Theorem limsupvaluz2 47021

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 46669	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	1, 2, 4	limsupvaluz 46991	. 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 46709	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
8	7	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍)
9	6, 8	feqresmpt 6985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	rneqd 5932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)))
11	10	supeq1d 9475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ))
12		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
13		limsupvaluz2.r	. . . . . . . . . . 11 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14	13	renepnfd 11399	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
15	12, 2, 3, 14	limsupubuz 46996	. . . . . . . . 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 3982	. . . . . . . . . . 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 3181	. . . . . . . 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 1949	. . . . . . . 8 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
23	2	eluzelz2 46686	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
24		uzid 13020	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
25		ne0i 4290	. . . . . . . . . 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 3913	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
30	28, 29	ffvelcdmd 7117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
31	22, 27, 30	supxrre3rnmpt 46712	. . . . . . 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 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) ∈ ℝ)
34	33	fmpttd 7147	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )):𝑍⟶ℝ)
35	34	frnd 6748	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ⊆ ℝ)
36		nfv 1949	. . . 4 ⊢ Ⅎ𝑛𝜑
37		eqid 2763	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
38	1, 2	uzn0d 46708	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
39	36, 33, 37, 38	rnmptn0 6264	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) ≠ ∅)
40		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐹
41	40, 1, 2, 4	limsupre3uz 47019	. . . . . . . 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 11398	. . . . . . . . . 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 13024	. . . . . . . . . . . . 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 7117	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ*)
50	49	ad5ant134 1393	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ*)
51		rnresss 6023	. . . . . . . . . . . . . 14 ⊢ ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ran 𝐹
52	3	frnd 6748	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
53	52	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
54	51, 53	sstrid 3924	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ)
55	54	ssrexr 46715	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
56	55	supxrcld 46395	. . . . . . . . . . 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 6935	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ≥‘𝑖) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
61	60	eqcomd 2769	. . . . . . . . . . . . . 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 6740	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑍)
64	2	uzssd3 46709	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑍 → (ℤ≥‘𝑖) ⊆ 𝑍)
65		fnssres 6692	. . . . . . . . . . . . . . 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 7112	. . . . . . . . . . . . . 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 2864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖)))
70		eqid 2763	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
71	59, 69, 70	supxrubd 46401	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
72	71	ad5ant134 1393	. . . . . . . . . 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 13331	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
74	73	rexlimdva2 3169	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
75	74	ralimdva 3178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )))
76	75	reximdva 3179	. . . . . 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 6916	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) = (ℤ≥‘𝑖))
79	78	reseq2d 5986	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑖)))
80	79	rneqd 5932	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑖)))
81	80	supeq1d 9475	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ))
82		eqcom 2770	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 ↔ 𝑖 = 𝑛)
83		eqcom 2770	. . . . . . . . 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 5120	. . . . . . 7 ⊢ (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )))
86	85	cbvralvw 3244	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < ) ↔ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
87	86	rexbii 3113	. . . . 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 46534	. . . 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 13507	. . 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 1399	. 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 6916	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
94	93	reseq2d 5986	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘)))
95	94	rneqd 5932	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘)))
96	95	supeq1d 9475	. . . . . 6 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
97	96	cbvmptv 5215	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
98	97	rneqi 5931	. . . 4 ⊢ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )) = ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
99	98	infeq1i 9508	. . 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 2802	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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281   class class class wbr 5108   ↦ cmpt 5192  ran crn 5659   ↾ cres 5660   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  supcsup 9469  infcinf 9470  ℝcr 11238  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383  ℤcz 12733  ℤ_≥cuz 13005  lim supclsp 15681

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-ico 13522  df-fz 13680  df-fl 13972  df-ceil 13973  df-limsup 15682

This theorem is used by:  supcnvlimsup  47023