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Theorem cau3 12161
Description: Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurence of  j in the assertion, so it can be used with rexanuz 12151 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 10507 . . . 4  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3380 . . 3  |-  Z  C_  ZZ
4 id 21 . . 3  |-  ( ( F `  k )  e.  CC  ->  ( F `  k )  e.  CC )
5 eleq1 2498 . . 3  |-  ( ( F `  k )  =  ( F `  j )  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
6 eleq1 2498 . . 3  |-  ( ( F `  k )  =  ( F `  m )  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
7 abssub 12132 . . . 4  |-  ( ( ( F `  j
)  e.  CC  /\  ( F `  k )  e.  CC )  -> 
( abs `  (
( F `  j
)  -  ( F `
 k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j )
) ) )
873adant1 976 . . 3  |-  ( (  T.  /\  ( F `
 j )  e.  CC  /\  ( F `
 k )  e.  CC )  ->  ( abs `  ( ( F `
 j )  -  ( F `  k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j ) ) ) )
9 abssub 12132 . . . 4  |-  ( ( ( F `  m
)  e.  CC  /\  ( F `  j )  e.  CC )  -> 
( abs `  (
( F `  m
)  -  ( F `
 j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m )
) ) )
1093adant1 976 . . 3  |-  ( (  T.  /\  ( F `
 m )  e.  CC  /\  ( F `
 j )  e.  CC )  ->  ( abs `  ( ( F `
 m )  -  ( F `  j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m ) ) ) )
11 abs3lem 12144 . . . 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 976 . . 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 12160 . 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 1333 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    <-> wb 178    /\ wa 360    T. wtru 1326    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991    < clt 9122    - cmin 9293    / cdiv 9679   2c2 10051   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   abscabs 12041
This theorem is referenced by:  cau4  12162  serf0  12476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043
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