Theorem xlimxrre 40057

Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis F e. dom ~~> is probably not enough, since in principle we could have +oo e. CC and -oo e. CC). (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
xlimxrre.m	|- ( ph -> M e. ZZ )
xlimxrre.z	|- Z = ( ZZ>= ` M )
xlimxrre.f	|- ( ph -> F : Z --> RR* )
xlimxrre.a	|- ( ph -> A e. RR )
xlimxrre.c	|- ( ph -> F~~>* A )

Assertion

Ref	Expression
xlimxrre	|- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR )

Distinct variable groups:   A, j   j, F   j, M   j, Z   ph, j

Proof of Theorem xlimxrre

Dummy variables k u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elioore 12205	. . . . . . 7 |- ( ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) -> ( F ` k ) e. RR )
2	1	anim2i 593	. . . . . 6 |- ( ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) -> ( k e. dom F /\ ( F ` k ) e. RR ) )
3	2	ralimi 2952	. . . . 5 |- ( A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) -> A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. RR ) )
4	3	adantl 482	. . . 4 |- ( ( ph /\ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) -> A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. RR ) )
5		xlimxrre.f	. . . . . . 7 |- ( ph -> F : Z --> RR* )
6	5	ffund 6049	. . . . . 6 |- ( ph -> Fun F )
7		ffvresb 6394	. . . . . 6 |- ( Fun F -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR <-> A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. RR ) ) )
8	6, 7	syl 17	. . . . 5 |- ( ph -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR <-> A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. RR ) ) )
9	8	adantr 481	. . . 4 |- ( ( ph /\ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR <-> A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. RR ) ) )
10	4, 9	mpbird 247	. . 3 |- ( ( ph /\ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) -> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR )
11	10	adantrl 752	. 2 |- ( ( ph /\ ( j e. Z /\ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) ) -> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR )
12		xlimxrre.a	. . . . . 6 |- ( ph -> A e. RR )
13		peano2rem 10348	. . . . . 6 |- ( A e. RR -> ( A - 1 ) e. RR )
14	12, 13	syl 17	. . . . 5 |- ( ph -> ( A - 1 ) e. RR )
15	14	rexrd 10089	. . . 4 |- ( ph -> ( A - 1 ) e. RR* )
16		peano2re 10209	. . . . . 6 |- ( A e. RR -> ( A + 1 ) e. RR )
17	12, 16	syl 17	. . . . 5 |- ( ph -> ( A + 1 ) e. RR )
18	17	rexrd 10089	. . . 4 |- ( ph -> ( A + 1 ) e. RR* )
19	12	ltm1d 10956	. . . 4 |- ( ph -> ( A - 1 ) < A )
20	12	ltp1d 10954	. . . 4 |- ( ph -> A < ( A + 1 ) )
21	15, 18, 12, 19, 20	eliood 39720	. . 3 |- ( ph -> A e. ( ( A - 1 ) (,) ( A + 1 ) ) )
22		iooordt 21021	. . . 4 |- ( ( A - 1 ) (,) ( A + 1 ) ) e. (ordTop ` <_ )
23		xlimxrre.c	. . . . . 6 |- ( ph -> F~~>* A )
24		nfcv 2764	. . . . . . 7 |- F/_ k F
25		xlimxrre.m	. . . . . . 7 |- ( ph -> M e. ZZ )
26		xlimxrre.z	. . . . . . 7 |- Z = ( ZZ>= ` M )
27		eqid 2622	. . . . . . 7 |- (ordTop ` <_ ) = (ordTop ` <_ )
28	24, 25, 26, 5, 27	xlimbr 40053	. . . . . 6 |- ( ph -> ( F~~>* A <-> ( A e. RR* /\ A. u e. (ordTop ` <_ ) ( A e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) ) )
29	23, 28	mpbid 222	. . . . 5 |- ( ph -> ( A e. RR* /\ A. u e. (ordTop ` <_ ) ( A e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) )
30	29	simprd 479	. . . 4 |- ( ph -> A. u e. (ordTop ` <_ ) ( A e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) )
31		eleq2 2690	. . . . . 6 |- ( u = ( ( A - 1 ) (,) ( A + 1 ) ) -> ( A e. u <-> A e. ( ( A - 1 ) (,) ( A + 1 ) ) ) )
32		eleq2 2690	. . . . . . . 8 |- ( u = ( ( A - 1 ) (,) ( A + 1 ) ) -> ( ( F ` k ) e. u <-> ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) )
33	32	anbi2d 740	. . . . . . 7 |- ( u = ( ( A - 1 ) (,) ( A + 1 ) ) -> ( ( k e. dom F /\ ( F ` k ) e. u ) <-> ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) )
34	33	rexralbidv 3058	. . . . . 6 |- ( u = ( ( A - 1 ) (,) ( A + 1 ) ) -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) )
35	31, 34	imbi12d 334	. . . . 5 |- ( u = ( ( A - 1 ) (,) ( A + 1 ) ) -> ( ( A e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) <-> ( A e. ( ( A - 1 ) (,) ( A + 1 ) ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) ) )
36	35	rspcva 3307	. . . 4 |- ( ( ( ( A - 1 ) (,) ( A + 1 ) ) e. (ordTop ` <_ ) /\ A. u e. (ordTop ` <_ ) ( A e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) -> ( A e. ( ( A - 1 ) (,) ( A + 1 ) ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) )
37	22, 30, 36	sylancr 695	. . 3 |- ( ph -> ( A e. ( ( A - 1 ) (,) ( A + 1 ) ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) ) )
38	21, 37	mpd 15	. 2 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. ( ( A - 1 ) (,) ( A + 1 ) ) ) )
39	11, 38	reximddv 3018	1 |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> RR )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   (,)cioo 12175  ordTopcordt 16159  ~~>*clsxlim 40044

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-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-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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-topgen 16104  df-ordt 16161  df-ps 17200  df-tsr 17201  df-top 20699  df-topon 20716  df-bases 20750  df-lm 21033  df-xlim 40045

This theorem is referenced by:  xlimclim2  40066