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Theorem cau3 13199
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 13189 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 11125 . . . 4  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3529 . . 3  |-  Z  C_  ZZ
4 id 22 . . 3  |-  ( ( F `  k )  e.  CC  ->  ( F `  k )  e.  CC )
5 eleq1 2529 . . 3  |-  ( ( F `  k )  =  ( F `  j )  ->  (
( F `  k
)  e.  CC  <->  ( F `  j )  e.  CC ) )
6 eleq1 2529 . . 3  |-  ( ( F `  k )  =  ( F `  m )  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
7 abssub 13170 . . . 4  |-  ( ( ( F `  j
)  e.  CC  /\  ( F `  k )  e.  CC )  -> 
( abs `  (
( F `  j
)  -  ( F `
 k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j )
) ) )
873adant1 1014 . . 3  |-  ( ( T.  /\  ( F `
 j )  e.  CC  /\  ( F `
 k )  e.  CC )  ->  ( abs `  ( ( F `
 j )  -  ( F `  k ) ) )  =  ( abs `  ( ( F `  k )  -  ( F `  j ) ) ) )
9 abssub 13170 . . . 4  |-  ( ( ( F `  m
)  e.  CC  /\  ( F `  j )  e.  CC )  -> 
( abs `  (
( F `  m
)  -  ( F `
 j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m )
) ) )
1093adant1 1014 . . 3  |-  ( ( T.  /\  ( F `
 m )  e.  CC  /\  ( F `
 j )  e.  CC )  ->  ( abs `  ( ( F `
 m )  -  ( F `  j ) ) )  =  ( abs `  ( ( F `  j )  -  ( F `  m ) ) ) )
11 abs3lem 13182 . . . 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 1014 . . 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 13198 . 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 1404 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 1395   T. wtru 1396    e. wcel 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508    < clt 9645    - cmin 9824    / cdiv 10227   2c2 10606   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   abscabs 13078
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  ax-pre-sup 9587
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-rmo 2815  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-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  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-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080
This theorem is referenced by:  cau4  13200  serf0  13514
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