Theorem limsupvaluz2 46567

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 46215	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	1, 2, 4	limsupvaluz 46537	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )), ℝ*, < ))
6	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
7	2	uzssd3 46255	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
8	7	adantl 487	. . . . . . . . 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 5922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
11	10	supeq1d 9417	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ))
12		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
13		limsupvaluz2.r	. . . . . . . . . . 11 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14	13	renepnfd 11285	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
15	12, 2, 3, 14	limsupubuz 46542	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
16	15	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
17		ssralv 4000	. . . . . . . . . . 11 ⊢ ((ℤ≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
18	7, 17	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	18	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
20	19	reximdv 3177	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
21	16, 20	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)
22		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
23	2	eluzelz2 46232	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
24		uzid 12903	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
25		ne0i 4287	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅)
26	23, 24, 25	3syl 19	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
27	26	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≠ ∅)
28	6	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
29	8	sselda 3931	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
30	28, 29	ffvelcdmd 7079	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
31	22, 27, 30	supxrre3rnmpt 46258	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈ ℝ)
33	11, 32	eqeltrd 2860	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) ∈ ℝ)
34	33	fmpttd 7109	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )):𝑍⟶ℝ)
35	34	frnd 6712	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ⊆ ℝ)
36		nfv 1947	. . . 4 ⊢ Ⅎ𝑛𝜑
37		eqid 2760	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
38	1, 2	uzn0d 46254	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
39	36, 33, 37, 38	rnmptn0 6240	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ≠ ∅)
40		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐹
41	40, 1, 2, 4	limsupre3uz 46565	. . . . . . . 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 235	. . . . . . 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 500	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
44		simp-4r 796	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
45	44	rexrd 11284	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ*)
46	4	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝐹:𝑍⟶ℝ*)
47	2	uztrn2 12907	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍)
48	47	3adant1 1148	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍)
49	46, 48	ffvelcdmd 7079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ*)
50	49	ad5ant134 1392	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ*)
51		rnresss 6010	. . . . . . . . . . . . . 14 ⊢ ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ran 𝐹
52	3	frnd 6712	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
53	52	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
54	51, 53	sstrid 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ)
55	54	ssrexr 46261	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
56	55	supxrcld 45940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ∈ ℝ*)
57	56	ad5ant13 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ∈ ℝ*)
58		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
59	55	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
60		fvres 6898	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
61	60	eqcomd 2766	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗))
62	61	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗))
63	3	ffnd 6704	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑍)
64	2	uzssd3 46255	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑍 → (ℤ≥‘𝑖) ⊆ 𝑍)
65		fnssres 6656	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖))
66	63, 64, 65	syl2an 608	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖))
67		fnfvelrn 7074	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ↾ (ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
68	66, 67	stoic3 1809	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
69	62, 68	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
70		eqid 2760	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
71	59, 69, 70	supxrubd 45946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
72	71	ad5ant134 1392	. . . . . . . . . 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 13214	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
74	73	rexlimdva2 3165	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
75	74	ralimdva 3174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
76	75	reximdva 3175	. . . . . 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 16	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
78		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) = (ℤ≥‘𝑖))
79	78	reseq2d 5972	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑖)))
80	79	rneqd 5922	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑖)))
81	80	supeq1d 9417	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
82		eqcom 2767	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 ↔ 𝑖 = 𝑛)
83		eqcom 2767	. . . . . . . . 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 294	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
85	84	breq2d 5115	. . . . . . 7 ⊢ (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
86	85	cbvralvw 3240	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
87	86	rexbii 3109	. . . . 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 221	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
89	36, 33	rnmptbd2 46079	. . . 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 235	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))𝑥 ≤ 𝑦)
91		infxrre 13390	. . 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 1398	. 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 6879	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
94	93	reseq2d 5972	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
95	94	rneqd 5922	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
96	95	supeq1d 9417	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
97	96	cbvmptv 5209	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
98	97	rneqi 5921	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
99	98	infeq1i 9450	. . 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 2799	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   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  supcsup 9411  infcinf 9412  ℝcr 11124  ℝ^*cxr 11267   < clt 11268   ≤ cle 11269  ℤcz 12616  ℤ_≥cuz 12888  lim supclsp 15558

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202  ax-pre-sup 11203

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9413  df-inf 9414  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-n0 12530  df-z 12617  df-uz 12889  df-ico 13405  df-fz 13563  df-fl 13854  df-ceil 13855  df-limsup 15559

This theorem is used by:  supcnvlimsup  46569