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Theorem iscauf 19186
Description: Express the property " F is a Cauchy sequence of metric  D " presupposing  F is a function. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
iscau3.2  |-  Z  =  ( ZZ>= `  M )
iscau3.3  |-  ( ph  ->  D  e.  ( * Met `  X ) )
iscau3.4  |-  ( ph  ->  M  e.  ZZ )
iscau4.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
iscau4.6  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  B )
iscauf.7  |-  ( ph  ->  F : Z --> X )
Assertion
Ref Expression
iscauf  |-  ( ph  ->  ( F  e.  ( Cau `  D )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B D A )  <  x ) )
Distinct variable groups:    j, k, x, D    j, F, k, x    ph, j, k, x   
j, X, k, x   
j, M    j, Z, k, x
Allowed substitution hints:    A( x, j, k)    B( x, j, k)    M( x, k)

Proof of Theorem iscauf
StepHypRef Expression
1 iscau3.3 . . . . . 6  |-  ( ph  ->  D  e.  ( * Met `  X ) )
2 elfvdm 5716 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
31, 2syl 16 . . . . 5  |-  ( ph  ->  X  e.  dom  * Met )
4 cnex 9027 . . . . 5  |-  CC  e.  _V
53, 4jctir 525 . . . 4  |-  ( ph  ->  ( X  e.  dom  * Met  /\  CC  e.  _V ) )
6 iscauf.7 . . . . 5  |-  ( ph  ->  F : Z --> X )
7 iscau3.2 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
8 uzssz 10461 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
9 zsscn 10246 . . . . . . 7  |-  ZZ  C_  CC
108, 9sstri 3317 . . . . . 6  |-  ( ZZ>= `  M )  C_  CC
117, 10eqsstri 3338 . . . . 5  |-  Z  C_  CC
126, 11jctir 525 . . . 4  |-  ( ph  ->  ( F : Z --> X  /\  Z  C_  CC ) )
13 elpm2r 6993 . . . 4  |-  ( ( ( X  e.  dom  * Met  /\  CC  e.  _V )  /\  ( F : Z --> X  /\  Z  C_  CC ) )  ->  F  e.  ( X  ^pm  CC )
)
145, 12, 13syl2anc 643 . . 3  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
1514biantrurd 495 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x )  <-> 
( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x ) ) ) )
161adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  D  e.  ( * Met `  X
) )
17 iscau4.6 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  B )
1817adantrr 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  j )  =  B )
196adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  F : Z --> X )
20 simprl 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  j  e.  Z )
2119, 20ffvelrnd 5830 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  j )  e.  X )
2218, 21eqeltrrd 2479 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  B  e.  X )
237uztrn2 10459 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
24 iscau4.5 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
2523, 24sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  k )  =  A )
26 ffvelrn 5827 . . . . . . . . . . 11  |-  ( ( F : Z --> X  /\  k  e.  Z )  ->  ( F `  k
)  e.  X )
276, 23, 26syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  k )  e.  X )
2825, 27eqeltrrd 2479 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  A  e.  X )
29 xmetsym 18330 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
3016, 22, 28, 29syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( B D A )  =  ( A D B ) )
3130breq1d 4182 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( B D A )  <  x  <->  ( A D B )  <  x
) )
32 fdm 5554 . . . . . . . . . . . . 13  |-  ( F : Z --> X  ->  dom  F  =  Z )
3332eleq2d 2471 . . . . . . . . . . . 12  |-  ( F : Z --> X  -> 
( k  e.  dom  F  <-> 
k  e.  Z ) )
3433biimpar 472 . . . . . . . . . . 11  |-  ( ( F : Z --> X  /\  k  e.  Z )  ->  k  e.  dom  F
)
356, 23, 34syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  k  e.  dom  F )
3635, 28jca 519 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
k  e.  dom  F  /\  A  e.  X
) )
3736biantrurd 495 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( A D B )  <  x  <->  ( (
k  e.  dom  F  /\  A  e.  X
)  /\  ( A D B )  <  x
) ) )
38 df-3an 938 . . . . . . . 8  |-  ( ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x )  <-> 
( ( k  e. 
dom  F  /\  A  e.  X )  /\  ( A D B )  < 
x ) )
3937, 38syl6bbr 255 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( A D B )  <  x  <->  ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  < 
x ) ) )
4031, 39bitrd 245 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( B D A )  <  x  <->  ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  < 
x ) ) )
4140anassrs 630 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( B D A )  < 
x  <->  ( k  e. 
dom  F  /\  A  e.  X  /\  ( A D B )  < 
x ) ) )
4241ralbidva 2682 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( B D A )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x ) ) )
4342rexbidva 2683 . . 3  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B D A )  <  x  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x ) ) )
4443ralbidv 2686 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B D A )  <  x  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x ) ) )
45 iscau3.4 . . 3  |-  ( ph  ->  M  e.  ZZ )
467, 1, 45, 24, 17iscau4 19185 . 2  |-  ( ph  ->  ( F  e.  ( Cau `  D )  <-> 
( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x ) ) ) )
4715, 44, 463bitr4rd 278 1  |-  ( ph  ->  ( F  e.  ( Cau `  D )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B D A )  <  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   class class class wbr 4172   dom cdm 4837   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^pm cpm 6978   CCcc 8944    < clt 9076   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   * Metcxmt 16641   Caucca 19159
This theorem is referenced by:  iscmet3lem1  19197  causs  19204  caubl  19213  minvecolem3  22331  h2hcau  22435  geomcau  26355  caushft  26357  rrncmslem  26431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-z 10239  df-uz 10445  df-rp 10569  df-xneg 10666  df-xadd 10667  df-psmet 16649  df-xmet 16650  df-bl 16652  df-cau 19162
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