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Theorem rlimclim 11985
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
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 5331 . . . . 5  |-  ( F : Z --> CC  ->  dom 
F  =  Z )
7 eqimss2 3206 . . . . 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 11984 . 2  |-  ( (
ph  /\  F  ~~> r  A
)  ->  F  ~~>  A )
11 climcl 11938 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
1211adantl 454 . . 3  |-  ( (
ph  /\  F  ~~>  A )  ->  A  e.  CC )
132ad2antrr 709 . . . . . 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 2259 . . . . . 6  |-  ( ( ( ( ph  /\  F 
~~>  A )  /\  y  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
16 simplr 734 . . . . . 6  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  F  ~~>  A )
171, 13, 14, 15, 16climi2 11950 . . . . 5  |-  ( ( ( ph  /\  F  ~~>  A )  /\  y  e.  RR+ )  ->  E. z  e.  Z  A. k  e.  ( ZZ>= `  z )
( abs `  (
( F `  k
)  -  A ) )  <  y )
18 uzssz 10214 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  M )  C_  ZZ
191, 18eqsstri 3183 . . . . . . . . . . . . 13  |-  Z  C_  ZZ
20 simplrl 739 . . . . . . . . . . . . 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 3153 . . . . . . . . . . . 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 735 . . . . . . . . . . . . 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 3153 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 10203 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  z
)  <->  ( z  e.  ZZ  /\  w  e.  ZZ  /\  z  <_  w ) )
2621, 23, 24, 25syl3anbrc 1141 . . . . . . . . . . 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 740 . . . . . . . . . . 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 5458 . . . . . . . . . . . . . . 15  |-  ( k  =  w  ->  ( F `  k )  =  ( F `  w ) )
2928oveq1d 5807 . . . . . . . . . . . . . 14  |-  ( k  =  w  ->  (
( F `  k
)  -  A )  =  ( ( F `
 w )  -  A ) )
3029fveq2d 5462 . . . . . . . . . . . . 13  |-  ( k  =  w  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  (
( F `  w
)  -  A ) ) )
3130breq1d 4007 . . . . . . . . . . . 12  |-  ( k  =  w  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  y  <->  ( abs `  ( ( F `  w )  -  A
) )  <  y
) )
3231rcla4v 2855 . . . . . . . . . . 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 601 . . . . . . . . 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 2601 . . . . . . . 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 601 . . . . . . 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 2630 . . . . . 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 9998 . . . . . . . 8  |-  ZZ  C_  RR
3919, 38sstri 3163 . . . . . . 7  |-  Z  C_  RR
40 ssrexv 3213 . . . . . . 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 2601 . . 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 2259 . . . 4  |-  ( ( ( ph  /\  F  ~~>  A )  /\  w  e.  Z )  ->  ( F `  w )  =  ( F `  w ) )
4845, 46, 47rlim 11934 . . 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 893 . 2  |-  ( (
ph  /\  F  ~~>  A )  ->  F  ~~> r  A
)
5010, 49impbida 808 1  |-  ( ph  ->  ( F  ~~> r  A  <->  F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   E.wrex 2519    C_ wss 3127   class class class wbr 3997   dom cdm 4661   -->wf 4669   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704    < clt 8835    <_ cle 8836    - cmin 9005   ZZcz 9991   ZZ>=cuz 10197   RR+crp 10321   abscabs 11684    ~~> cli 11923    ~~> r crli 11924
This theorem is referenced by:  climmpt2  12012  climrecl  12022  climge0  12023  caurcvg  12114  caucvg  12116  climfsum  12243  divcnv  12274  dfef2  20227
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-n0 9933  df-z 9992  df-uz 10198  df-fl 10891  df-clim 11927  df-rlim 11928
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