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Theorem climreeq 31110
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 9538 . . . . . 6  |-  RR  C_  CC
43a1i 11 . . . . 5  |-  ( ph  ->  RR  C_  CC )
5 fss 5730 . . . . 5  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
62, 4, 5syl2anc 661 . . . 4  |-  ( ph  ->  F : Z --> CC )
7 eqid 2460 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
8 climreeq.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
97, 8lmclimf 21470 . . . 4  |-  ( ( M  e.  ZZ  /\  F : Z --> CC )  ->  ( F ( ~~> t `  ( TopOpen ` fld )
) A  <->  F  ~~>  A ) )
101, 6, 9syl2anc 661 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F  ~~>  A ) )
117tgioo2 21036 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
12 reex 9572 . . . . . . 7  |-  RR  e.  _V
1312a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  RR  e.  _V )
147cnfldtop 21019 . . . . . . 7  |-  ( TopOpen ` fld )  e.  Top
1514a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  ( TopOpen ` fld )  e.  Top )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  A  e.  RR )
171adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  M  e.  ZZ )
182adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  F : Z
--> RR )
1911, 8, 13, 15, 16, 17, 18lmss 19558 . . . . 5  |-  ( (
ph  /\  A  e.  RR )  ->  ( F ( ~~> t `  ( TopOpen
` fld
) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
2019pm5.32da 641 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) ) )
21 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F ( ~~> t `  ( TopOpen ` fld ) ) A )
221adantr 465 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  M  e.  ZZ )
2310biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F  ~~>  A )
242ffvelrnda 6012 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  RR )
2524adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  F
( ~~> t `  ( TopOpen
` fld
) ) A )  /\  n  e.  Z
)  ->  ( F `  n )  e.  RR )
268, 22, 23, 25climrecl 13355 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  A  e.  RR )
2726ex 434 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  ->  A  e.  RR )
)
2827ancrd 554 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  -> 
( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A ) ) )
2921, 28impbid2 204 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  F ( ~~> t `  ( TopOpen ` fld ) ) A ) )
30 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  F ( ~~> t `  ( topGen `  ran  (,) )
) A )
31 retopon 20998 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
3231a1i 11 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR ) )
33 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
34 lmcl 19557 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  A  e.  RR )
3532, 33, 34syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  A  e.  RR )
3635ex 434 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  A  e.  RR ) )
3736ancrd 554 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  ( A  e.  RR  /\  F
( ~~> t `  ( topGen `
 ran  (,) )
) A ) ) )
3830, 37impbid2 204 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  <->  F ( ~~> t `  ( topGen `  ran  (,) )
) A ) )
3920, 29, 383bitr3d 283 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
4010, 39bitr3d 255 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
41 climreeq.1 . . 3  |-  R  =  ( ~~> t `  ( topGen `
 ran  (,) )
)
4241breqi 4446 . 2  |-  ( F R A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
4340, 42syl6rbbr 264 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   _Vcvv 3106    C_ wss 3469   class class class wbr 4440   ran crn 4993   -->wf 5575   ` cfv 5579   CCcc 9479   RRcr 9480   ZZcz 10853   ZZ>=cuz 11071   (,)cioo 11518    ~~> cli 13256   TopOpenctopn 14666   topGenctg 14682  ℂfldccnfld 18184   Topctop 19154  TopOnctopon 19155   ~~> tclm 19486
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-rep 4551  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-int 4276  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-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fi 7860  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-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-fz 11662  df-fl 11886  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-rest 14667  df-topn 14668  df-topgen 14688  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-lm 19489  df-xms 20551  df-ms 20552
This theorem is referenced by:  stirlingr  31345
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