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Theorem caubl 20830
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 5703 . . . . . . . . . . . . . 14  |-  ( r  =  n  ->  ( F `  r )  =  ( F `  n ) )
32fveq2d 5707 . . . . . . . . . . . . 13  |-  ( r  =  n  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  n ) ) )
43sseq1d 3395 . . . . . . . . . . . 12  |-  ( r  =  n  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  n ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
54imbi2d 316 . . . . . . . . . . 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 5703 . . . . . . . . . . . . . 14  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
76fveq2d 5707 . . . . . . . . . . . . 13  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
87sseq1d 3395 . . . . . . . . . . . 12  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
98imbi2d 316 . . . . . . . . . . 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 5703 . . . . . . . . . . . . . 14  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1110fveq2d 5707 . . . . . . . . . . . . 13  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
1211sseq1d 3395 . . . . . . . . . . . 12  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
1312imbi2d 316 . . . . . . . . . . 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 3387 . . . . . . . . . . . 12  |-  ( (
ball `  D ) `  ( F `  n
) )  C_  (
( ball `  D ) `  ( F `  n
) )
1514a1ii 27 . . . . . . . . . . 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 10937 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  k  e.  ( ZZ>= `  n ) )  -> 
k  e.  NN )
18 oveq1 6110 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1918fveq2d 5707 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2019fveq2d 5707 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
21 fveq2 5703 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
2221fveq2d 5707 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
2320, 22sseq12d 3397 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
2423rspccva 3084 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  NN  /\  k  e.  ( ZZ>= `  n )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
2625anassrs 648 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
27 sstr2 3375 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 435 . . . . . . . . . . . 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 26 . . . . . . . . . . 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 10924 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3231com12 31 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3332ad2ant2r 746 . . . . . . . 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 4959 . . . . . . . . . . . . . . . 16  |-  Rel  ( X  X.  RR+ )
35 caubl.3 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
3635ad3antrrr 729 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  F : NN
--> ( X  X.  RR+ ) )
37 simplrl 759 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  n  e.  NN )
3836, 37ffvelrnd 5856 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  e.  ( X  X.  RR+ )
)
39 1st2nd 6632 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 n )  e.  ( X  X.  RR+ ) )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
4034, 38, 39sylancr 663 . . . . . . . . . . . . . . 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 5707 . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) ( 2nd `  ( F `  n )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  n
) ) ,  ( 2nd `  ( F `
 n ) )
>. )
4341, 42syl6eqr 2493 . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  D  e.  ( *Met `  X
) )
46 xp1st 6618 . . . . . . . . . . . . . . 15  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  n
) )  e.  X
)
4738, 46syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  n
) )  e.  X
)
48 xp2nd 6619 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
4938, 48syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
5049rpxrd 11040 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR* )
51 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR+ )
5251rpxrd 11040 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR* )
53 simplrr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <  r
)
54 rpre 11009 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR+  ->  ( 2nd `  ( F `  n
) )  e.  RR )
55 rpre 11009 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
56 ltle 9475 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR  /\  r  e.  RR )  ->  ( ( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5754, 55, 56syl2an 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR+  /\  r  e.  RR+ )  ->  (
( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5849, 51, 57syl2anc 661 . . . . . . . . . . . . . . 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 20010 . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . 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 3402 . . . . . . . . . . . 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 3375 . . . . . . . . . . . 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 30 . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  n  e.  NN )
6665, 17sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  k  e.  NN )
6736, 66ffvelrnd 5856 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
68 xp1st 6618 . . . . . . . . . . . . . . 15  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  k
) )  e.  X
)
6967, 68syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  X
)
70 xp2nd 6619 . . . . . . . . . . . . . . 15  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
7167, 70syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
72 blcntr 20000 . . . . . . . . . . . . . 14  |-  ( ( 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 1218 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 6632 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 k )  e.  ( X  X.  RR+ ) )  ->  ( F `  k )  =  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
7534, 67, 74sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
7675fveq2d 5707 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 6106 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
7876, 77syl6eqr 2493 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 2520 . . . . . . . . . . . 12  |-  ( ( ( ( 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 3362 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) )
8179, 80syl5com 30 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( (
( ball `  D ) `  ( F `  k
) )  C_  (
( 1st `  ( F `  n )
) ( ball `  D
) r )  -> 
( 1st `  ( F `  k )
)  e.  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) r ) ) )
8264, 81syld 44 . . . . . . . . . 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 ) ) )
83 elbl2 19977 . . . . . . . . . . 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
) )
8445, 52, 47, 69, 83syl22anc 1219 . . . . . . . . . 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
) )
8582, 84sylibd 214 . . . . . . . . 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 ) )
8685ex 434 . . . . . . . 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
) ) )
8733, 86mpdd 40 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  ( k  e.  ( ZZ>= `  n )  ->  ( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
8887ralrimiv 2810 . . . . . 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 )
8988expr 615 . . . . 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
) )
9089reximdva 2840 . . . 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 ) )
9190ralimdva 2806 . . 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 ) )
921, 91mpd 15 . 2  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( 1st `  ( F `
 n ) ) D ( 1st `  ( F `  k )
) )  <  r
)
93 nnuz 10908 . . 3  |-  NN  =  ( ZZ>= `  1 )
94 1zzd 10689 . . 3  |-  ( ph  ->  1  e.  ZZ )
95 fvco3 5780 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
9635, 95sylan 471 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1st  o.  F ) `
 k )  =  ( 1st `  ( F `  k )
) )
97 fvco3 5780 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  n  e.  NN )  ->  (
( 1st  o.  F
) `  n )  =  ( 1st `  ( F `  n )
) )
9835, 97sylan 471 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  F ) `
 n )  =  ( 1st `  ( F `  n )
) )
99 1stcof 6616 . . . 4  |-  ( F : NN --> ( X  X.  RR+ )  ->  ( 1st  o.  F ) : NN --> X )
10035, 99syl 16 . . 3  |-  ( ph  ->  ( 1st  o.  F
) : NN --> X )
10193, 44, 94, 96, 98, 100iscauf 20803 . 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 ) )
10292, 101mpbird 232 1  |-  ( ph  ->  ( 1st  o.  F
)  e.  ( Cau `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   E.wrex 2728    C_ wss 3340   <.cop 3895   class class class wbr 4304    X. cxp 4850    o. ccom 4856   Rel wrel 4857   -->wf 5426   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588   RRcr 9293   1c1 9295    + caddc 9297   RR*cxr 9429    < clt 9430    <_ cle 9431   NNcn 10334   ZZcz 10658   ZZ>=cuz 10873   RR+crp 11003   *Metcxmt 17813   ballcbl 17815   Caucca 20776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-psmet 17821  df-xmet 17822  df-bl 17824  df-cau 20779
This theorem is referenced by:  bcthlem4  20850  heiborlem9  28730
  Copyright terms: Public domain W3C validator