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Theorem caubl 22163
Description: Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
caubl.2  |-  ( ph  ->  D  e.  ( *Met `  X ) )
caubl.3  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
caubl.4  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )
caubl.5  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( F `  n ) )  < 
r )
Assertion
Ref Expression
caubl  |-  ( ph  ->  ( 1st  o.  F
)  e.  ( Cau `  D ) )
Distinct variable groups:    n, r, D    n, F, r    ph, r    n, X, r    ph, n

Proof of Theorem caubl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 caubl.5 . . 3  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( F `  n ) )  < 
r )
2 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( r  =  n  ->  ( F `  r )  =  ( F `  n ) )
32fveq2d 5876 . . . . . . . . . . . . 13  |-  ( r  =  n  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  n ) ) )
43sseq1d 3488 . . . . . . . . . . . 12  |-  ( r  =  n  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  n ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
54imbi2d 317 . . . . . . . . . . 11  |-  ( r  =  n  ->  (
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )  <-> 
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  n ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) ) ) )
6 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
76fveq2d 5876 . . . . . . . . . . . . 13  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
87sseq1d 3488 . . . . . . . . . . . 12  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
98imbi2d 317 . . . . . . . . . . 11  |-  ( r  =  k  ->  (
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )  <-> 
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  k ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) ) ) )
10 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1110fveq2d 5876 . . . . . . . . . . . . 13  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
1211sseq1d 3488 . . . . . . . . . . . 12  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
1312imbi2d 317 . . . . . . . . . . 11  |-  ( r  =  ( k  +  1 )  ->  (
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )  <-> 
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) ) ) )
14 ssid 3480 . . . . . . . . . . . 12  |-  ( (
ball `  D ) `  ( F `  n
) )  C_  (
( ball `  D ) `  ( F `  n
) )
15142a1i 12 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
( ph  /\  n  e.  NN )  ->  (
( ball `  D ) `  ( F `  n
) )  C_  (
( ball `  D ) `  ( F `  n
) ) ) )
16 caubl.4 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )
17 eluznn 11218 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  k  e.  ( ZZ>= `  n ) )  -> 
k  e.  NN )
18 oveq1 6303 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1918fveq2d 5876 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2019fveq2d 5876 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
21 fveq2 5872 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
2221fveq2d 5876 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
2320, 22sseq12d 3490 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
2423rspccva 3178 . . . . . . . . . . . . . . . 16  |-  ( ( A. n  e.  NN  ( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  /\  k  e.  NN )  ->  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  k ) ) )
2516, 17, 24syl2an 479 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  NN  /\  k  e.  ( ZZ>= `  n )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
2625anassrs 652 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
27 sstr2 3468 . . . . . . . . . . . . . 14  |-  ( ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  k ) )  -> 
( ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  n ) )  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
2826, 27syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  ( ZZ>= `  n )
)  ->  ( (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  n
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  n
) ) ) )
2928expcom 436 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ph  /\  n  e.  NN )  ->  ( ( (
ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  n
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  n
) ) ) ) )
3029a2d 29 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  n
)  ->  ( (
( ph  /\  n  e.  NN )  ->  (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  n
) ) )  -> 
( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) ) ) )
315, 9, 13, 9, 15, 30uzind4 11206 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3231com12 32 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3332ad2ant2r 751 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  ( k  e.  ( ZZ>= `  n )  ->  ( ( ball `  D
) `  ( F `  k ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) ) )
34 relxp 4953 . . . . . . . . . . . . . . . 16  |-  Rel  ( X  X.  RR+ )
35 caubl.3 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
3635ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  F : NN
--> ( X  X.  RR+ ) )
37 simplrl 768 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  n  e.  NN )
3836, 37ffvelrnd 6029 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  e.  ( X  X.  RR+ )
)
39 1st2nd 6844 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 n )  e.  ( X  X.  RR+ ) )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
4034, 38, 39sylancr 667 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
4140fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  n ) )  =  ( (
ball `  D ) `  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
)
42 df-ov 6299 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) ( 2nd `  ( F `  n )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  n
) ) ,  ( 2nd `  ( F `
 n ) )
>. )
4341, 42syl6eqr 2479 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  n ) )  =  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) ( 2nd `  ( F `  n )
) ) )
44 caubl.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  ( *Met `  X ) )
4544ad3antrrr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  D  e.  ( *Met `  X
) )
46 xp1st 6828 . . . . . . . . . . . . . . 15  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  n
) )  e.  X
)
4738, 46syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  n
) )  e.  X
)
48 xp2nd 6829 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
4938, 48syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
5049rpxrd 11331 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR* )
51 simpllr 767 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR+ )
5251rpxrd 11331 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR* )
53 simplrr 769 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <  r
)
54 rpre 11297 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR+  ->  ( 2nd `  ( F `  n
) )  e.  RR )
55 rpre 11297 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
56 ltle 9711 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR  /\  r  e.  RR )  ->  ( ( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5754, 55, 56syl2an 479 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR+  /\  r  e.  RR+ )  ->  (
( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5849, 51, 57syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( 2nd `  ( F `  n ) )  < 
r  ->  ( 2nd `  ( F `  n
) )  <_  r
) )
5953, 58mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <_  r
)
60 ssbl 21362 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( *Met `  X
)  /\  ( 1st `  ( F `  n
) )  e.  X
)  /\  ( ( 2nd `  ( F `  n ) )  e. 
