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Theorem rlimclim 13022
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 465 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  M  e.  ZZ )
4 simpr 461 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~> r  A
)
5 rlimclim.3 . . . . 5  |-  ( ph  ->  F : Z --> CC )
6 fdm 5561 . . . . 5  |-  ( F : Z --> CC  ->  dom 
F  =  Z )
7 eqimss2 3407 . . . . 5  |-  ( dom 
F  =  Z  ->  Z  C_  dom  F )
85, 6, 73syl 20 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
98adantr 465 . . 3  |-  ( (
ph  /\  F  ~~> r  A
)  ->  Z  C_  dom  F )
101, 3, 4, 9rlimclim1 13021 . 2  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~>  A )
11 climcl 12975 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
1211adantl 466 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A  e.  CC )
132ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  M  e.  ZZ )
14 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
15 eqidd 2442 . . . . . 6  |-  ( ( ( ( ph  /\  F 
~~>  A )  /\  y  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
16 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  F  ~~>  A )
171, 13, 14, 15, 16climi2 12987 . . . . 5  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  Z  A. k  e.  ( ZZ>= `  z )
( abs `  (
( F `  k
)  -  A ) )  <  y )
18 uzssz 10878 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  ZZ
191, 18eqsstri 3384 . . . . . . . . . . . . 13  |-  Z  C_  ZZ
20 simplrl 759 . . . . . . . . . . . . 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 3352 . . . . . . . . . . . 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 755 . . . . . . . . . . . . 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 3352 . . . . . . . . . . . 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 756 . . . . . . . . . . . 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 10865 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  z
)  <->  ( z  e.  ZZ  /\  w  e.  ZZ  /\  z  <_  w ) )
2621, 23, 24, 25syl3anbrc 1172 . . . . . . . . . . 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 760 . . . . . . . . . . 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 5689 . . . . . . . . . . . . . . 15  |-  ( k  =  w  ->  ( F `  k )  =  ( F `  w ) )
2928oveq1d 6104 . . . . . . . . . . . . . 14  |-  ( k  =  w  ->  (
( F `  k
)  -  A )  =  ( ( F `
 w )  -  A ) )
3029fveq2d 5693 . . . . . . . . . . . . 13  |-  ( k  =  w  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  (
( F `  w
)  -  A ) ) )
3130breq1d 4300 . . . . . . . . . . . 12  |-  ( k  =  w  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  y  <->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3231rspcv 3067 . . . . . . . . . . 11  |-  ( w  e.  ( ZZ>= `  z
)  ->  ( A. k  e.  ( ZZ>= `  z ) ( abs `  ( ( F `  k )  -  A
) )  <  y  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) )
3326, 27, 32sylc 60 . . . . . . . . . 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 615 . . . . . . . . 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 2797 . . . . . . . 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 615 . . . . . . 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 2826 . . . . . 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 10651 . . . . . . . 8  |-  ZZ  C_  RR
3919, 38sstri 3363 . . . . . . 7  |-  Z  C_  RR
40 ssrexv 3415 . . . . . . 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 5 . . . . . 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 33 . . . . 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 15 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  RR  A. w  e.  Z  ( z  <_  w  ->  ( abs `  (
( F `  w
)  -  A ) )  <  y ) )
4443ralrimiva 2797 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A. y  e.  RR+  E. z  e.  RR  A. w  e.  Z  (
z  <_  w  ->  ( abs `  ( ( F `  w )  -  A ) )  <  y ) )
455adantr 465 . . . 4  |-  ( (
ph  /\  F  ~~>  A )  ->  F : Z --> CC )
4639a1i 11 . . . 4  |-  ( (
ph  /\  F  ~~>  A )  ->  Z  C_  RR )
47 eqidd 2442 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  w  e.  Z )  ->  ( F `  w )  =  ( F `  w ) )
4845, 46, 47rlim 12971 . . 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 913 . 2  |-  ( (
ph  /\  F  ~~>  A )  ->  F  ~~> r  A
)
5010, 49impbida 828 1  |-  ( ph  ->  ( F  ~~> r  A  <->  F  ~~>  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714    C_ wss 3326   class class class wbr 4290   dom cdm 4838   -->wf 5412   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279    < clt 9416    <_ cle 9417    - cmin 9593   ZZcz 10644   ZZ>=cuz 10859   RR+crp 10989   abscabs 12721    ~~> cli 12960    ~~> r crli 12961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fl 11640  df-clim 12964  df-rlim 12965
This theorem is referenced by:  climmpt2  13049  climrecl  13059  climge0  13060  caurcvg  13152  caucvg  13154  climfsum  13281  divcnv  13314  dfef2  22362
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