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Theorem cau3 12841
Description: Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of  j in the assertion, so it can be used with rexanuz 12831 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
Hypothesis
Ref Expression
cau3.1  |-  Z  =  ( ZZ>= `  M )
Assertion
Ref Expression
cau3  |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  ( F `
 j ) ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\ 
A. m  e.  (
ZZ>= `  k ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) )
Distinct variable groups:    j, k, m, x, F    j, M, k, x    j, Z, k, x
Allowed substitution hints:    M( m)    Z( m)

Proof of Theorem cau3
StepHypRef Expression
1 cau3.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
2 uzssz 10878 . . . 4  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3384 . . 3  |-  Z  C_  ZZ
4 id 22 . . 3  |-  ( ( F `  k )  e.  CC  ->  ( F `  k )  e.  CC )
5 eleq1 2501 . . 3  |-  ( ( F `  k )  =  ( F `  j )  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
6 eleq1 2501 . . 3  |-  ( ( F `  k )  =  ( F `  m )  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
7 abssub 12812 . . . 4  |-  ( ( ( F `  j
)  e.  CC  /\  ( F `  k )  e.  CC )  -> 
( abs `  (
( F `  j
)  -  ( F `
 k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j )
) ) )
873adant1 1006 . . 3  |-  ( ( T.  /\  ( F `
 j )  e.  CC  /\  ( F `
 k )  e.  CC )  ->  ( abs `  ( ( F `
 j )  -  ( F `  k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j ) ) ) )
9 abssub 12812 . . . 4  |-  ( ( ( F `  m
)  e.  CC  /\  ( F `  j )  e.  CC )  -> 
( abs `  (
( F `  m
)  -  ( F `
 j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m )
) ) )
1093adant1 1006 . . 3  |-  ( ( T.  /\  ( F `
 m )  e.  CC  /\  ( F `
 j )  e.  CC )  ->  ( abs `  ( ( F `
 m )  -  ( F `  j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m ) ) ) )
11 abs3lem 12824 . . . 4  |-  ( ( ( ( F `  k )  e.  CC  /\  ( F `  m
)  e.  CC )  /\  ( ( F `
 j )  e.  CC  /\  x  e.  RR ) )  -> 
( ( ( abs `  ( ( F `  k )  -  ( F `  j )
) )  <  (
x  /  2 )  /\  ( abs `  (
( F `  j
)  -  ( F `
 m ) ) )  <  ( x  /  2 ) )  ->  ( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x ) )
12113adant1 1006 . . 3  |-  ( ( T.  /\  ( ( F `  k )  e.  CC  /\  ( F `  m )  e.  CC )  /\  (
( F `  j
)  e.  CC  /\  x  e.  RR )
)  ->  ( (
( abs `  (
( F `  k
)  -  ( F `
 j ) ) )  <  ( x  /  2 )  /\  ( abs `  ( ( F `  j )  -  ( F `  m ) ) )  <  ( x  / 
2 ) )  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x ) )
133, 4, 5, 6, 8, 10, 12cau3lem 12840 . 2  |-  ( T. 
->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( ( F `  k
)  e.  CC  /\  ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\ 
A. m  e.  (
ZZ>= `  k ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) ) )
1413trud 1378 1  |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  ( F `
 j ) ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\ 
A. m  e.  (
ZZ>= `  k ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756   A.wral 2713   E.wrex 2714   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279    < clt 9416    - cmin 9593    / cdiv 9991   2c2 10369   ZZcz 10644   ZZ>=cuz 10859   RR+crp 10989   abscabs 12721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  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-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-seq 11805  df-exp 11864  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723
This theorem is referenced by:  cau4  12842  serf0  13156
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