Theorem limsupvaluz2 45753

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 45396	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	1, 2, 4	limsupvaluz 45723	. 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 45437	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍)
9	6, 8	feqresmpt 6978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	rneqd 5949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
11	10	supeq1d 9486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ))
12		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
13		limsupvaluz2.r	. . . . . . . . . . 11 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14	13	renepnfd 11312	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
15	12, 2, 3, 14	limsupubuz 45728	. . . . . . . . 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 4052	. . . . . . . . . . 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 3170	. . . . . . . 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 45414	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
24		uzid 12893	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
25		ne0i 4341	. . . . . . . . . 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 3983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
30	28, 29	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
31	22, 27, 30	supxrre3rnmpt 45440	. . . . . . 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 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) ∈ ℝ)
34	33	fmpttd 7135	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )):𝑍⟶ℝ)
35	34	frnd 6744	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ⊆ ℝ)
36		nfv 1914	. . . 4 ⊢ Ⅎ𝑛𝜑
37		eqid 2737	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
38	1, 2	uzn0d 45436	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
39	36, 33, 37, 38	rnmptn0 6264	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ≠ ∅)
40		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐹
41	40, 1, 2, 4	limsupre3uz 45751	. . . . . . . 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 784	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
45	44	rexrd 11311	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ*)
46	4	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝐹:𝑍⟶ℝ*)
47	2	uztrn2 12897	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍)
48	47	3adant1 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍)
49	46, 48	ffvelcdmd 7105	. . . . . . . . . . 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 6035	. . . . . . . . . . . . . 14 ⊢ ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ran 𝐹
52	3	frnd 6744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
53	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
54	51, 53	sstrid 3995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ)
55	54	ssrexr 45443	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
56	55	supxrcld 45112	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ∈ ℝ*)
57	56	ad5ant13 757	. . . . . . . . . 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 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
60		fvres 6925	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
61	60	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗))
62	61	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗))
63	3	ffnd 6737	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑍)
64	2	uzssd3 45437	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑍 → (ℤ≥‘𝑖) ⊆ 𝑍)
65		fnssres 6691	. . . . . . . . . . . . . . 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 7100	. . . . . . . . . . . . . 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 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
70		eqid 2737	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
71	59, 69, 70	supxrubd 45118	. . . . . . . . . . 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 13204	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
74	73	rexlimdva2 3157	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
75	74	ralimdva 3167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
76	75	reximdva 3168	. . . . . 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 6906	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) = (ℤ≥‘𝑖))
79	78	reseq2d 5997	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑖)))
80	79	rneqd 5949	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑖)))
81	80	supeq1d 9486	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
82		eqcom 2744	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 ↔ 𝑖 = 𝑛)
83		eqcom 2744	. . . . . . . . 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 5155	. . . . . . 7 ⊢ (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
86	85	cbvralvw 3237	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
87	86	rexbii 3094	. . . . 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 45256	. . . 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 13378	. . 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 6906	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
94	93	reseq2d 5997	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
95	94	rneqd 5949	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
96	95	supeq1d 9486	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
97	96	cbvmptv 5255	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
98	97	rneqi 5948	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
99	98	infeq1i 9518	. . 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 2781	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ, < ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686   ↾ cres 5687   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  supcsup 9480  infcinf 9481  ℝcr 11154  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  ℤcz 12613  ℤ_≥cuz 12878  lim supclsp 15506

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-ico 13393  df-fz 13548  df-fl 13832  df-ceil 13833  df-limsup 15507

This theorem is referenced by:  supcnvlimsup  45755