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Theorem climreeq 31523
Description: If  F is a real function, then  F converges to  A with respect to the standard topology on the reals if and only if it converges to  A with respect to the standard topology on complex numbers. In the theorem,  R is defined to be convergence w.r.t. the standard topology on the reals and then  F R A represents the statement " F converges to  A, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that  A is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.)
Hypotheses
Ref Expression
climreeq.1  |-  R  =  ( ~~> t `  ( topGen `
 ran  (,) )
)
climreeq.2  |-  Z  =  ( ZZ>= `  M )
climreeq.3  |-  ( ph  ->  M  e.  ZZ )
climreeq.4  |-  ( ph  ->  F : Z --> RR )
Assertion
Ref Expression
climreeq  |-  ( ph  ->  ( F R A  <-> 
F  ~~>  A ) )

Proof of Theorem climreeq
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 climreeq.3 . . . 4  |-  ( ph  ->  M  e.  ZZ )
2 climreeq.4 . . . . 5  |-  ( ph  ->  F : Z --> RR )
3 ax-resscn 9547 . . . . . 6  |-  RR  C_  CC
43a1i 11 . . . . 5  |-  ( ph  ->  RR  C_  CC )
52, 4fssd 5726 . . . 4  |-  ( ph  ->  F : Z --> CC )
6 eqid 2441 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7 climreeq.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
86, 7lmclimf 21608 . . . 4  |-  ( ( M  e.  ZZ  /\  F : Z --> CC )  ->  ( F ( ~~> t `  ( TopOpen ` fld )
) A  <->  F  ~~>  A ) )
91, 5, 8syl2anc 661 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F  ~~>  A ) )
106tgioo2 21174 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
11 reex 9581 . . . . . . 7  |-  RR  e.  _V
1211a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  RR  e.  _V )
136cnfldtop 21157 . . . . . . 7  |-  ( TopOpen ` fld )  e.  Top
1413a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  ( TopOpen ` fld )  e.  Top )
15 simpr 461 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  A  e.  RR )
161adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  M  e.  ZZ )
172adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  F : Z
--> RR )
1810, 7, 12, 14, 15, 16, 17lmss 19665 . . . . 5  |-  ( (
ph  /\  A  e.  RR )  ->  ( F ( ~~> t `  ( TopOpen
` fld
) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
1918pm5.32da 641 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) ) )
20 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F ( ~~> t `  ( TopOpen ` fld ) ) A )
211adantr 465 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  M  e.  ZZ )
229biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F  ~~>  A )
232ffvelrnda 6012 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  RR )
2423adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  F
( ~~> t `  ( TopOpen
` fld
) ) A )  /\  n  e.  Z
)  ->  ( F `  n )  e.  RR )
257, 21, 22, 24climrecl 13380 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  A  e.  RR )
2625ex 434 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  ->  A  e.  RR )
)
2726ancrd 554 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  -> 
( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A ) ) )
2820, 27impbid2 204 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  F ( ~~> t `  ( TopOpen ` fld ) ) A ) )
29 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  F ( ~~> t `  ( topGen `  ran  (,) )
) A )
30 retopon 21136 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
3130a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR ) )
32 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
33 lmcl 19664 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  A  e.  RR )
3431, 32, 33syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  A  e.  RR )
3534ex 434 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  A  e.  RR ) )
3635ancrd 554 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  ( A  e.  RR  /\  F
( ~~> t `  ( topGen `
 ran  (,) )
) A ) ) )
3729, 36impbid2 204 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  <->  F ( ~~> t `  ( topGen `  ran  (,) )
) A ) )
3819, 28, 373bitr3d 283 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
399, 38bitr3d 255 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
40 climreeq.1 . . 3  |-  R  =  ( ~~> t `  ( topGen `
 ran  (,) )
)
4140breqi 4439 . 2  |-  ( F R A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
4239, 41syl6rbbr 264 1  |-  ( ph  ->  ( F R A  <-> 
F  ~~>  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093    C_ wss 3458   class class class wbr 4433   ran crn 4986   -->wf 5570   ` cfv 5574   CCcc 9488   RRcr 9489   ZZcz 10865   ZZ>=cuz 11085   (,)cioo 11533    ~~> cli 13281   TopOpenctopn 14691   topGenctg 14707  ℂfldccnfld 18288   Topctop 19261  TopOnctopon 19262   ~~> tclm 19593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fi 7869  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-fz 11677  df-fl 11903  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-plusg 14582  df-mulr 14583  df-starv 14584  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-rest 14692  df-topn 14693  df-topgen 14713  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-lm 19596  df-xms 20689  df-ms 20690
This theorem is referenced by:  stirlingr  31757
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