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Theorem caubl 18733
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 5525 . . . . . . . . . . . . . 14  |-  ( r  =  n  ->  ( F `  r )  =  ( F `  n ) )
32fveq2d 5529 . . . . . . . . . . . . 13  |-  ( r  =  n  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  n ) ) )
43sseq1d 3205 . . . . . . . . . . . 12  |-  ( r  =  n  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  n ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
54imbi2d 307 . . . . . . . . . . 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 5525 . . . . . . . . . . . . . 14  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
76fveq2d 5529 . . . . . . . . . . . . 13  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
87sseq1d 3205 . . . . . . . . . . . 12  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
98imbi2d 307 . . . . . . . . . . 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 5525 . . . . . . . . . . . . . 14  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1110fveq2d 5529 . . . . . . . . . . . . 13  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
1211sseq1d 3205 . . . . . . . . . . . 12  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
1312imbi2d 307 . . . . . . . . . . 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 3197 . . . . . . . . . . . 12  |-  ( (
ball `  D ) `  ( F `  n
) )  C_  (
( ball `  D ) `  ( F `  n
) )
1514a1ii 24 . . . . . . . . . . 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 10263 . . . . . . . . . . . . . . . . 17  |-  NN  =  ( ZZ>= `  1 )
1817uztrn2 10245 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  k  e.  ( ZZ>= `  n ) )  -> 
k  e.  NN )
19 oveq1 5865 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2019fveq2d 5529 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2120fveq2d 5529 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
22 fveq2 5525 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
2322fveq2d 5529 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
2421, 23sseq12d 3207 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
2524rspccva 2883 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  NN  /\  k  e.  ( ZZ>= `  n )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
2726anassrs 629 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
28 sstr2 3186 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 23 . . . . . . . . . . 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 10276 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ph  /\  n  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3332com12 27 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( k  e.  ( ZZ>= `  n
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) ) )
3433ad2ant2r 727 . . . . . . . 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 4794 . . . . . . . . . . . . . . . 16  |-  Rel  ( X  X.  RR+ )
36 caubl.3 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
3736ad3antrrr 710 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  F : NN
--> ( X  X.  RR+ ) )
38 simplrl 736 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  n  e.  NN )
39 ffvelrn 5663 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( X  X.  RR+ ) )
4037, 38, 39syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  e.  ( X  X.  RR+ )
)
41 1st2nd 6166 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 n )  e.  ( X  X.  RR+ ) )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
4235, 40, 41sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
4342fveq2d 5529 . . . . . . . . . . . . . 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
) ) >. )
)
44 df-ov 5861 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) ( ball `  D
) ( 2nd `  ( F `  n )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  n
) ) ,  ( 2nd `  ( F `
 n ) )
>. )
4543, 44syl6eqr 2333 . . . . . . . . . . . . 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 )
) ) )
46 caubl.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  ( * Met `  X ) )
4746ad3antrrr 710 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  D  e.  ( * Met `  X
) )
48 xp1st 6149 . . . . . . . . . . . . . . 15  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  n
) )  e.  X
)
4940, 48syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  n
) )  e.  X
)
50 xp2nd 6150 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
5140, 50syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR+ )
5251rpxrd 10391 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  e.  RR* )
53 simpllr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR+ )
5453rpxrd 10391 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  r  e.  RR* )
55 simplrr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <  r
)
56 rpre 10360 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR+  ->  ( 2nd `  ( F `  n
) )  e.  RR )
57 rpre 10360 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
58 ltle 8910 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR  /\  r  e.  RR )  ->  ( ( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
5956, 57, 58syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( F `  n )
