Theorem limsupvaluz 42038

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 2821	. . 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 6684	. . . . . 6 ⊢ 𝑍 ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ V)
6	2, 5	fexd 41428	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
7	6	elexd 3514	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
8		uzssre 41718	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
9	3, 8	eqsstri 4001	. . . 4 ⊢ 𝑍 ⊆ ℝ
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
11		limsupvaluz.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12	3	uzsup 13232	. . . 4 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
14	1, 7, 10, 13	limsupval2 14837	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
15	10	mptima2 41566	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
16		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
17	16	imaeq2d 5929	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
18	17	ineq1d 4188	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
19	18	supeq1d 8910	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
20	19	cbvmptv 5169	. . . . . . 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 6555	. . . . . . . . . . . 12 ⊢ (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
23	2, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
24		df-ss 3952	. . . . . . . . . . . 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 5568	. . . . . . . . . 10 ⊢ (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
30	2	freld 41542	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
31		resindm 5900	. . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . 15 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞))
35	3	ineq1i 4185	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
36	34, 35	eqtri 2844	. . . . . . . . . . . . . 14 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
38	2	fdmd 6523	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐹 = 𝑍)
39	38	ineq2d 4189	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
40	39	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
41	3	eleq2i 2904	. . . . . . . . . . . . . . . 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 41887	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
45	37, 40, 44	3eqtr4d 2866	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ≥‘𝑛))
46	45	reseq2d 5853	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ≥‘𝑛)))
47	33, 46	eqtr3d 2858	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ≥‘𝑛)))
48	47	rneqd 5808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
49	27, 29, 48	3eqtrd 2860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
50	49	supeq1d 8910	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
51	50	mpteq2dva 5161	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
52	21, 51	eqtrd 2856	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
53	52	rneqd 5808	. . . 4 ⊢ (𝜑 → ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
54	15, 53	eqtrd 2856	. . 3 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
55	54	infeq1d 8941	. 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 6670	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
57	56	reseq2d 5853	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
58	57	rneqd 5808	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
59	58	supeq1d 8910	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
60	59	cbvmptv 5169	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
61	60	rneqi 5807	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
62	61	infeq1i 8942	. . 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 2860	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936   ↦ cmpt 5146  dom cdm 5555  ran crn 5556   ↾ cres 5557   “ cima 5558  Rel wrel 5560  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  supcsup 8904  infcinf 8905  ℝcr 10536  +∞cpnf 10672  ℝ^*cxr 10674   < clt 10675  ℤcz 11982  ℤ_≥cuz 12244  [,)cico 12741  lim supclsp 14827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-ico 12745  df-fl 13163  df-limsup 14828

This theorem is referenced by:  limsupvaluzmpt  42047  limsupvaluz2  42068  limsupgtlem  42107