Theorem limsupvaluz 41995

Description: The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -∞ and the r.h.s. would be +∞). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupvaluz.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupvaluz.z	⊢ 𝑍 = (ℤ≥‘𝑀)
limsupvaluz.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)

Assertion

Ref	Expression
limsupvaluz	⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))

Distinct variable groups:   𝑘,𝐹   𝑘,𝑍

Allowed substitution hints:   𝜑(𝑘)   𝑀(𝑘)

Proof of Theorem limsupvaluz

Dummy variables 𝑖 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2824	. . 3 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))
2		limsupvaluz.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
3		limsupvaluz.z	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
4	3	fvexi 6687	. . . . . 6 ⊢ 𝑍 ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ V)
6	2, 5	fexd 41385	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
7	6	elexd 3517	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
8		uzssre 41675	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
9	3, 8	eqsstri 4004	. . . 4 ⊢ 𝑍 ⊆ ℝ
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
11		limsupvaluz.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12	3	uzsup 13234	. . . 4 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
14	1, 7, 10, 13	limsupval2 14840	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
15	10	mptima2 41523	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
16		oveq1 7166	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
17	16	imaeq2d 5932	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
18	17	ineq1d 4191	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
19	18	supeq1d 8913	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
20	19	cbvmptv 5172	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
21	20	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )))
22		fimass 6558	. . . . . . . . . . . 12 ⊢ (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
23	2, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
24		df-ss 3955	. . . . . . . . . . . 12 ⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
25	24	biimpi 218	. . . . . . . . . . 11 ⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
26	23, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
27	26	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
28		df-ima 5571	. . . . . . . . . 10 ⊢ (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
30	2	freld 41499	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
31		resindm 5903	. . . . . . . . . . . . 13 ⊢ (Rel 𝐹 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
33	32	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
34		incom 4181	. . . . . . . . . . . . . . 15 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞))
35	3	ineq1i 4188	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
36	34, 35	eqtri 2847	. . . . . . . . . . . . . 14 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
38	2	fdmd 6526	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐹 = 𝑍)
39	38	ineq2d 4192	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
40	39	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
41	3	eleq2i 2907	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
42	41	biimpi 218	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
43	42	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀))
44	43	uzinico2 41844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
45	37, 40, 44	3eqtr4d 2869	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ≥‘𝑛))
46	45	reseq2d 5856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ≥‘𝑛)))
47	33, 46	eqtr3d 2861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ≥‘𝑛)))
48	47	rneqd 5811	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
49	27, 29, 48	3eqtrd 2863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
50	49	supeq1d 8913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
51	50	mpteq2dva 5164	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
52	21, 51	eqtrd 2859	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
53	52	rneqd 5811	. . . 4 ⊢ (𝜑 → ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
54	15, 53	eqtrd 2859	. . 3 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
55	54	infeq1d 8944	. 2 ⊢ (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ))
56		fveq2 6673	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
57	56	reseq2d 5856	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
58	57	rneqd 5811	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
59	58	supeq1d 8913	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
60	59	cbvmptv 5172	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
61	60	rneqi 5810	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
62	61	infeq1i 8945	. . 3 ⊢ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < )
63	62	a1i 11	. 2 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))
64	14, 55, 63	3eqtrd 2863	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939   ↦ cmpt 5149  dom cdm 5558  ran crn 5559   ↾ cres 5560   “ cima 5561  Rel wrel 5563  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  supcsup 8907  infcinf 8908  ℝcr 10539  +∞cpnf 10675  ℝ^*cxr 10677   < clt 10678  ℤcz 11984  ℤ_≥cuz 12246  [,)cico 12743  lim supclsp 14830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-sup 8909  df-inf 8910  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-n0 11901  df-z 11985  df-uz 12247  df-ico 12747  df-fl 13165  df-limsup 14831

This theorem is referenced by:  limsupvaluzmpt  42004  limsupvaluz2  42025  limsupgtlem  42064