MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caun0 Structured version   Unicode version

Theorem caun0 21450
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 11215 . . . 4  |-  1  e.  RR+
2 ne0i 3786 . . . 4  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
31, 2ax-mp 5 . . 3  |-  RR+  =/=  (/)
4 iscau2 21446 . . . 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 624 . . 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 3912 . . 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 663 . 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 11087 . . . . . 6  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
9 ne0i 3786 . . . . . 6  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( ZZ>= `  j )  =/=  (/) )
10 r19.2z 3912 . . . . . . 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 3786 . . . . . . 7  |-  ( ( F `  k )  e.  X  ->  X  =/=  (/) )
14133ad2ant2 1013 . . . . . 6  |-  ( ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) )
1514rexlimivw 2947 . . . . 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 2944 . . 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 2947 . 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 968    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   (/)c0 3780   class class class wbr 4442   dom cdm 4994   ` cfv 5581  (class class class)co 6277    ^pm cpm 7413   CCcc 9481   1c1 9484    < clt 9619   ZZcz 10855   ZZ>=cuz 11073   RR+crp 11211   *Metcxmt 18169   Caucca 21422
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-z 10856  df-uz 11074  df-rp 11212  df-xadd 11310  df-psmet 18177  df-xmet 18178  df-bl 18180  df-cau 21425
This theorem is referenced by:  cmetcau  21458
  Copyright terms: Public domain W3C validator