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Theorem caubl 19252
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 5720 . . . . . . . . . . . . . 14  |-  ( r  =  n  ->  ( F `  r )  =  ( F `  n ) )
32fveq2d 5724 . . . . . . . . . . . . 13  |-  ( r  =  n  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  n ) ) )
43sseq1d 3367 . . . . . . . . . . . 12  |-  ( r  =  n  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  n ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
54imbi2d 308 . . . . . . . . . . 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 5720 . . . . . . . . . . . . . 14  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
76fveq2d 5724 . . . . . . . . . . . . 13  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
87sseq1d 3367 . . . . . . . . . . . 12  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
98imbi2d 308 . . . . . . . . . . 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 5720 . . . . . . . . . . . . . 14  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1110fveq2d 5724 . . . . . . . . . . . . 13  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
1211sseq1d 3367 . . . . . . . . . . . 12  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
1312imbi2d 308 . . . . . . . . . . 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 3359 . . . . . . . . . . . 12  |-  ( (
ball `  D ) `  ( F `  n
) )  C_  (
( ball `  D ) `  ( F `  n
) )
1514a1ii 25 . . . . . . . . . . 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 nnuz 10513 . . . . . . . . . . . . . . . . 17  |-  NN  =  ( ZZ>= `  1 )
1817uztrn2 10495 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  k  e.  ( ZZ>= `  n ) )  -> 
k  e.  NN )
19 oveq1 6080 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2019fveq2d 5724 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2120fveq2d 5724 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
22 fveq2 5720 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
2322fveq2d 5724 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
2421, 23sseq12d 3369 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
2524rspccva 3043 . . . . . . . . . . . . . . . 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 ) ) )
2616, 18, 25syl2an 464 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  NN  /\  k  e.  ( ZZ>= `  n )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
2726anassrs 630 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
28 sstr2 3347 . . . . . . . . . . . . . 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 ) ) ) )
2927, 28syl 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
) ) ) )
3029expcom 425 . . . . . . . . . . . 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
) ) ) ) )
3130a2d 24 . . . . . . . . . . 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 ) ) ) ) )
325, 9, 13, 9, 15, 31uzind4 10526 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3332com12 29 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3433ad2ant2r 728 . . . . . . . 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 ) ) ) )
35 relxp 4975 . . . . . . . . . . . . . . . 16  |-  Rel  ( X  X.  RR+ )
36 caubl.3 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
3736ad3antrrr 711 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  F : NN
--> ( X  X.  RR+ ) )
38 simplrl 737 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  n  e.  NN )
3937, 38ffvelrnd 5863 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  e.  ( X  X.  RR+ )
)
40 1st2nd 6385 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 n )  e.  ( X  X.  RR+ ) )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
4135, 39, 40sylancr 645 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
4241fveq2d 5724 . . . . . . . . . . . . . 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
) ) >. )
)
43 df-ov 6076 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) ( 2nd `  ( F `  n )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  n
) ) ,  ( 2nd `  ( F `
 n ) )
>. )
4442, 43syl6eqr 2485 . . . . . . . . . . . . 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 )
) ) )
45 caubl.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  ( * Met `  X ) )
4645ad3antrrr 711 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  D  e.  ( * Met `  X
) )
47 xp1st 6368 . . . . . . . . . . . . . . 15  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  n
) )  e.  X
)
4839, 47syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  n
) )  e.  X
)
49 xp2nd 6369 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
5039, 49syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
5150rpxrd 10641 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR* )
52 simpllr 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR+ )
5352rpxrd 10641 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR* )
54 simplrr 738 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <  r
)
55 rpre 10610 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR+  ->  ( 2nd `  ( F `  n
) )  e.  RR )
56 rpre 10610 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