RR*  /\  r  e.  RR* )  /\  ( 2nd `  ( F `  n
) )  <_  r
)  ->  ( ( 1st `  ( F `  n ) ) (
ball `  D )
( 2nd `  ( F `  n )
) )  C_  (
( 1st `  ( F `  n )
) ( ball `  D
) r ) )
6145, 47, 50, 52, 59, 60syl221anc 1275 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( 1st `  ( F `  n ) ) (
ball `  D )
( 2nd `  ( F `  n )
) )  C_  (
( 1st `  ( F `  n )
) ( ball `  D
) r ) )
6243, 61eqsstrd 3495 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  n ) )  C_  ( ( 1st `  ( F `  n ) ) (
ball `  D )
r ) )
63 sstr2 3468 . . . . . . . . . . . 12  |-  ( ( ( ball `  D
) `  ( F `  k ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  -> 
( ( ( ball `  D ) `  ( F `  n )
)  C_  ( ( 1st `  ( F `  n ) ) (
ball `  D )
r )  ->  (
( ball `  D ) `  ( F `  k
) )  C_  (
( 1st `  ( F `  n )
) ( ball `  D
) r ) ) )
6462, 63syl5com 31 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  n
) )  ->  (
( ball `  D ) `  ( F `  k
) )  C_  (
( 1st `  ( F `  n )
) ( ball `  D
) r ) ) )
65 simprl 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  n  e.  NN )
6665, 17sylan 473 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  k  e.  NN )
6736, 66ffvelrnd 6029 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
68 xp1st 6828 . . . . . . . . . . . . . 14  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  k
) )  e.  X
)
6967, 68syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  X
)
70 xp2nd 6829 . . . . . . . . . . . . . 14  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
7167, 70syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
72 blcntr 21352 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( *Met `  X )  /\  ( 1st `  ( F `  k )
)  e.  X  /\  ( 2nd `  ( F `
 k ) )  e.  RR+ )  ->  ( 1st `  ( F `  k ) )  e.  ( ( 1st `  ( F `  k )
) ( ball `  D
) ( 2nd `  ( F `  k )
) ) )
7345, 69, 71, 72syl3anc 1264 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  ( ( 1st `  ( F `  k )
) ( ball `  D
) ( 2nd `  ( F `  k )
) ) )
74 1st2nd 6844 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 k )  e.  ( X  X.  RR+ ) )  ->  ( F `  k )  =  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
7534, 67, 74sylancr 667 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
7675fveq2d 5876 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  =  ( (
ball `  D ) `  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
)
77 df-ov 6299 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
7876, 77syl6eqr 2479 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  =  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) ) )
7973, 78eleqtrrd 2511 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  ( ( ball `  D
) `  ( F `  k ) ) )
80 ssel 3455 . . . . . . . . . . 11  |-  ( ( ( ball `  D
) `  ( F `  k ) )  C_  ( ( 1st `  ( F `  n )
) ( ball `  D
) r )  -> 
( ( 1st `  ( F `  k )
)  e.  ( (
ball `  D ) `  ( F `  k
) )  ->  ( 1st `  ( F `  k ) )  e.  ( ( 1st `  ( F `  n )
) ( ball `  D
) r ) ) )
8164, 79, 80syl6ci 67 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  n
) )  ->  ( 1st `  ( F `  k ) )  e.  ( ( 1st `  ( F `  n )
) ( ball `  D
) r ) ) )
82 elbl2 21329 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  r  e.  RR* )  /\  ( ( 1st `  ( F `
 n ) )  e.  X  /\  ( 1st `  ( F `  k ) )  e.  X ) )  -> 
( ( 1st `  ( F `  k )
)  e.  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) r )  <->  ( ( 1st `  ( F `  n ) ) D ( 1st `  ( F `  k )
