Theorem limsupvaluz 46158

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 2740	. . 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 6848	. . . . 5 ⊢ 𝑍 ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
6	2, 5	fexd 7178	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
7		uzssre 12808	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
8	3, 7	eqsstri 3968	. . . 4 ⊢ 𝑍 ⊆ ℝ
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
10		limsupvaluz.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
11	3	uzsup 13820	. . . 4 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
13	1, 6, 9, 12	limsupval2 15440	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ))
14	9	mptimass 6032	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )))
15		oveq1 7370	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞))
16	15	imaeq2d 6019	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞)))
17	16	ineq1d 4155	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*))
18	17	supeq1d 9356	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
19	18	cbvmptv 5183	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )))
21		fimass 6682	. . . . . . . . . . . 12 ⊢ (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
22	2, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*)
23		dfss2 3908	. . . . . . . . . . . 12 ⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
24	23	biimpi 217	. . . . . . . . . . 11 ⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ* → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
25	22, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
26	25	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑛[,)+∞)))
27		df-ima 5638	. . . . . . . . . 10 ⊢ (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))
28	27	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)))
29	2	freld 6668	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
30		resindm 5989	. . . . . . . . . . . . 13 ⊢ (Rel 𝐹 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
31	29, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
32	31	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞)))
33		incom 4145	. . . . . . . . . . . . . . 15 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞))
34	3	ineq1i 4152	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∩ (𝑛[,)+∞)) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
35	33, 34	eqtri 2763	. . . . . . . . . . . . . 14 ⊢ ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))
36	35	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
37	2	fdmd 6672	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐹 = 𝑍)
38	37	ineq2d 4156	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
39	38	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍))
40	3	eleq2i 2832	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
41	40	bilani 505	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀))
42	41	uzinico2 46013	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)))
43	36, 39, 42	3eqtr4d 2785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ≥‘𝑛))
44	43	reseq2d 5938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ≥‘𝑛)))
45	32, 44	eqtr3d 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ≥‘𝑛)))
46	45	rneqd 5887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
47	26, 28, 46	3eqtrd 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) = ran (𝐹 ↾ (ℤ≥‘𝑛)))
48	47	supeq1d 9356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
49	48	mpteq2dva 5172	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
50	20, 49	eqtrd 2775	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
51	50	rneqd 5887	. . . 4 ⊢ (𝜑 → ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
52	14, 51	eqtrd 2775	. . 3 ⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
53	52	infeq1d 9388	. 2 ⊢ (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ))
54		fveq2 6834	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
55	54	reseq2d 5938	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
56	55	rneqd 5887	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
57	56	supeq1d 9356	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
58	57	cbvmptv 5183	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
59	58	rneqi 5886	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
60	59	infeq1i 9389	. . 3 ⊢ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < )
61	60	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*, < ))
62	13, 53, 61	3eqtrd 2779	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890   ↦ cmpt 5160  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Rel wrel 5630  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  supcsup 9350  infcinf 9351  ℝcr 11035  +∞cpnf 11174  ℝ^*cxr 11176   < clt 11177  ℤcz 12522  ℤ_≥cuz 12786  [,)cico 13298  lim supclsp 15430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9352  df-inf 9353  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-ico 13302  df-fl 13749  df-limsup 15431

This theorem is referenced by:  limsupvaluzmpt  46167  limsupvaluz2  46188  limsupgtlem  46227