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Theorem iscauf 18722
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 5570 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
31, 2syl 15 . . . . 5  |-  ( ph  ->  X  e.  dom  * Met )
4 cnex 8834 . . . . 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 10263 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
9 zsscn 10048 . . . . . . 7  |-  ZZ  C_  CC
108, 9sstri 3201 . . . . . 6  |-  ( ZZ>= `  M )  C_  CC
117, 10eqsstri 3221 . . . . 5  |-  Z  C_  CC
126, 11jctir 524 . . . 4  |-  ( ph  ->  ( F : Z --> X  /\  Z  C_  CC ) )
13 elpm2r 6804 . . . 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 5679 . . . . . . . . . . 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 2371 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  B  e.  X )
247uztrn2 10261 . . . . . . . . . . 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 5679 . . . . . . . . . . 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 2371 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  A  e.  X )
30 xmetsym 17928 . . . . . . . . 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 4049 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( B D A )  <  x  <->  ( A D B )  <  x
) )
33 fdm 5409 . . . . . . . . . . . . 13  |-  ( F : Z --> X  ->  dom  F  =  Z )
3433eleq2d 2363 . . . . . . . . . . . 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 2572 . . . 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 2573 . . 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 2576 . 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 18721 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   class class class wbr 4039   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751    < clt 8883   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   * Metcxmt 16385   Caucca 18695
This theorem is referenced by:  iscmet3lem1  18733  causs  18740  caubl  18749  minvecolem3  21471  h2hcau  21575  geomcau  26578  caushft  26580  rrncmslem  26659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-z 10041  df-uz 10247  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmet 16389  df-bl 16391  df-cau 18698
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