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Theorem caun0 19234
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 10616 . . . 4  |-  1  e.  RR+
2 ne0i 3634 . . . 4  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
31, 2ax-mp 8 . . 3  |-  RR+  =/=  (/)
4 iscau2 19230 . . . 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 608 . . 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 3717 . . 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 645 . 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 10500 . . . . . 6  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
9 ne0i 3634 . . . . . 6  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( ZZ>= `  j )  =/=  (/) )
10 r19.2z 3717 . . . . . . 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 424 . . . . . 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 19 . . . . 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 3634 . . . . . . 7  |-  ( ( F `  k )  e.  X  ->  X  =/=  (/) )
14133ad2ant2 979 . . . . . 6  |-  ( ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) )
1514rexlimivw 2826 . . . . 5  |-  ( E. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D ( F `
 j ) )  <  x )  ->  X  =/=  (/) )
1612, 15syl6 31 . . . 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 2824 . . 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 2826 . 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 16 1  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  X  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   (/)c0 3628   class class class wbr 4212   dom cdm 4878   ` cfv 5454  (class class class)co 6081    ^pm cpm 7019   CCcc 8988   1c1 8991    < clt 9120   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   * Metcxmt 16686   Caucca 19206
This theorem is referenced by:  cmetcau  19242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-z 10283  df-uz 10489  df-rp 10613  df-xadd 10711  df-psmet 16694  df-xmet 16695  df-bl 16697  df-cau 19209
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