Theorem climliminflimsupd 40033

Description: If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
climliminflimsupd.1	|- ( ph -> M e. ZZ )
climliminflimsupd.2	|- Z = ( ZZ>= ` M )
climliminflimsupd.3	|- ( ph -> F : Z --> RR )
climliminflimsupd.4	|- ( ph -> F e. dom ~~> )

Assertion

Ref	Expression
climliminflimsupd	|- ( ph -> (liminf ` F ) = ( limsup ` F ) )

Proof of Theorem climliminflimsupd

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climliminflimsupd.3	. . . . . . 7 |- ( ph -> F : Z --> RR )
2	1	feqmptd 6249	. . . . . 6 |- ( ph -> F = ( k e. Z |-> ( F ` k ) ) )
3	2	fveq2d 6195	. . . . 5 |- ( ph -> (liminf ` F ) = (liminf ` ( k e. Z |-> ( F ` k ) ) ) )
4		climliminflimsupd.2	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
5	4	fvexi 6202	. . . . . . . 8 |- Z e. _V
6	5	mptex 6486	. . . . . . 7 |- ( k e. Z |-> ( F ` k ) ) e. _V
7		liminfcl 39995	. . . . . . 7 |- ( ( k e. Z |-> ( F ` k ) ) e. _V -> (liminf ` ( k e. Z |-> ( F ` k ) ) ) e. RR* )
8	6, 7	ax-mp 5	. . . . . 6 |- (liminf ` ( k e. Z |-> ( F ` k ) ) ) e. RR*
9	8	a1i 11	. . . . 5 |- ( ph -> (liminf ` ( k e. Z |-> ( F ` k ) ) ) e. RR* )
10	3, 9	eqeltrd 2701	. . . 4 |- ( ph -> (liminf ` F ) e. RR* )
11		nfv 1843	. . . . . . 7 |- F/ k ph
12		climliminflimsupd.1	. . . . . . 7 |- ( ph -> M e. ZZ )
13	1	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
14	13	renegcld 10457	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> -u ( F ` k ) e. RR )
15	11, 12, 4, 14	limsupvaluz4 40032	. . . . . 6 |- ( ph -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = -e (liminf ` ( k e. Z |-> -u -u ( F ` k ) ) ) )
16		climrel 14223	. . . . . . . . . 10 |- Rel ~~>
17	16	a1i 11	. . . . . . . . 9 |- ( ph -> Rel ~~> )
18		nfcv 2764	. . . . . . . . . 10 |- F/_ k F
19		climliminflimsupd.4	. . . . . . . . . . 11 |- ( ph -> F e. dom ~~> )
20	12, 4, 1	climlimsup 39992	. . . . . . . . . . 11 |- ( ph -> ( F e. dom ~~> <-> F ~~> ( limsup ` F ) ) )
21	19, 20	mpbid 222	. . . . . . . . . 10 |- ( ph -> F ~~> ( limsup ` F ) )
22	13	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
23	11, 18, 4, 12, 21, 22	climneg 39842	. . . . . . . . 9 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) ~~> -u ( limsup ` F ) )
24		releldm 5358	. . . . . . . . 9 |- ( ( Rel ~~> /\ ( k e. Z |-> -u ( F ` k ) ) ~~> -u ( limsup ` F ) ) -> ( k e. Z |-> -u ( F ` k ) ) e. dom ~~> )
25	17, 23, 24	syl2anc 693	. . . . . . . 8 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) e. dom ~~> )
26		eqid 2622	. . . . . . . . . 10 |- ( k e. Z |-> -u ( F ` k ) ) = ( k e. Z |-> -u ( F ` k ) )
27	14, 26	fmptd 6385	. . . . . . . . 9 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) : Z --> RR )
28	12, 4, 27	climlimsup 39992	. . . . . . . 8 |- ( ph -> ( ( k e. Z |-> -u ( F ` k ) ) e. dom ~~> <-> ( k e. Z |-> -u ( F ` k ) ) ~~> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) ) )
29	25, 28	mpbid 222	. . . . . . 7 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) ~~> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) )
30		climuni 14283	. . . . . . 7 |- ( ( ( k e. Z |-> -u ( F ` k ) ) ~~> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) /\ ( k e. Z |-> -u ( F ` k ) ) ~~> -u ( limsup ` F ) ) -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = -u ( limsup ` F ) )
31	29, 23, 30	syl2anc 693	. . . . . 6 |- ( ph -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = -u ( limsup ` F ) )
32	22	negnegd 10383	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> -u -u ( F ` k ) = ( F ` k ) )
33	32	mpteq2dva 4744	. . . . . . . . 9 |- ( ph -> ( k e. Z |-> -u -u ( F ` k ) ) = ( k e. Z |-> ( F ` k ) ) )
34	33, 2	eqtr4d 2659	. . . . . . . 8 |- ( ph -> ( k e. Z |-> -u -u ( F ` k ) ) = F )
35	34	fveq2d 6195	. . . . . . 7 |- ( ph -> (liminf ` ( k e. Z |-> -u -u ( F ` k ) ) ) = (liminf ` F ) )
36	35	xnegeqd 39664	. . . . . 6 |- ( ph -> -e (liminf ` ( k e. Z |-> -u -u ( F ` k ) ) ) = -e (liminf ` F ) )
37	15, 31, 36	3eqtr3d 2664	. . . . 5 |- ( ph -> -u ( limsup ` F ) = -e (liminf ` F ) )
38	4, 12, 21, 13	climrecl 14314	. . . . . 6 |- ( ph -> ( limsup ` F ) e. RR )
39	38	renegcld 10457	. . . . 5 |- ( ph -> -u ( limsup ` F ) e. RR )
40	37, 39	eqeltrrd 2702	. . . 4 |- ( ph -> -e (liminf ` F ) e. RR )
41		xnegrecl2 39690	. . . 4 |- ( ( (liminf ` F ) e. RR* /\ -e (liminf ` F ) e. RR ) -> (liminf ` F ) e. RR )
42	10, 40, 41	syl2anc 693	. . 3 |- ( ph -> (liminf ` F ) e. RR )
43	42	recnd 10068	. 2 |- ( ph -> (liminf ` F ) e. CC )
44	38	recnd 10068	. 2 |- ( ph -> ( limsup ` F ) e. CC )
45	42	rexnegd 39334	. . 3 |- ( ph -> -e (liminf ` F ) = -u (liminf ` F ) )
46	37, 45	eqtr2d 2657	. 2 |- ( ph -> -u (liminf ` F ) = -u ( limsup ` F ) )
47	43, 44, 46	neg11d 10404	1 |- ( ph -> (liminf ` F ) = ( limsup ` F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Rel wrel 5119   -->wf 5884   ` cfv 5888   RRcr 9935   RR*cxr 10073   -ucneg 10267   ZZcz 11377   ZZ>=cuz 11687   -ecxne 11943   limsupclsp 14201   ~~> cli 14215  liminfclsi 39983

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-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-isom 5897  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-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-ico 12181  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-limsup 14202  df-clim 14219  df-rlim 14220  df-liminf 39984

This theorem is referenced by:  climliminf  40038  climliminflimsup  40040  climliminflimsup2  40041