57 ltle 9155 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR  /\  r  e.  RR )  ->  ( ( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5855, 56, 57syl2an 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR+  /\  r  e.  RR+ )  ->  (
( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5950, 52, 58syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( 2nd `  ( F `  n ) )  < 
r  ->  ( 2nd `  ( F `  n
) )  <_  r
) )
6054, 59mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <_  r
)
61 ssbl 18445 . . . . . . . . . . . . . 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 ) )
6246, 48, 51, 53, 60, 61syl221anc 1195 . . . . . . . . . . . . 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 ) )
6344, 62eqsstrd 3374 . . . . . . . . . . . 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 ) )
64 sstr2 3347 . . . . . . . . . . . 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 ) ) )
6563, 64syl5com 28 . . . . . . . . . . 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 ) ) )
66 simprl 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  n  e.  NN )
6766, 18sylan 458 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  k  e.  NN )
6837, 67ffvelrnd 5863 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
69 xp1st 6368 . . . . . . . . . . . . . . 15  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  k
) )  e.  X
)
7068, 69syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  X
)
71 xp2nd 6369 . . . . . . . . . . . . . . 15  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
7268, 71syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
73 blcntr 18435 . . . . . . . . . . . . . 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 )
) ) )
7446, 70, 72, 73syl3anc 1184 . . . . . . . . . . . . 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 )
) ) )
75 1st2nd 6385 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 k )  e.  ( X  X.  RR+ ) )  ->  ( F `  k )  =  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
7635, 68, 75sylancr 645 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
7776fveq2d 5724 . . . . . . . . . . . . . 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
) ) >. )
)
78 df-ov 6076 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
7977, 78syl6eqr 2485 . . . . . . . . . . . . 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 )
) ) )
8074, 79eleqtrrd 2512 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  ( ( ball `  D
) `  ( F `  k ) ) )
81 ssel 3334 . . . . . . . . . . . 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 ) ) )
8280, 81syl5com 28 . . . . . . . . . . 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 ) ) )
8365, 82syld 42 . . . . . . . . . 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 ) ) )
84 elbl2 18412 . . . . . . . . . . 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
) )
8546, 53, 48, 70, 84syl22anc 1185 . . . . . . . . . 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
) )
8683, 85sylibd 206 . . . . . . . . 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 ) )
8786ex 424 . . . . . . . 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
) ) )
8834, 87mpdd 38 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  ( k  e.  ( ZZ>= `  n )  ->  ( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
8988ralrimiv 2780 . . . . . 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 )
9089expr 599 . . . . 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
) )
9190reximdva 2810 . . . 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 ) )
9291ralimdva 2776 . . 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 ) )
931, 92mpd 15 . 2  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( 1st `  ( F `
 n ) ) D ( 1st `  ( F `  k )
) )  <  r
)
94 1z 10303 . . . 4  |-  1  e.  ZZ
9594a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
96 fvco3 5792 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
9736, 96sylan 458 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1st  o.  F ) `
 k )  =  ( 1st `  ( F `  k )
) )
98 fvco3 5792 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  n  e.  NN )  ->  (
( 1st  o.  F
) `  n )  =  ( 1st `  ( F `  n )
) )
9936, 98sylan 458 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  F ) `
 n )  =  ( 1st `  ( F `  n )
) )
100 1stcof 6366 . . . 4  |-  ( F : NN --> ( X  X.  RR+ )  ->  ( 1st  o.  F ) : NN --> X )
10136, 100syl 16 . . 3  |-  ( ph  ->  ( 1st  o.  F
) : NN --> X )
10217, 45, 95, 97, 99, 101iscauf 19225 . 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 ) )
10393, 102mpbird 224 1  |-  ( ph  ->  ( 1st  o.  F
)  e.  ( Cau `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   <.cop 3809   class class class wbr 4204    X. cxp 4868    o. ccom 4874   Rel wrel 4875   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   RRcr 8981   1c1 8983    + caddc 8985   RR*cxr 9111    < clt 9112    <_ cle 9113   NNcn 9992   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   * Metcxmt 16678   ballcbl 16680   Caucca 19198
This theorem is referenced by:  bcthlem4  19272  heiborlem9  26509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-psmet 16686  df-xmet 16687  df-bl 16689  df-cau 19201
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