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Theorem iscauf 18804
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 5634 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
31, 2syl 15 . . . . 5  |-  ( ph  ->  X  e.  dom  * Met )
4 cnex 8905 . . . . 5  |-  CC  e.  _V
53, 4jctir 524 . . . 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 10336 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
9 zsscn 10121 . . . . . . 7  |-  ZZ  C_  CC
108, 9sstri 3264 . . . . . 6  |-  ( ZZ>= `  M )  C_  CC
117, 10eqsstri 3284 . . . . 5  |-  Z  C_  CC
126, 11jctir 524 . . . 4  |-  ( ph  ->  ( F : Z --> X  /\  Z  C_  CC ) )
13 elpm2r 6873 . . . 4  |-  ( ( ( X  e.  dom  * Met  /\  CC  e.  _V )  /\  ( F : Z --> X  /\  Z  C_  CC ) )  ->  F  e.  ( X  ^pm  CC )
)
145, 12, 13syl2anc 642 . . 3  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
1514biantrurd 494 . 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 451 . . . . . . . . 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 697 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  j )  =  B )
196adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  F : Z --> X )
20 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  j  e.  Z )
21 ffvelrn 5743 . . . . . . . . . . 11  |-  ( ( F : Z --> X  /\  j  e.  Z )  ->  ( F `  j
)  e.  X )
2219, 20, 21syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  j )  e.  X )
2318, 22eqeltrrd 2433 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  B  e.  X )
247uztrn2 10334 . . . . . . . . . . 11  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
25 iscau4.5 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
2624, 25sylan2 460 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  k )  =  A )
27 ffvelrn 5743 . . . . . . . . . . 11  |-  ( ( F : Z --> X  /\  k  e.  Z )  ->  ( F `  k
)  e.  X )
286, 24, 27syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( F `  k )  e.  X )
2926, 28eqeltrrd 2433 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  A  e.  X )
30 xmetsym 18008 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
3116, 23, 29, 30syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  ( B D A )  =  ( A D B ) )
3231breq1d 4112 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( B D A )  <  x  <->  ( A D B )  <  x
) )
33 fdm 5473 . . . . . . . . . . . . 13  |-  ( F : Z --> X  ->  dom  F  =  Z )
3433eleq2d 2425 . . . . . . . . . . . 12  |-  ( F : Z --> X  -> 
( k  e.  dom  F  <-> 
k  e.  Z ) )
3534biimpar 471 . . . . . . . . . . 11  |-  ( ( F : Z --> X  /\  k  e.  Z )  ->  k  e.  dom  F
)
366, 24, 35syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  k  e.  dom  F )
3736, 29jca 518 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
k  e.  dom  F  /\  A  e.  X
) )
3837biantrurd 494 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( A D B )  <  x  <->  ( (
k  e.  dom  F  /\  A  e.  X
)  /\  ( A D B )  <  x
) ) )
39 df-3an 936 . . . . . . . 8  |-  ( ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  <  x )  <-> 
( ( k  e. 
dom  F  /\  A  e.  X )  /\  ( A D B )  < 
x ) )
4038, 39syl6bbr 254 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( A D B )  <  x  <->  ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  < 
x ) ) )
4132, 40bitrd 244 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( B D A )  <  x  <->  ( k  e.  dom  F  /\  A  e.  X  /\  ( A D B )  < 
x ) ) )
4241anassrs 629 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( B D A )  < 
x  <->  ( k  e. 
dom  F  /\  A  e.  X  /\  ( A D B )  < 
x ) ) )
4342ralbidva 2635 . . . 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 ) ) )
4443rexbidva 2636 . . 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 ) ) )
4544ralbidv 2639 . 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 ) ) )
46 iscau3.4 . . 3  |-  ( ph  ->  M  e.  ZZ )
477, 1, 46, 25, 17iscau4 18803 . 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 ) ) ) )
4815, 45, 473bitr4rd 277 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   _Vcvv 2864    C_ wss 3228   class class class wbr 4102   dom cdm 4768   -->wf 5330   ` cfv 5334  (class class class)co 5942    ^pm cpm 6858   CCcc 8822    < clt 8954   ZZcz 10113   ZZ>=cuz 10319   RR+crp 10443   * Metcxmt 16462   Caucca 18777
This theorem is referenced by:  iscmet3lem1  18815  causs  18822  caubl  18831  minvecolem3  21563  h2hcau  21667  geomcau  25799  caushft  25801  rrncmslem  25879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-2 9891  df-z 10114  df-uz 10320  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmet 16469  df-bl 16471  df-cau 18780
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