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Theorem rlimclim 12020
Description: A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)
Hypotheses
Ref Expression
rlimclim.1  |-  Z  =  ( ZZ>= `  M )
rlimclim.2  |-  ( ph  ->  M  e.  ZZ )
rlimclim.3  |-  ( ph  ->  F : Z --> CC )
Assertion
Ref Expression
rlimclim  |-  ( ph  ->  ( F  ~~> r  A  <->  F  ~~>  A ) )

Proof of Theorem rlimclim
Dummy variables  w  k  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimclim.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 rlimclim.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
32adantr 451 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  M  e.  ZZ )
4 simpr 447 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~> r  A
)
5 rlimclim.3 . . . . 5  |-  ( ph  ->  F : Z --> CC )
6 fdm 5393 . . . . 5  |-  ( F : Z --> CC  ->  dom 
F  =  Z )
7 eqimss2 3231 . . . . 5  |-  ( dom 
F  =  Z  ->  Z  C_  dom  F )
85, 6, 73syl 18 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
98adantr 451 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  Z  C_  dom  F )
101, 3, 4, 9rlimclim1 12019 . 2  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~>  A )
11 climcl 11973 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
1211adantl 452 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A  e.  CC )
132ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  M  e.  ZZ )
14 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
15 eqidd 2284 . . . . . 6  |-  ( ( ( ( ph  /\  F 
~~>  A )  /\  y  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
16 simplr 731 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  F  ~~>  A )
171, 13, 14, 15, 16climi2 11985 . . . . 5  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  Z  A. k  e.  ( ZZ>= `  z )
( abs `  (
( F `  k
)  -  A ) )  <  y )
18 uzssz 10247 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  ZZ
191, 18eqsstri 3208 . . . . . . . . . . . . 13  |-  Z  C_  ZZ
20 simplrl 736 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  -> 
z  e.  Z )
2119, 20sseldi 3178 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  -> 
z  e.  ZZ )
22 simprl 732 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  ->  w  e.  Z )
2319, 22sseldi 3178 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  ->  w  e.  ZZ )
24 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  -> 
z  <_  w )
25 eluz2 10236 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  z
)  <->  ( z  e.  ZZ  /\  w  e.  ZZ  /\  z  <_  w ) )
2621, 23, 24, 25syl3anbrc 1136 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  ->  w  e.  ( ZZ>= `  z ) )
27 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  ->  A. k  e.  ( ZZ>=
`  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y )
28 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( k  =  w  ->  ( F `  k )  =  ( F `  w ) )
2928oveq1d 5873 . . . . . . . . . . . . . 14  |-  ( k  =  w  ->  (
( F `  k
)  -  A )  =  ( ( F `
 w )  -  A ) )
3029fveq2d 5529 . . . . . . . . . . . . 13  |-  ( k  =  w  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  (
( F `  w
)  -  A ) ) )
3130breq1d 4033 . . . . . . . . . . . 12  |-  ( k  =  w  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  y  <->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3231rspcv 2880 . . . . . . . . . . 11  |-  ( w  e.  ( ZZ>= `  z
)  ->  ( A. k  e.  ( ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A
) )  <  y  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) )
3326, 27, 32sylc 56 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  ( w  e.  Z  /\  z  <_  w ) )  -> 
( abs `  (
( F `  w
)  -  A ) )  <  y )
3433expr 598 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  /\  ( z  e.  Z  /\  A. k  e.  (
ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y ) )  /\  w  e.  Z
)  ->  ( z  <_  w  ->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3534ralrimiva 2626 . . . . . . . 8  |-  ( ( ( ( ph  /\  F 
~~>  A )  /\  y  e.  RR+ )  /\  (
z  e.  Z  /\  A. k  e.  ( ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A
) )  <  y
) )  ->  A. w  e.  Z  ( z  <_  w  ->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3635expr 598 . . . . . . 7  |-  ( ( ( ( ph  /\  F 
~~>  A )  /\  y  e.  RR+ )  /\  z  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y  ->  A. w  e.  Z  ( z  <_  w  ->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) ) )
3736reximdva 2655 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  ( E. z  e.  Z  A. k  e.  ( ZZ>=
`  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y  ->  E. z  e.  Z  A. w  e.  Z  ( z  <_  w  ->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) ) )
38 zssre 10031 . . . . . . . 8  |-  ZZ  C_  RR
3919, 38sstri 3188 . . . . . . 7  |-  Z  C_  RR
40 ssrexv 3238 . . . . . . 7  |-  ( Z 
C_  RR  ->  ( E. z  e.  Z  A. w  e.  Z  (
z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y )  ->  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y ) ) )
4139, 40ax-mp 8 . . . . . 6  |-  ( E. z  e.  Z  A. w  e.  Z  (
z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y )  ->  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y ) )
4237, 41syl6 29 . . . . 5  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  ( E. z  e.  Z  A. k  e.  ( ZZ>=
`  z ) ( abs `  ( ( F `  k )  -  A ) )  <  y  ->  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) ) )
4317, 42mpd 14 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) )
4443ralrimiva 2626 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A. y  e.  RR+  E. z  e.  RR  A. w  e.  Z  (
z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y ) )
455adantr 451 . . . 4  |-  ( (
ph  /\  F  ~~>  A )  ->  F : Z --> CC )
4639a1i 10 . . . 4  |-  ( (
ph  /\  F  ~~>  A )  ->  Z  C_  RR )
47 eqidd 2284 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  w  e.  Z )  ->  ( F `  w )  =  ( F `  w ) )
4845, 46, 47rlim 11969 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  ( F  ~~> r  A  <->  ( A  e.  CC  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) ) ) )
4912, 44, 48mpbir2and 888 . 2  |-  ( (
ph  /\  F  ~~>  A )  ->  F  ~~> r  A
)
5010, 49impbida 805 1  |-  ( ph  ->  ( F  ~~> r  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    < clt 8867    <_ cle 8868    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   abscabs 11719    ~~> cli 11958    ~~> r crli 11959
This theorem is referenced by:  climmpt2  12047  climrecl  12057  climge0  12058  caurcvg  12149  caucvg  12151  climfsum  12278  divcnv  12312  dfef2  20265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fl 10925  df-clim 11962  df-rlim 11963
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