Theorem limsupvaluz 46463

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 2766	. . 3 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))
2		limsupvaluz.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
3		limsupvaluz.z	. . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀)
4	3	fvexi 6902	. . . . 5 ⊢ 𝑍 ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
6	2, 5	fexd 7232	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
7	3	uzssre2 46162	. . . 4 ⊢ 𝑍 ⊆ ℝ
8	7	a1i 11	. . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
9		limsupvaluz.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
10	3	uzsup 13916	. . . 4 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
11	9, 10	syl 18	. . 3 ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
12	1, 6, 8, 11	limsupval2 15557	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
13	8	mptimass 6080	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
14		oveq1 7430	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
15	14	imaeq2d 6067	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
16	15	ineq1d 4175	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
17	16	supeq1d 9416	. . . . . . 7 ⊢ (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
18	17	cbvmptv 5220	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
19	2	fimassd 6734	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
20		dfss2 3926	. . . . . . . . . . 11 ⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
21	19, 20	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
22	21	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
23		df-ima 5679	. . . . . . . . . 10 ⊢ (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
24	23	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
25		resindm 6034	. . . . . . . . . . 11 ⊢ (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞))
26	3	ineq1i 4172	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
27	26	ineqcomi 4167	. . . . . . . . . . . . 13 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
28	2	fdmd 6723	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐹 = 𝑍)
29	28	ineq2d 4176	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
30	29	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
31	3	eleq2i 2858	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
32	31	bilani 510	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀))
33	32	uzinico2 46318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
34	27, 30, 33	3eqtr4a 2827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ≥‘𝑛))
35	34	reseq2d 5983	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ≥‘𝑛)))
36	25, 35	eqtr3id 2815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ≥‘𝑛)))
37	36	rneqd 5933	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
38	22, 24, 37	3eqtrd 2805	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
39	38	supeq1d 9416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
40	39	mpteq2dva 5209	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
41	18, 40	eqtrid 2813	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
42	41	rneqd 5933	. . . 4 ⊢ (𝜑 → ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
43	13, 42	eqtrd 2801	. . 3 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
44	43	infeq1d 9448	. 2 ⊢ (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ))
45		fveq2 6888	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
46	45	reseq2d 5983	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
47	46	rneqd 5933	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
48	47	supeq1d 9416	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
49	48	cbvmptv 5220	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
50	49	rneqi 5932	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
51	50	infeq1i 9449	. . 3 ⊢ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < )
52	51	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*, < ))
53	12, 44, 52	3eqtrd 2805	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908   ↦ cmpt 5197  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  supcsup 9410  infcinf 9411  ℝcr 11117  +∞cpnf 11258  ℝ^*cxr 11260   < clt 11261  ℤcz 12609  ℤ_≥cuz 12880  [,)cico 13392  lim supclsp 15547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-inf 9413  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-ico 13396  df-fl 13845  df-limsup 15548

This theorem is used by:  limsupvaluzmpt  46472  limsupvaluz2  46493  limsupgtlem  46532