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Theorem cau3 10220
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 10093 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 8789 . . . 4  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3038 . . 3  |-  Z  C_  ZZ
4 id 19 . . 3  |-  ( ( F `  k )  e.  CC  ->  ( F `  k )  e.  CC )
5 eleq1 2145 . . 3  |-  ( ( F `  k )  =  ( F `  j )  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
6 eleq1 2145 . . 3  |-  ( ( F `  k )  =  ( F `  m )  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
7 abssub 10206 . . . 4  |-  ( ( ( F `  j
)  e.  CC  /\  ( F `  k )  e.  CC )  -> 
( abs `  (
( F `  j
)  -  ( F `
 k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j )
) ) )
873adant1 957 . . 3  |-  ( ( T.  /\  ( F `
 j )  e.  CC  /\  ( F `
 k )  e.  CC )  ->  ( abs `  ( ( F `
 j )  -  ( F `  k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j ) ) ) )
9 abssub 10206 . . . 4  |-  ( ( ( F `  m
)  e.  CC  /\  ( F `  j )  e.  CC )  -> 
( abs `  (
( F `  m
)  -  ( F `
 j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m )
) ) )
1093adant1 957 . . 3  |-  ( ( T.  /\  ( F `
 m )  e.  CC  /\  ( F `
 j )  e.  CC )  ->  ( abs `  ( ( F `
 m )  -  ( F `  j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m ) ) ) )
11 abs3lem 10216 . . . 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 957 . . 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 10219 . 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 1294 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 set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   T. wtru 1286    e. wcel 1434   A.wral 2353   E.wrex 2354   class class class wbr 3805   ` cfv 4952  (class class class)co 5564   CCcc 7111   RRcr 7112    < clt 7285    - cmin 7416    / cdiv 7897   2c2 8226   ZZcz 8502   ZZ>=cuz 8770   RR+crp 8885   abscabs 10102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226  ax-arch 7227  ax-caucvg 7228
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-2 8235  df-3 8236  df-4 8237  df-n0 8426  df-z 8503  df-uz 8771  df-rp 8886  df-iseq 9592  df-iexp 9643  df-cj 9948  df-re 9949  df-im 9950  df-rsqrt 10103  df-abs 10104
This theorem is referenced by:  cau4  10221  serif0  10408
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