2nd `  C
)  e.  RR+ )
)
4038, 39bi2anan9 845 . . . . . . 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 4224 . . . . . . . 8  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
r  <  ( 1  /  A )  <->  ( 2nd `  C )  <  (
1  /  A ) ) )
43 oveq12 6092 . . . . . . . . . 10  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
x ( ball `  D
) r )  =  ( ( 1st `  C
) ( ball `  D
) ( 2nd `  C
) ) )
4443fveq2d 5734 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  C )  /\  r  =  ( 2nd `  C
) )  ->  (
( cls `  J
) `  ( x
( ball `  D )
r ) )  =  ( ( cls `  J
) `  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) ) ) )
4544sseq1d 3377 . . . . . . . 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 693 . . . . . . 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 693 . . . . . 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 4473 . . . . 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 2504 . . . . . . 7  |-  ( C  e.  ( X  X.  RR+ )  ->  ( C  e.  ( X  X.  RR+ ) 
<-> 
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  ( X  X.  RR+ )
) )
51 opelxp 4910 . . . . . . 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 5734 . . . . . . . . . 10  |-  ( C  e.  ( X  X.  RR+ )  ->  ( ( ball `  D ) `  C )  =  ( ( ball `  D
) `  <. ( 1st `  C ) ,  ( 2nd `  C )
>. ) )
54 df-ov 6086 . . . . . . . . . 10  |-  ( ( 1st `  C ) ( ball `  D
) ( 2nd `  C
) )  =  ( ( ball `  D
) `  <. ( 1st `  C ) ,  ( 2nd `  C )
>. )
5553, 54syl6reqr 2489 . . . . . . . . 9  |-  ( C  e.  ( X  X.  RR+ )  ->  ( ( 1st `  C ) (
ball `  D )
( 2nd `  C
) )  =  ( ( ball `  D
) `  C )
)
5655fveq2d 5734 . . . . . . . 8  |-  ( C  e.  ( X  X.  RR+ )  ->  ( ( cls `  J ) `  ( ( 1st `  C
) ( ball `  D
) ( 2nd `  C
) ) )  =  ( ( cls `  J
) `  ( ( ball `  D ) `  C ) ) )
5756sseq1d 3377 . . . . . . 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 686 . . . . . 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 693 . . . . 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 941 . . . . 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 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    C_ wss 3322   <.cop 3819   class class class wbr 4214   {copab 4267    X. cxp 4878   dom cdm 4880   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   RRcr 8991   1c1 8993    < clt 9122    / cdiv 9679   NNcn 10002   RR+crp 10614   ballcbl 16690   MetOpencmopn 16693   clsccl 17084   CMetcms 19209
This theorem is referenced by:  bcthlem2  19280  bcthlem3  19281  bcthlem4  19282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-rp 10615
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