Theorem limsupvaluz2 45767

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 45412	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	1, 2, 4	limsupvaluz 45737	. 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	2	uzssd3 45453	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍)
9	6, 8	feqresmpt 6948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	rneqd 5918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
11	10	supeq1d 9458	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ))
12		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
13		limsupvaluz2.r	. . . . . . . . . . 11 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14	13	renepnfd 11286	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
15	12, 2, 3, 14	limsupubuz 45742	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
17		ssralv 4027	. . . . . . . . . . 11 ⊢ ((ℤ≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
18	7, 17	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
20	19	reximdv 3155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
21	16, 20	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)
22		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
23	2	eluzelz2 45430	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
24		uzid 12867	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
25		ne0i 4316	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅)
26	23, 24, 25	3syl 18	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≠ ∅)
28	6	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
29	8	sselda 3958	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
30	28, 29	ffvelcdmd 7075	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
31	22, 27, 30	supxrre3rnmpt 45456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
32	21, 31	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈ ℝ)
33	11, 32	eqeltrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) ∈ ℝ)
34	33	fmpttd 7105	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )):𝑍⟶ℝ)
35	34	frnd 6714	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ⊆ ℝ)
36		nfv 1914	. . . 4 ⊢ Ⅎ𝑛𝜑
37		eqid 2735	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
38	1, 2	uzn0d 45452	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
39	36, 33, 37, 38	rnmptn0 6233	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ≠ ∅)
40		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐹
41	40, 1, 2, 4	limsupre3uz 45765	. . . . . . . 8 ⊢ (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)))
42	13, 41	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥))
43	42	simpld 494	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
44		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
45	44	rexrd 11285	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ*)
46	4	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝐹:𝑍⟶ℝ*)
47	2	uztrn2 12871	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍)
48	47	3adant1 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍)
49	46, 48	ffvelcdmd 7075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ*)
50	49	ad5ant134 1369	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ*)
51		rnresss 6004	. . . . . . . . . . . . . 14 ⊢ ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ran 𝐹
52	3	frnd 6714	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
53	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
54	51, 53	sstrid 3970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ)
55	54	ssrexr 45459	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
56	55	supxrcld 45131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ∈ ℝ*)
57	56	ad5ant13 756	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ∈ ℝ*)
58		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
59	55	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
60		fvres 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
61	60	eqcomd 2741	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗))
62	61	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗))
63	3	ffnd 6707	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑍)
64	2	uzssd3 45453	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑍 → (ℤ≥‘𝑖) ⊆ 𝑍)
65		fnssres 6661	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖))
66	63, 64, 65	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖))
67		fnfvelrn 7070	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ↾ (ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
68	66, 67	stoic3 1776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
69	62, 68	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
70		eqid 2735	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
71	59, 69, 70	supxrubd 45137	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
72	71	ad5ant134 1369	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
73	45, 50, 57, 58, 72	xrletrd 13178	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
74	73	rexlimdva2 3143	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
75	74	ralimdva 3152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
76	75	reximdva 3153	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
77	43, 76	mpd 15	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
78		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) = (ℤ≥‘𝑖))
79	78	reseq2d 5966	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑖)))
80	79	rneqd 5918	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑖)))
81	80	supeq1d 9458	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
82		eqcom 2742	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 ↔ 𝑖 = 𝑛)
83		eqcom 2742	. . . . . . . . 9 ⊢ (sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
84	81, 82, 83	3imtr3i 291	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
85	84	breq2d 5131	. . . . . . 7 ⊢ (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
86	85	cbvralvw 3220	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
87	86	rexbii 3083	. . . . 5 ⊢ (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
88	77, 87	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
89	36, 33	rnmptbd2 45273	. . . 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))
90	88, 89	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))𝑥 ≤ 𝑦)
91		infxrre 13353	. . 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, < ))
92	35, 39, 90, 91	syl3anc 1373	. 2 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ, < ))
93		fveq2 6876	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
94	93	reseq2d 5966	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
95	94	rneqd 5918	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
96	95	supeq1d 9458	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
97	96	cbvmptv 5225	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
98	97	rneqi 5917	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
99	98	infeq1i 9491	. . 3 ⊢ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ, < )
100	99	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, < ))
101	5, 92, 100	3eqtrd 2774	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ, < ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119   ↦ cmpt 5201  ran crn 5655   ↾ cres 5656   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  supcsup 9452  infcinf 9453  ℝcr 11128  ℝ^*cxr 11268   < clt 11269   ≤ cle 11270  ℤcz 12588  ℤ_≥cuz 12852  lim supclsp 15486

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-ico 13368  df-fz 13525  df-fl 13809  df-ceil 13810  df-limsup 15487

This theorem is referenced by:  supcnvlimsup  45769