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Theorem rlimclim 13318
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 5726 . . . . 5  |-  ( F : Z --> CC  ->  dom 
F  =  Z )
7 eqimss2 3550 . . . . 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 13317 . 2  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~>  A )
11 climcl 13271 . . . 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 2461 . . . . . 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 13283 . . . . 5  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  Z  A. k  e.  ( ZZ>= `  z )
( abs `  (
( F `  k
)  -  A ) )  <  y )
18 uzssz 11090 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  ZZ
191, 18eqsstri 3527 . . . . . . . . . . . . 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 3495 . . . . . . . . . . . 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 3495 . . . . . . . . . . . 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 11077 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  z
)  <->  ( z  e.  ZZ  /\  w  e.  ZZ  /\  z  <_  w ) )
2621, 23, 24, 25syl3anbrc 1175 . . . . . . . . . . 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 5857 . . . . . . . . . . . . . . 15  |-  ( k  =  w  ->  ( F `  k )  =  ( F `  w ) )
2928oveq1d 6290 . . . . . . . . . . . . . 14  |-  ( k  =  w  ->  (
( F `  k
)  -  A )  =  ( ( F `
 w )  -  A ) )
3029fveq2d 5861 . . . . . . . . . . . . 13  |-  ( k  =  w  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  (
( F `  w
)  -  A ) ) )
3130breq1d 4450 . . . . . . . . . . . 12  |-  ( k  =  w  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  y  <->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3231rspcv 3203 . . . . . . . . . . 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 2871 . . . . . . . 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 2931 . . . . . 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 10860 . . . . . . . 8  |-  ZZ  C_  RR
3919, 38sstri 3506 . . . . . . 7  |-  Z  C_  RR
40 ssrexv 3558 . . . . . . 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 2871 . . 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 2461 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  w  e.  Z )  ->  ( F `  w )  =  ( F `  w ) )
4845, 46, 47rlim 13267 . . 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 915 . 2  |-  ( (
ph  /\  F  ~~>  A )  ->  F  ~~> r  A
)
5010, 49impbida 829 1  |-  ( ph  ->  ( F  ~~> r  A  <->  F  ~~>  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   class class class wbr 4440   dom cdm 4992   -->wf 5575   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480    < clt 9617    <_ cle 9618    - cmin 9794   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209   abscabs 13017    ~~> cli 13256    ~~> r crli 13257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fl 11886  df-clim 13260  df-rlim 13261
This theorem is referenced by:  climmpt2  13345  climrecl  13355  climge0  13356  caurcvg  13448  caucvg  13450  climfsum  13583  divcnv  13617  dfef2  23021
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