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Theorem climreeq 27408
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 8981 . . . . . 6  |-  RR  C_  CC
43a1i 11 . . . . 5  |-  ( ph  ->  RR  C_  CC )
5 fss 5540 . . . . 5  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
62, 4, 5syl2anc 643 . . . 4  |-  ( ph  ->  F : Z --> CC )
7 eqid 2388 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
8 climreeq.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
97, 8lmclimf 19128 . . . 4  |-  ( ( M  e.  ZZ  /\  F : Z --> CC )  ->  ( F ( ~~> t `  ( TopOpen ` fld )
) A  <->  F  ~~>  A ) )
101, 6, 9syl2anc 643 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F  ~~>  A ) )
117tgioo2 18706 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
12 reex 9015 . . . . . . 7  |-  RR  e.  _V
1312a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  RR  e.  _V )
147cnfldtop 18690 . . . . . . 7  |-  ( TopOpen ` fld )  e.  Top
1514a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  ( TopOpen ` fld )  e.  Top )
16 simpr 448 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  A  e.  RR )
171adantr 452 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  M  e.  ZZ )
182adantr 452 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  F : Z
--> RR )
1911, 8, 13, 15, 16, 17, 18lmss 17285 . . . . 5  |-  ( (
ph  /\  A  e.  RR )  ->  ( F ( ~~> t `  ( TopOpen
` fld
) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
2019pm5.32da 623 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) ) )
21 simpr 448 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F ( ~~> t `  ( TopOpen ` fld ) ) A )
221adantr 452 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  M  e.  ZZ )
2310biimpa 471 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F  ~~>  A )
242ffvelrnda 5810 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  RR )
2524adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  F
( ~~> t `  ( TopOpen
` fld
) ) A )  /\  n  e.  Z
)  ->  ( F `  n )  e.  RR )
268, 22, 23, 25climrecl 12305 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  A  e.  RR )
2726ex 424 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  ->  A  e.  RR )
)
2827ancrd 538 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  -> 
( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A ) ) )
2921, 28impbid2 196 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  F ( ~~> t `  ( TopOpen ` fld ) ) A ) )
30 simpr 448 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  F ( ~~> t `  ( topGen `  ran  (,) )
) A )
31 retopon 18669 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
3231a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR ) )
33 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
34 lmcl 17284 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  A  e.  RR )
3532, 33, 34syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  A  e.  RR )
3635ex 424 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  A  e.  RR ) )
3736ancrd 538 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  ( A  e.  RR  /\  F
( ~~> t `  ( topGen `
 ran  (,) )
) A ) ) )
3830, 37impbid2 196 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  <->  F ( ~~> t `  ( topGen `  ran  (,) )
) A ) )
3920, 29, 383bitr3d 275 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
4010, 39bitr3d 247 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
41 climreeq.1 . . 3  |-  R  =  ( ~~> t `  ( topGen `
 ran  (,) )
)
4241breqi 4160 . 2  |-  ( F R A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
4340, 42syl6rbbr 256 1  |-  ( ph  ->  ( F R A  <-> 
F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900    C_ wss 3264   class class class wbr 4154   ran crn 4820   -->wf 5391   ` cfv 5395   CCcc 8922   RRcr 8923   ZZcz 10215   ZZ>=cuz 10421   (,)cioo 10849    ~~> cli 12206   TopOpenctopn 13577   topGenctg 13593  ℂfldccnfld 16627   Topctop 16882  TopOnctopon 16883   ~~> tclm 17213
This theorem is referenced by:  stirlingr  27508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-fz 10977  df-fl 11130  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-plusg 13470  df-mulr 13471  df-starv 13472  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-rest 13578  df-topn 13579  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-lm 17216  df-xms 18260  df-ms 18261
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