limsupvaluz2 - Mathbox for Glauco Siliprandi

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Theorem limsupvaluz2 44754

Description: The superior limit, when the domain of a real-valued function is a set of upper integers, and the superior limit is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
limsupvaluz2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupvaluz2.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
limsupvaluz2.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
limsupvaluz2.r	⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

**Assertion**
Ref	Expression
limsupvaluz2	⊢ (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 limsupvaluz2 Dummy variables 𝑖 𝑗 𝑥 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupvaluz2.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		limsupvaluz2.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3		limsupvaluz2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
4	3	frexr 44395	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
5	1, 2, 4	limsupvaluz 44724	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ^, < ))
6	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
7		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
8	2, 7	uzssd2 44427	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
9	8	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
10	6, 9	feqresmpt 6962	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
11	10	rneqd 5938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
12	11	supeq1d 9444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^, < ))
13		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
14		limsupvaluz2.r	. . . . . . . . . . 11 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
15	14	renepnfd 11270	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
16	13, 2, 3, 15	limsupubuz 44729	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
18		ssralv 4051	. . . . . . . . . . 11 ⊢ ((ℤ_≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	8, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
20	19	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
21	20	reximdv 3169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
22	17, 21	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)
23		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
24	2	eluzelz2 44413	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
25		uzid 12842	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
26		ne0i 4335	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑛) ≠ ∅)
27	24, 25, 26	3syl 18	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
28	27	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ≠ ∅)
29	6	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
30	9	sselda 3983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
31	29, 30	ffvelcdmd 7088	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
32	23, 28, 31	supxrre3rnmpt 44439	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
33	22, 32	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^*, < ) ∈ ℝ)
34	12, 33	eqeltrd 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ) ∈ ℝ)
35	34	fmpttd 7117	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )):𝑍⟶ℝ)
36	35	frnd 6726	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )) ⊆ ℝ)
37		nfv 1916	. . . 4 ⊢ Ⅎ𝑛𝜑
38		eqid 2731	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))
39	1, 2	uzn0d 44435	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
40	37, 34, 38, 39	rnmptn0 6244	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )) ≠ ∅)
41		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹
42	41, 1, 2, 4	limsupre3uz 44752	. . . . . . . . 9 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥)))
43	14, 42	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥))
44	43	simpld 494	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
45		simp-4r 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
46	45	rexrd 11269	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
47	4	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝐹:𝑍⟶ℝ^*)
48	2	uztrn2 12846	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
49	48	3adant1 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
50	47, 49	ffvelcdmd 7088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ^*)
51	50	ad5ant134 1366	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
52		rnresss 6018	. . . . . . . . . . . . . . . 16 ⊢ ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹
53	52	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹)
54	3	frnd 6726	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
55	54	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
56	53, 55	sstrd 3993	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ)
57		ressxr 11263	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℝ^*
58	57	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ℝ ⊆ ℝ^*)
59	56, 58	sstrd 3993	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
60	59	supxrcld 44099	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
61	60	ad5ant13 754	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
62		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
63	59	3adant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
64		fvres 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
65	64	eqcomd 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
66	65	3ad2ant3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
67	3	ffnd 6719	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝑍)
68	67	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹 Fn 𝑍)
69		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ 𝑍)
70	2, 69	uzssd2 44427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
71	70	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ 𝑍)
72		fnssres 6674	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
73	68, 71, 72	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
74	73	3adant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
75		simp3 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ (ℤ_≥‘𝑖))
76		fnfvelrn 7083	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
77	74, 75, 76	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
78	66, 77	eqeltrd 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
79		eqid 2731	. . . . . . . . . . . . 13 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )
80	63, 78, 79	supxrubd 44105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
81	80	ad5ant134 1366	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
82	46, 51, 61, 62, 81	xrletrd 13146	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
83	82	rexlimdva2 3156	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
84	83	ralimdva 3166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
85	84	reximdva 3167	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
86	44, 85	mpd 15	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
87	86	idi 1	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
88		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑖))
89	88	reseq2d 5982	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑖)))
90	89	rneqd 5938	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑖)))
91	90	supeq1d 9444	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
92		eqcom 2738	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 ↔ 𝑖 = 𝑛)
93	92	imbi1i 348	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )) ↔ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )))
94		eqcom 2738	. . . . . . . . . . 11 ⊢ (sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ↔ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))
95	94	imbi2i 335	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )) ↔ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )))
96	93, 95	bitri 274	. . . . . . . . 9 ⊢ ((𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )) ↔ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )))
97	91, 96	mpbi 229	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))
98	97	breq2d 5161	. . . . . . 7 ⊢ (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )))
99	98	cbvralvw 3233	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ↔ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))
100	99	rexbii 3093	. . . . 5 ⊢ (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))
101	87, 100	sylib 217	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ))
102	34	elexd 3494	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ) ∈ V)
103	37, 102	rnmptbd2 44253	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))𝑥 ≤ 𝑦))
104	101, 103	mpbid 231	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ))𝑥 ≤ 𝑦)
105		infxrre 13320	. . 3 ⊢ ((ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))𝑥 ≤ 𝑦) → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ^, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ, < ))
106	36, 40, 104, 105	syl3anc 1370	. 2 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ^, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ))
107		fveq2 6892	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
108	107	reseq2d 5982	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑘)))
109	108	rneqd 5938	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑘)))
110	109	supeq1d 9444	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
111	110	cbvmptv 5262	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
112	111	rneqi 5937	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
113	112	infeq1i 9476	. . 3 ⊢ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < )), ℝ, < )
114	113	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, < ))
115	5, 106, 114	3eqtrd 2775	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^*, < )), ℝ, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3473 ⊆ wss 3949 ∅c0 4323 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 ↾ cres 5679 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 supcsup 9438 infcinf 9439 ℝcr 11112 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 ℤcz 12563 ℤ_≥cuz 12827 lim supclsp 15419

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-ico 13335 df-fz 13490 df-fl 13762 df-ceil 13763 df-limsup 15420

This theorem is referenced by: supcnvlimsup 44756

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