Theorem climreeq 39845

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.

Step	Hyp	Ref	Expression
1		climreeq.3	. . . 4 |- ( ph -> M e. ZZ )
2		climreeq.4	. . . . 5 |- ( ph -> F : Z --> RR )
3		ax-resscn 9993	. . . . . 6 |- RR C_ CC
4	3	a1i 11	. . . . 5 |- ( ph -> RR C_ CC )
5	2, 4	fssd 6057	. . . 4 |- ( ph -> F : Z --> CC )
6		eqid 2622	. . . . 5 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
7		climreeq.2	. . . . 5 |- Z = ( ZZ>= ` M )
8	6, 7	lmclimf 23102	. . . 4 |- ( ( M e. ZZ /\ F : Z --> CC ) -> ( F ( ~~> t ` ( TopOpen ` ℂfld ) ) A <-> F ~~> A ) )
9	1, 5, 8	syl2anc 693	. . 3 |- ( ph -> ( F ( ~~> t ` ( TopOpen ` ℂfld ) ) A <-> F ~~> A ) )
10	6	tgioo2 22606	. . . . . 6 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
11		reex 10027	. . . . . . 7 |- RR e. _V
12	11	a1i 11	. . . . . 6 |- ( ( ph /\ A e. RR ) -> RR e. _V )
13	6	cnfldtop 22587	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. Top
14	13	a1i 11	. . . . . 6 |- ( ( ph /\ A e. RR ) -> ( TopOpen ` ℂfld ) e. Top )
15		simpr 477	. . . . . 6 |- ( ( ph /\ A e. RR ) -> A e. RR )
16	1	adantr 481	. . . . . 6 |- ( ( ph /\ A e. RR ) -> M e. ZZ )
17	2	adantr 481	. . . . . 6 |- ( ( ph /\ A e. RR ) -> F : Z --> RR )
18	10, 7, 12, 14, 15, 16, 17	lmss 21102	. . . . 5 |- ( ( ph /\ A e. RR ) -> ( F ( ~~> t ` ( TopOpen ` ℂfld ) ) A <-> F ( ~~> t ` ( topGen ` ran (,) ) ) A ) )
19	18	pm5.32da 673	. . . 4 |- ( ph -> ( ( A e. RR /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) <-> ( A e. RR /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) ) )
20		simpr 477	. . . . 5 |- ( ( A e. RR /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) -> F ( ~~> t ` ( TopOpen ` ℂfld ) ) A )
21	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) -> M e. ZZ )
22	9	biimpa 501	. . . . . . . 8 |- ( ( ph /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) -> F ~~> A )
23	2	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. RR )
24	23	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) /\ n e. Z ) -> ( F ` n ) e. RR )
25	7, 21, 22, 24	climrecl 14314	. . . . . . 7 |- ( ( ph /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) -> A e. RR )
26	25	ex 450	. . . . . 6 |- ( ph -> ( F ( ~~> t ` ( TopOpen ` ℂfld ) ) A -> A e. RR ) )
27	26	ancrd 577	. . . . 5 |- ( ph -> ( F ( ~~> t ` ( TopOpen ` ℂfld ) ) A -> ( A e. RR /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) ) )
28	20, 27	impbid2 216	. . . 4 |- ( ph -> ( ( A e. RR /\ F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) <-> F ( ~~> t ` ( TopOpen ` ℂfld ) ) A ) )
29		simpr 477	. . . . 5 |- ( ( A e. RR /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) -> F ( ~~> t ` ( topGen ` ran (,) ) ) A )
30		retopon 22567	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
31	30	a1i 11	. . . . . . . 8 |- ( ( ph /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) -> ( topGen ` ran (,) ) e. (TopOn ` RR ) )
32		simpr 477	. . . . . . . 8 |- ( ( ph /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) -> F ( ~~> t ` ( topGen ` ran (,) ) ) A )
33		lmcl 21101	. . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) -> A e. RR )
34	31, 32, 33	syl2anc 693	. . . . . . 7 |- ( ( ph /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) -> A e. RR )
35	34	ex 450	. . . . . 6 |- ( ph -> ( F ( ~~> t ` ( topGen ` ran (,) ) ) A -> A e. RR ) )
36	35	ancrd 577	. . . . 5 |- ( ph -> ( F ( ~~> t ` ( topGen ` ran (,) ) ) A -> ( A e. RR /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) ) )
37	29, 36	impbid2 216	. . . 4 |- ( ph -> ( ( A e. RR /\ F ( ~~> t ` ( topGen ` ran (,) ) ) A ) <-> F ( ~~> t ` ( topGen ` ran (,) ) ) A ) )
38	19, 28, 37	3bitr3d 298	. . 3 |- ( ph -> ( F ( ~~> t ` ( TopOpen ` ℂfld ) ) A <-> F ( ~~> t ` ( topGen ` ran (,) ) ) A ) )
39	9, 38	bitr3d 270	. 2 |- ( ph -> ( F ~~> A <-> F ( ~~> t ` ( topGen ` ran (,) ) ) A ) )
40		climreeq.1	. . 3 |- R = ( ~~> t ` ( topGen ` ran (,) ) )
41	40	breqi 4659	. 2 |- ( F R A <-> F ( ~~> t ` ( topGen ` ran (,) ) ) A )
42	39, 41	syl6rbbr 279	1 |- ( ph -> ( F R A <-> F ~~> A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   ran crn 5115   -->wf 5884   ` cfv 5888   CCcc 9934   RRcr 9935   ZZcz 11377   ZZ>=cuz 11687   (,)cioo 12175   ~~> cli 14215   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   ~~> tclm 21030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-lm 21033  df-xms 22125  df-ms 22126

This theorem is referenced by:  xlimclim  40050  stirlingr  40307