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Theorem caun0 21845
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 11249 . . . 4  |-  1  e.  RR+
21ne0ii 3800 . . 3  |-  RR+  =/=  (/)
3 iscau2 21841 . . . 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 ) ) ) )
43simplbda 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 ) )
5 r19.2z 3921 . . 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 ) )
62, 4, 5sylancr 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 ) )
7 uzid 11120 . . . . . 6  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
8 ne0i 3799 . . . . . 6  |-  ( j  e.  ( ZZ>= `  j
)  ->  ( ZZ>= `  j )  =/=  (/) )
9 r19.2z 3921 . . . . . . 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 ) )
109ex 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 ) ) )
117, 8, 103syl 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 ) ) )
12 ne0i 3799 . . . . . . 7  |-  ( ( F `  k )  e.  X  ->  X  =/=  (/) )
13123ad2ant2 1018 . . . . . 6  |-  ( ( k  e.  dom  F  /\  ( F `  k
)  e.  X  /\  ( ( F `  k ) D ( F `  j ) )  <  x )  ->  X  =/=  (/) )
1413rexlimivw 2946 . . . . 5  |-  ( E. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
( F `  k
) D ( F `
 j ) )  <  x )  ->  X  =/=  (/) )
1511, 14syl6 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  =/=  (/) ) )
1615rexlimiv 2943 . . 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  =/=  (/) )
1716rexlimivw 2946 . 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  =/=  (/) )
186, 17syl 16 1  |-  ( ( D  e.  ( *Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  X  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   (/)c0 3793   class class class wbr 4456   dom cdm 5008   ` cfv 5594  (class class class)co 6296    ^pm cpm 7439   CCcc 9507   1c1 9510    < clt 9645   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   *Metcxmt 18529   Caucca 21817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-z 10886  df-uz 11107  df-rp 11246  df-xadd 11344  df-psmet 18537  df-xmet 18538  df-bl 18540  df-cau 21820
This theorem is referenced by:  cmetcau  21853
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