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Theorem climreeq 27739
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 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
2 climreeq.4 . . . . . . 7  |-  ( ph  ->  F : Z --> RR )
3 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
43a1i 10 . . . . . . 7  |-  ( ph  ->  RR  C_  CC )
52, 4jca 518 . . . . . 6  |-  ( ph  ->  ( F : Z --> RR  /\  RR  C_  CC ) )
6 fss 5397 . . . . . 6  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
75, 6syl 15 . . . . 5  |-  ( ph  ->  F : Z --> CC )
81, 7jca 518 . . . 4  |-  ( ph  ->  ( M  e.  ZZ  /\  F : Z --> CC ) )
9 eqid 2283 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
10 climreeq.2 . . . . 5  |-  Z  =  ( ZZ>= `  M )
119, 10lmclimf 18729 . . . 4  |-  ( ( M  e.  ZZ  /\  F : Z --> CC )  ->  ( F ( ~~> t `  ( TopOpen ` fld )
) A  <->  F  ~~>  A ) )
128, 11syl 15 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F  ~~>  A ) )
139tgioo2 18309 . . . . . 6  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
14 reex 8828 . . . . . . 7  |-  RR  e.  _V
1514a1i 10 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  RR  e.  _V )
169cnfldtop 18293 . . . . . . 7  |-  ( TopOpen ` fld )  e.  Top
1716a1i 10 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  ( TopOpen ` fld )  e.  Top )
18 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  A  e.  RR )
191adantr 451 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  M  e.  ZZ )
202adantr 451 . . . . . 6  |-  ( (
ph  /\  A  e.  RR )  ->  F : Z
--> RR )
2113, 10, 15, 17, 18, 19, 20lmss 17026 . . . . 5  |-  ( (
ph  /\  A  e.  RR )  ->  ( F ( ~~> t `  ( TopOpen
` fld
) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
2221pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) ) )
23 simpr 447 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F ( ~~> t `  ( TopOpen ` fld ) ) A )
241adantr 451 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  M  e.  ZZ )
2512biimpa 470 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  F  ~~>  A )
262ffvelrnda 5665 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  RR )
2726adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  F
( ~~> t `  ( TopOpen
` fld
) ) A )  /\  n  e.  Z
)  ->  ( F `  n )  e.  RR )
2810, 24, 25, 27climrecl 12057 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A )  ->  A  e.  RR )
2928ex 423 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  ->  A  e.  RR )
)
3029ancrd 537 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  -> 
( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld ) ) A ) ) )
3123, 30impbid2 195 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( TopOpen ` fld )
) A )  <->  F ( ~~> t `  ( TopOpen ` fld ) ) A ) )
32 simpr 447 . . . . 5  |-  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  F ( ~~> t `  ( topGen `  ran  (,) )
) A )
33 retopon 18272 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
3433a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR ) )
35 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
3634, 35jca 518 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  ( ( topGen `
 ran  (,) )  e.  (TopOn `  RR )  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A ) )
37 lmcl 17025 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  F ( ~~> t `  ( topGen `  ran  (,) )
) A )  ->  A  e.  RR )
3836, 37syl 15 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  ->  A  e.  RR )
3938ex 423 . . . . . 6  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  A  e.  RR ) )
4039ancrd 537 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  ( topGen `  ran  (,) )
) A  ->  ( A  e.  RR  /\  F
( ~~> t `  ( topGen `
 ran  (,) )
) A ) ) )
4132, 40impbid2 195 . . . 4  |-  ( ph  ->  ( ( A  e.  RR  /\  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )  <->  F ( ~~> t `  ( topGen `  ran  (,) )
) A ) )
4222, 31, 413bitr3d 274 . . 3  |-  ( ph  ->  ( F ( ~~> t `  ( TopOpen ` fld ) ) A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
4312, 42bitr3d 246 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A ) )
44 climreeq.1 . . 3  |-  R  =  ( ~~> t `  ( topGen `
 ran  (,) )
)
4544breqi 4029 . 2  |-  ( F R A  <->  F ( ~~> t `  ( topGen ` 
ran  (,) ) ) A )
4643, 45syl6rbbr 255 1  |-  ( ph  ->  ( F R A  <-> 
F  ~~>  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   ran crn 4690   -->wf 5251   ` cfv 5255   CCcc 8735   RRcr 8736   ZZcz 10024   ZZ>=cuz 10230   (,)cioo 10656    ~~> cli 11958   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632   ~~> tclm 16956
This theorem is referenced by:  stirlingr  27839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-fz 10783  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-lm 16959  df-xms 17885  df-ms 17886
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