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Theorem caun0 18723
Description: A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
Assertion
Ref Expression
caun0  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  X  =/=  (/) )

Proof of Theorem caun0
Dummy variables  j 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1rp 10374 . . . 4  |-  1  e.  RR+
2 ne0i 3474 . . . 4  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
31, 2ax-mp 8 . . 3  |-  RR+  =/=  (/)
4 iscau2 18719 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D ( F `
 j ) )  <  x ) ) ) )
54simplbda 607 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) )
6 r19.2z 3556 . . 3  |-  ( (
RR+  =/=  (/)  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) )  ->  E. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) )
73, 5, 6sylancr 644 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  E. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) )
8 uzid 10258 . . . . . 6  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
9 ne0i 3474 . . . . . 6  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( ZZ>= `  j )  =/=  (/) )
10 r19.2z 3556 . . . . . . 7  |-  ( ( ( ZZ>= `  j )  =/=  (/)  /\  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) )  ->  E. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) )
1110ex 423 . . . . . 6  |-  ( (
ZZ>= `  j )  =/=  (/)  ->  ( A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  E. k  e.  (
ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) ) )
128, 9, 113syl 18 . . . . 5  |-  ( j  e.  ZZ  ->  ( A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  E. k  e.  (
ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x ) ) )
13 ne0i 3474 . . . . . . 7  |-  ( ( F `  k )  e.  X  ->  X  =/=  (/) )
14133ad2ant2 977 . . . . . 6  |-  ( ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) )
1514rexlimivw 2676 . . . . 5  |-  ( E. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D ( F `
 j ) )  <  x )  ->  X  =/=  (/) )
1612, 15syl6 29 . . . 4  |-  ( j  e.  ZZ  ->  ( A. k  e.  ( ZZ>=
`  j ) ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) ) )
1716rexlimiv 2674 . . 3  |-  ( E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D ( F `
 j ) )  <  x )  ->  X  =/=  (/) )
1817rexlimivw 2676 . 2  |-  ( E. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) )
197, 18syl 15 1  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  X  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   1c1 8754    < clt 8883   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   * Metcxmt 16385   Caucca 18695
This theorem is referenced by:  cmetcau  18731
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-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-z 10041  df-uz 10247  df-rp 10371  df-xadd 10469  df-xmet 16389  df-bl 16391  df-cau 18698
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