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Theorem rlimclim 12016
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 ) )
Dummy variables  w  k  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem rlimclim
StepHypRef Expression
1 rlimclim.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 rlimclim.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
32adantr 453 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  M  e.  ZZ )
4 simpr 449 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~> r  A
)
5 rlimclim.3 . . . . 5  |-  ( ph  ->  F : Z --> CC )
6 fdm 5360 . . . . 5  |-  ( F : Z --> CC  ->  dom 
F  =  Z )
7 eqimss2 3234 . . . . 5  |-  ( dom 
F  =  Z  ->  Z  C_  dom  F )
85, 6, 73syl 20 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
98adantr 453 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  Z  C_  dom  F )
101, 3, 4, 9rlimclim1 12015 . 2  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~>  A )
11 climcl 11969 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
1211adantl 454 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A  e.  CC )
132ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  M  e.  ZZ )
14 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
15 eqidd 2287 . . . . . 6  |-  ( ( ( ( ph  /\  F 
~~>  A )  /\  y  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
16 simplr 733 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  F  ~~>  A )
171, 13, 14, 15, 16climi2 11981 . . . . 5  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  Z  A. k  e.  ( ZZ>= `  z )
( abs `  (
( F `  k
)  -  A ) )  <  y )
18 uzssz 10244 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  ZZ
191, 18eqsstri 3211 . . . . . . . . . . . . 13  |-  Z  C_  ZZ
20 simplrl 738 . . . . . . . . . . . . 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 3181 . . . . . . . . . . . 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 734 . . . . . . . . . . . . 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 3181 . . . . . . . . . . . 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 735 . . . . . . . . . . . 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 10233 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  z
)  <->  ( z  e.  ZZ  /\  w  e.  ZZ  /\  z  <_  w ) )
2621, 23, 24, 25syl3anbrc 1138 . . . . . . . . . . 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 739 . . . . . . . . . . 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 5487 . . . . . . . . . . . . . . 15  |-  ( k  =  w  ->  ( F `  k )  =  ( F `  w ) )
2928oveq1d 5836 . . . . . . . . . . . . . 14  |-  ( k  =  w  ->  (
( F `  k
)  -  A )  =  ( ( F `
 w )  -  A ) )
3029fveq2d 5491 . . . . . . . . . . . . 13  |-  ( k  =  w  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  (
( F `  w
)  -  A ) ) )
3130breq1d 4036 . . . . . . . . . . . 12  |-  ( k  =  w  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  y  <->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3231rspcv 2883 . . . . . . . . . . 11  |-  ( w  e.  ( ZZ>= `  z
)  ->  ( A. k  e.  ( ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A
) )  <  y  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) )
3326, 27, 32sylc 58 . . . . . . . . . 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 600 . . . . . . . . 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 2629 . . . . . . . 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 600 . . . . . . 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 2658 . . . . . 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 10028 . . . . . . . 8  |-  ZZ  C_  RR
3919, 38sstri 3191 . . . . . . 7  |-  Z  C_  RR
40 ssrexv 3241 . . . . . . 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 10 . . . . . 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 31 . . . . 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 16 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) )
4443ralrimiva 2629 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A. y  e.  RR+  E. z  e.  RR  A. w  e.  Z  (
z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y ) )
455adantr 453 . . . 4  |-  ( (
ph  /\  F  ~~>  A )  ->  F : Z --> CC )
4639a1i 12 . . . 4  |-  ( (
ph  /\  F  ~~>  A )  ->  Z  C_  RR )
47 eqidd 2287 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  w  e.  Z )  ->  ( F `  w )  =  ( F `  w ) )
4845, 46, 47rlim 11965 . . 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 890 . 2  |-  ( (
ph  /\  F  ~~>  A )  ->  F  ~~> r  A
)
5010, 49impbida 807 1  |-  ( ph  ->  ( F  ~~> r  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687   A.wral 2546   E.wrex 2547    C_ wss 3155   class class class wbr 4026   dom cdm 4690   -->wf 5219   ` cfv 5223  (class class class)co 5821   CCcc 8732   RRcr 8733    < clt 8864    <_ cle 8865    - cmin 9034   ZZcz 10021   ZZ>=cuz 10227   RR+crp 10351   abscabs 11715    ~~> cli 11954    ~~> r crli 11955
This theorem is referenced by:  climmpt2  12043  climrecl  12053  climge0  12054  caurcvg  12145  caucvg  12147  climfsum  12274  divcnv  12308  dfef2  20261
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-er 6657  df-pm 6772  df-en 6861  df-dom 6862  df-sdom 6863  df-sup 7191  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-nn 9744  df-n0 9963  df-z 10022  df-uz 10228  df-fl 10921  df-clim 11958  df-rlim 11959
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