) )  <  r
) )
8345, 52, 47, 69, 82syl22anc 1265 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( 1st `  ( F `  k ) )  e.  ( ( 1st `  ( F `  n )
) ( ball `  D
) r )  <->  ( ( 1st `  ( F `  n ) ) D ( 1st `  ( F `  k )
) )  <  r
) )
8481, 83sylibd 217 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  n
) )  ->  (
( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
8584ex 435 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  ( k  e.  ( ZZ>= `  n )  ->  ( ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  n ) )  ->  ( ( 1st `  ( F `  n ) ) D ( 1st `  ( F `  k )
) )  <  r
) ) )
8633, 85mpdd 41 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  ( k  e.  ( ZZ>= `  n )  ->  ( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
8786ralrimiv 2835 . . . . . 6  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  A. k  e.  (
ZZ>= `  n ) ( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r )
8887expr 618 . . . . 5  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  n  e.  NN )  ->  (
( 2nd `  ( F `  n )
)  <  r  ->  A. k  e.  ( ZZ>= `  n ) ( ( 1st `  ( F `
 n ) ) D ( 1st `  ( F `  k )
) )  <  r
) )
8988reximdva 2898 . . . 4  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( E. n  e.  NN  ( 2nd `  ( F `  n ) )  < 
r  ->  E. n  e.  NN  A. k  e.  ( ZZ>= `  n )
( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
9089ralimdva 2831 . . 3  |-  ( ph  ->  ( A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( F `
 n ) )  <  r  ->  A. r  e.  RR+  E. n  e.  NN  A. k  e.  ( ZZ>= `  n )
( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
911, 90mpd 15 . 2  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( 1st `  ( F `
 n ) ) D ( 1st `  ( F `  k )
) )  <  r
)
92 nnuz 11183 . . 3  |-  NN  =  ( ZZ>= `  1 )
93 1zzd 10957 . . 3  |-  ( ph  ->  1  e.  ZZ )
94 fvco3 5949 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
9535, 94sylan 473 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1st  o.  F ) `
 k )  =  ( 1st `  ( F `  k )
) )
96 fvco3 5949 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  n  e.  NN )  ->  (
( 1st  o.  F
) `  n )  =  ( 1st `  ( F `  n )
) )
9735, 96sylan 473 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  F ) `
 n )  =  ( 1st `  ( F `  n )
) )
98 1stcof 6826 . . . 4  |-  ( F : NN --> ( X  X.  RR+ )  ->  ( 1st  o.  F ) : NN --> X )
9935, 98syl 17 . . 3  |-  ( ph  ->  ( 1st  o.  F
) : NN --> X )
10092, 44, 93, 95, 97, 99iscauf 22136 . 2  |-  ( ph  ->  ( ( 1st  o.  F )  e.  ( Cau `  D )  <->  A. r  e.  RR+  E. n  e.  NN  A. k  e.  ( ZZ>= `  n )
( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
10191, 100mpbird 235 1  |-  ( ph  ->  ( 1st  o.  F
)  e.  ( Cau `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774    C_ wss 3433   <.cop 3999   class class class wbr 4417    X. cxp 4843    o. ccom 4849   Rel wrel 4850   -->wf 5588   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   RRcr 9527   1c1 9529    + caddc 9531   RR*cxr 9663    < clt 9664    <_ cle 9665   NNcn 10598   ZZcz 10926   ZZ>=cuz 11148   RR+crp 11291   *Metcxmt 18883   ballcbl 18885   Caucca 22109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-psmet 18890  df-xmet 18891  df-bl 18893  df-cau 22112
This theorem is referenced by:  bcthlem4  22181  heiborlem9  31855
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