Theorem liminfltlem 40036

Description: Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfltlem.m	|- ( ph -> M e. ZZ )
liminfltlem.z	|- Z = ( ZZ>= ` M )
liminfltlem.f	|- ( ph -> F : Z --> RR )
liminfltlem.r	|- ( ph -> (liminf ` F ) e. RR )
liminfltlem.x	|- ( ph -> X e. RR+ )

Assertion

Ref	Expression
liminfltlem	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) (liminf ` F ) < ( ( F ` k ) + X ) )

Distinct variable groups:   j, F, k   k, M   j, X, k   j, Z, k   ph, j, k

Allowed substitution hint:   M( j)

Proof of Theorem liminfltlem

Step	Hyp	Ref	Expression
1		nfmpt1 4747	. . 3 |- F/_ k ( k e. Z |-> -u ( F ` k ) )
2		liminfltlem.m	. . 3 |- ( ph -> M e. ZZ )
3		liminfltlem.z	. . 3 |- Z = ( ZZ>= ` M )
4		liminfltlem.f	. . . . . 6 |- ( ph -> F : Z --> RR )
5	4	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
6	5	renegcld 10457	. . . 4 |- ( ( ph /\ k e. Z ) -> -u ( F ` k ) e. RR )
7	6	fmptd2 39460	. . 3 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) : Z --> RR )
8	3	fvexi 6202	. . . . . . 7 |- Z e. _V
9	8	mptex 6486	. . . . . 6 |- ( k e. Z |-> -u ( F ` k ) ) e. _V
10	9	limsupcli 39989	. . . . 5 |- ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) e. RR*
11	10	a1i 11	. . . 4 |- ( ph -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) e. RR* )
12		nfv 1843	. . . . . 6 |- F/ k ph
13		nfcv 2764	. . . . . 6 |- F/_ k F
14	12, 13, 2, 3, 4	liminfvaluz4 40031	. . . . 5 |- ( ph -> (liminf ` F ) = -e ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) )
15		liminfltlem.r	. . . . 5 |- ( ph -> (liminf ` F ) e. RR )
16	14, 15	eqeltrrd 2702	. . . 4 |- ( ph -> -e ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) e. RR )
17	11, 16	xnegrecl2d 39697	. . 3 |- ( ph -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) e. RR )
18		liminfltlem.x	. . 3 |- ( ph -> X e. RR+ )
19	1, 2, 3, 7, 17, 18	limsupgt 40010	. 2 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) < ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) )
20		simpll 790	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ph )
21	3	uztrn2 11705	. . . . . 6 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
22	21	adantll 750	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
23		negex 10279	. . . . . . . . . . 11 |- -u ( F ` k ) e. _V
24		fvmpt4 39446	. . . . . . . . . . 11 |- ( ( k e. Z /\ -u ( F ` k ) e. _V ) -> ( ( k e. Z |-> -u ( F ` k ) ) ` k ) = -u ( F ` k ) )
25	23, 24	mpan2 707	. . . . . . . . . 10 |- ( k e. Z -> ( ( k e. Z |-> -u ( F ` k ) ) ` k ) = -u ( F ` k ) )
26	25	adantl 482	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> -u ( F ` k ) ) ` k ) = -u ( F ` k ) )
27	26	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) = ( -u ( F ` k ) - X ) )
28	5	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
29	18	rpred 11872	. . . . . . . . . . 11 |- ( ph -> X e. RR )
30	29	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> X e. RR )
31	30	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> X e. CC )
32	28, 31	negdi2d 10406	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> -u ( ( F ` k ) + X ) = ( -u ( F ` k ) - X ) )
33	27, 32	eqtr4d 2659	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) = -u ( ( F ` k ) + X ) )
34	17	recnd 10068	. . . . . . . . . 10 |- ( ph -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) e. CC )
35	17	rexnegd 39334	. . . . . . . . . . 11 |- ( ph -> -e ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = -u ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) )
36	14, 35	eqtr2d 2657	. . . . . . . . . 10 |- ( ph -> -u ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = (liminf ` F ) )
37	34, 36	negcon1ad 10387	. . . . . . . . 9 |- ( ph -> -u (liminf ` F ) = ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) )
38	37	eqcomd 2628	. . . . . . . 8 |- ( ph -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = -u (liminf ` F ) )
39	38	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) = -u (liminf ` F ) )
40	33, 39	breq12d 4666	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) < ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) <-> -u ( ( F ` k ) + X ) < -u (liminf ` F ) ) )
41	15	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> (liminf ` F ) e. RR )
42	5, 30	readdcld 10069	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) + X ) e. RR )
43	41, 42	ltnegd 10605	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( (liminf ` F ) < ( ( F ` k ) + X ) <-> -u ( ( F ` k ) + X ) < -u (liminf ` F ) ) )
44	40, 43	bitr4d 271	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) < ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) <-> (liminf ` F ) < ( ( F ` k ) + X ) ) )
45	20, 22, 44	syl2anc 693	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) < ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) <-> (liminf ` F ) < ( ( F ` k ) + X ) ) )
46	45	ralbidva 2985	. . 3 |- ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) < ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) <-> A. k e. ( ZZ>= ` j ) (liminf ` F ) < ( ( F ` k ) + X ) ) )
47	46	rexbidva 3049	. 2 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ( ( k e. Z |-> -u ( F ` k ) ) ` k ) - X ) < ( limsup ` ( k e. Z |-> -u ( F ` k ) ) ) <-> E. j e. Z A. k e. ( ZZ>= ` j ) (liminf ` F ) < ( ( F ` k ) + X ) ) )
48	19, 47	mpbid 222	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) (liminf ` F ) < ( ( F ` k ) + X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   + caddc 9939   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   -ecxne 11943   limsupclsp 14201  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-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-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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-ceil 12594  df-limsup 14202  df-liminf 39984

This theorem is referenced by:  liminflt  40037