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Theorem climreeq 29783
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 9337 . . . . . 6  |-  RR  C_  CC
43a1i 11 . . . . 5  |-  ( ph  ->  RR  C_  CC )
5 fss 5565 . . . . 5  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
62, 4, 5syl2anc 661 . . . 4  |-  ( ph  ->  F : Z --> CC )
7 eqid 2441 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
8 climreeq.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
97, 8lmclimf 20812 . . . 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 20378 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
12 reex 9371 . . . . . . 7  |-  RR  e.  _V
1312a1i 11 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  RR  e.  _V )
147cnfldtop 20361 . . . . . . 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 18900 . . . . 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 5841 . . . . . . . . 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 13059 . . . . . . 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 20340 . . . . . . . . 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 18899 . . . . . . . 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 4296 . 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 1369    e. wcel 1756   _Vcvv 2970    C_ wss 3326   class class class wbr 4290   ran crn 4839   -->wf 5412   ` cfv 5416   CCcc 9278   RRcr 9279   ZZcz 10644   ZZ>=cuz 10859   (,)cioo 11298    ~~> cli 12960   TopOpenctopn 14358   topGenctg 14374  ℂfldccnfld 17816   Topctop 18496  TopOnctopon 18497   ~~> tclm 18828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fi 7659  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-fz 11436  df-fl 11640  df-seq 11805  df-exp 11864  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-rlim 12965  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-plusg 14249  df-mulr 14250  df-starv 14251  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-rest 14359  df-topn 14360  df-topgen 14380  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-cnfld 17817  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-lm 18831  df-xms 19893  df-ms 19894
This theorem is referenced by:  stirlingr  29882
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