)  e.  RR+  /\  r  e.  RR+ )  ->  (
( 2nd `  ( F `  n )
)  <  r  ->  ( 2nd `  ( F `
 n ) )  <_  r ) )
6051, 53, 59syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( ( 2nd `  ( F `  n ) )  < 
r  ->  ( 2nd `  ( F `  n
) )  <_  r
) )
6155, 60mpd 14 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  n
) )  <_  r
)
62 ssbl 17971 . . . . . . . . . . . . . 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 ) )
6347, 49, 52, 54, 61, 62syl221anc 1193 . . . . . . . . . . . . 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 ) )
6445, 63eqsstrd 3212 . . . . . . . . . . . 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 ) )
65 sstr2 3186 . . . . . . . . . . . 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 ) ) )
6664, 65syl5com 26 . . . . . . . . . . 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 ) ) )
67 simprl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  n  e.  NN )
6867, 18sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  k  e.  NN )
69 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  ( F `  k )  e.  ( X  X.  RR+ ) )
7037, 68, 69syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
71 xp1st 6149 . . . . . . . . . . . . . . 15  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  k
) )  e.  X
)
7270, 71syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  X
)
73 xp2nd 6150 . . . . . . . . . . . . . . 15  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
7470, 73syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
75 blcntr 17964 . . . . . . . . . . . . . 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 )
) ) )
7647, 72, 74, 75syl3anc 1182 . . . . . . . . . . . . 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 )
) ) )
77 1st2nd 6166 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( X  X.  RR+ )  /\  ( F `
 k )  e.  ( X  X.  RR+ ) )  ->  ( F `  k )  =  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
7835, 70, 77sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
7978fveq2d 5529 . . . . . . . . . . . . . 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
) ) >. )
)
80 df-ov 5861 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
8179, 80syl6eqr 2333 . . . . . . . . . . . . 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 )
) ) )
8276, 81eleqtrrd 2360 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  ( n  e.  NN  /\  ( 2nd `  ( F `  n )
)  <  r )
)  /\  k  e.  ( ZZ>= `  n )
)  ->  ( 1st `  ( F `  k
) )  e.  ( ( ball `  D
) `  ( F `  k ) ) )
83 ssel 3174 . . . . . . . . . . . 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 ) ) )
8482, 83syl5com 26 . . . . . . . . . . 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 ) ) )
8566, 84syld 40 . . . . . . . . . 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 ) ) )
86 elbl2 17950 . . . . . . . . . . 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
) )
8747, 54, 49, 72, 86syl22anc 1183 . . . . . . . . . 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
) )
8885, 87sylibd 205 . . . . . . . . 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 ) )
8988ex 423 . . . . . . . 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
) ) )
9034, 89mpdd 36 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  (
n  e.  NN  /\  ( 2nd `  ( F `
 n ) )  <  r ) )  ->  ( k  e.  ( ZZ>= `  n )  ->  ( ( 1st `  ( F `  n )
) D ( 1st `  ( F `  k
) ) )  < 
r ) )
9190ralrimiv 2625 . . . . . 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 )
9291expr 598 . . . . 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
) )
9392reximdva 2655 . . . 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 ) )
9493ralimdva 2621 . . 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 ) )
951, 94mpd 14 . 2  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( 1st `  ( F `
 n ) ) D ( 1st `  ( F `  k )
) )  <  r
)
96 1z 10053 . . . 4  |-  1  e.  ZZ
9796a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
98 fvco3 5596 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
9936, 98sylan 457 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1st  o.  F ) `
 k )  =  ( 1st `  ( F `  k )
) )
100 fvco3 5596 . . . 4  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  n  e.  NN )  ->  (
( 1st  o.  F
) `  n )  =  ( 1st `  ( F `  n )
) )
10136, 100sylan 457 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  F ) `
 n )  =  ( 1st `  ( F `  n )
) )
102 1stcof 6147 . . . 4  |-  ( F : NN --> ( X  X.  RR+ )  ->  ( 1st  o.  F ) : NN --> X )
10336, 102syl 15 . . 3  |-  ( ph  ->  ( 1st  o.  F
) : NN --> X )
10417, 46, 97, 99, 101, 103iscauf 18706 . 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 ) )
10595, 104mpbird 223 1  |-  ( ph  ->  ( 1st  o.  F
)  e.  ( Cau `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   <.cop 3643   class class class wbr 4023    X. cxp 4687    o. ccom 4693   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RRcr 8736   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   * Metcxmt 16369   ballcbl 16371   Caucca 18679
This theorem is referenced by:  bcthlem4  18749  heiborlem9  26543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-xmet 16373  df-bl 16375  df-cau 18682
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