Theorem limsupubuz 46159

Description: For a real-valued function on a set of upper integers, if the superior limit is not +∞, then the function is bounded above. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupubuz.j	⊢ Ⅎ𝑗𝐹
limsupubuz.z	⊢ 𝑍 = (ℤ≥‘𝑀)
limsupubuz.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
limsupubuz.n	⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)

Assertion

Ref	Expression
limsupubuz	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Distinct variable groups:   𝑥,𝐹   𝑥,𝑀   𝑗,𝑍,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑗)   𝐹(𝑗)   𝑀(𝑗)

Proof of Theorem limsupubuz

Dummy variables 𝑖 𝑘 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑙𝜑
2		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑙𝐹
3		limsupubuz.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		uzssre 12801	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
5	3, 4	eqsstri 3969	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
7		limsupubuz.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
8	7	frexr 45832	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
9		limsupubuz.n	. . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
10	1, 2, 6, 8, 9	limsupub 46150	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
12		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑙 𝑀 ∈ ℤ
13	1, 12	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝜑 ∧ 𝑀 ∈ ℤ)
14		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑙 𝑦 ∈ ℝ
15	13, 14	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑙((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ)
16		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑙 𝑘 ∈ ℝ
17	15, 16	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑙(((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ)
18		nfra1 3262	. . . . . . . 8 ⊢ Ⅎ𝑙∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)
19	17, 18	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑙((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
20		nfmpt1 5185	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
21	20	nfrn 5901	. . . . . . . . . 10 ⊢ Ⅎ𝑙ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
22		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑙ℝ
23		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑙 <
24	21, 22, 23	nfsup 9357	. . . . . . . . 9 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < )
25		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑙 ≤
26		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑙𝑦
27	24, 25, 26	nfbr 5133	. . . . . . . 8 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦
28	27, 26, 24	nfif 4498	. . . . . . 7 ⊢ Ⅎ𝑙if(sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ))
29		breq2 5090	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑖))
30		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → (𝐹‘𝑙) = (𝐹‘𝑖))
31	30	breq1d 5096	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑖) ≤ 𝑦))
32	29, 31	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)))
33	32	cbvralvw 3216	. . . . . . . . . 10 ⊢ (∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦))
34	33	biimpi 216	. . . . . . . . 9 ⊢ (∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) → ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦))
35	34	adantl 481	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦))
36		simp-4r 784	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑀 ∈ ℤ)
37	35, 36	syldan 592	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑀 ∈ ℤ)
38	7	ad4antr 733	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
39	35, 38	syldan 592	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
40		simpllr 776	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑦 ∈ ℝ)
41	35, 40	syldan 592	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑦 ∈ ℝ)
42		simplr 769	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑘 ∈ ℝ)
43	35, 42	syldan 592	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑘 ∈ ℝ)
44	33	biimpri 228	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦) → ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
45	35, 44	syl 17	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
46		eqid 2737	. . . . . . 7 ⊢ if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47		eqid 2737	. . . . . . 7 ⊢ sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) = sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < )
48		eqid 2737	. . . . . . 7 ⊢ if(sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < )) = if(sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ))
49	19, 28, 37, 3, 39, 41, 43, 45, 46, 47, 48	limsupubuzlem 46158	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
50	49	rexlimdva2 3141	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥))
51	50	rexlimdva 3139	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥))
52	11, 51	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
53	3	a1i 11	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → 𝑍 = (ℤ≥‘𝑀))
54		uz0 45858	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
55	53, 54	eqtrd 2772	. . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → 𝑍 = ∅)
56		0red 11138	. . . . . 6 ⊢ (𝑍 = ∅ → 0 ∈ ℝ)
57		rzal 4435	. . . . . 6 ⊢ (𝑍 = ∅ → ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 0)
58		brralrspcev 5146	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 0) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
59	56, 57, 58	syl2anc 585	. . . . 5 ⊢ (𝑍 = ∅ → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
60	55, 59	syl 17	. . . 4 ⊢ (¬ 𝑀 ∈ ℤ → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
61	60	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
62	52, 61	pm2.61dan 813	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
63		limsupubuz.j	. . . . . 6 ⊢ Ⅎ𝑗𝐹
64		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑗𝑙
65	63, 64	nffv 6844	. . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑙)
66		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗 ≤
67		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗𝑥
68	65, 66, 67	nfbr 5133	. . . 4 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
69		nfv 1916	. . . 4 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
70		fveq2 6834	. . . . 5 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
71	70	breq1d 5096	. . . 4 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
72	68, 69, 71	cbvralw 3280	. . 3 ⊢ (∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
73	72	rexbii 3085	. 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
74	62, 73	sylib 218	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274  ifcif 4467   class class class wbr 5086   ↦ cmpt 5167  ran crn 5625  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  supcsup 9346  ℝcr 11028  0cc0 11029  +∞cpnf 11167   < clt 11170   ≤ cle 11171  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  ⌈cceil 13741  lim supclsp 15423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-ico 13295  df-fz 13453  df-fl 13742  df-ceil 13743  df-limsup 15424

This theorem is referenced by:  limsupubuzmpt  46165  limsupvaluz2  46184  supcnvlimsup  46186