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Theorem caun0 20751
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 10991 . . . 4  |-  1  e.  RR+
2 ne0i 3640 . . . 4  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
31, 2ax-mp 5 . . 3  |-  RR+  =/=  (/)
4 iscau2 20747 . . . 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 621 . . 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 3766 . . 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 658 . 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 10871 . . . . . 6  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
9 ne0i 3640 . . . . . 6  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( ZZ>= `  j )  =/=  (/) )
10 r19.2z 3766 . . . . . . 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 434 . . . . . 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 20 . . . . 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 3640 . . . . . . 7  |-  ( ( F `  k )  e.  X  ->  X  =/=  (/) )
14133ad2ant2 1005 . . . . . 6  |-  ( ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) )
1514rexlimivw 2835 . . . . 5  |-  ( E. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D ( F `
 j ) )  <  x )  ->  X  =/=  (/) )
1612, 15syl6 33 . . . 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 2833 . . 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 2835 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   (/)c0 3634   class class class wbr 4289   dom cdm 4836   ` cfv 5415  (class class class)co 6090    ^pm cpm 7211   CCcc 9276   1c1 9279    < clt 9414   ZZcz 10642   ZZ>=cuz 10857   RR+crp 10987   *Metcxmt 17760   Caucca 20723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-z 10643  df-uz 10858  df-rp 10988  df-xadd 11086  df-psmet 17768  df-xmet 17769  df-bl 17771  df-cau 20726
This theorem is referenced by:  cmetcau  20759
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