Theorem limsupubuz 45751

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 1915	. . . . . 6 ⊢ Ⅎ𝑙𝜑
2		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑙𝐹
3		limsupubuz.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		uzssre 12749	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
5	3, 4	eqsstri 3976	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
7		limsupubuz.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
8	7	frexr 45423	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
9		limsupubuz.n	. . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
10	1, 2, 6, 8, 9	limsupub 45742	. . . . 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 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑙 𝑀 ∈ ℤ
13	1, 12	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝜑 ∧ 𝑀 ∈ ℤ)
14		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑙 𝑦 ∈ ℝ
15	13, 14	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑙((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ)
16		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑙 𝑘 ∈ ℝ
17	15, 16	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑙(((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ)
18		nfra1 3256	. . . . . . . 8 ⊢ Ⅎ𝑙∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)
19	17, 18	nfan 1900	. . . . . . 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 5887	. . . . . . . . . 10 ⊢ Ⅎ𝑙ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
22		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑙ℝ
23		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑙 <
24	21, 22, 23	nfsup 9330	. . . . . . . . 9 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < )
25		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑙 ≤
26		nfcv 2894	. . . . . . . . 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 4501	. . . . . . 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 6817	. . . . . . . . . . . . 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 3210	. . . . . . . . . 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 783	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑀 ∈ ℤ)
37	35, 36	syldan 591	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑀 ∈ ℤ)
38	7	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
39	35, 38	syldan 591	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
40		simpllr 775	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑦 ∈ ℝ)
41	35, 40	syldan 591	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑦 ∈ ℝ)
42		simplr 768	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑘 ∈ ℝ)
43	35, 42	syldan 591	. . . . . . 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 2731	. . . . . . 7 ⊢ if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47		eqid 2731	. . . . . . 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 2731	. . . . . . 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 45750	. . . . . 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 3135	. . . . 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 3133	. . . 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 45450	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
55	53, 54	eqtrd 2766	. . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → 𝑍 = ∅)
56		0red 11110	. . . . . 6 ⊢ (𝑍 = ∅ → 0 ∈ ℝ)
57		rzal 4454	. . . . . 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 584	. . . . 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 812	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
63		limsupubuz.j	. . . . . 6 ⊢ Ⅎ𝑗𝐹
64		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑗𝑙
65	63, 64	nffv 6827	. . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑙)
66		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑗 ≤
67		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑗𝑥
68	65, 66, 67	nfbr 5133	. . . 4 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
69		nfv 1915	. . . 4 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
70		fveq2 6817	. . . . 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 3274	. . 3 ⊢ (∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
73	72	rexbii 3079	. 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 1541   ∈ wcel 2111  Ⅎwnfc 2879   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ∅c0 4278  ifcif 4470   class class class wbr 5086   ↦ cmpt 5167  ran crn 5612  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  supcsup 9319  ℝcr 11000  0cc0 11001  +∞cpnf 11138   < clt 11141   ≤ cle 11142  ℤcz 12463  ℤ_≥cuz 12727  ...cfz 13402  ⌈cceil 13690  lim supclsp 15372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-n0 12377  df-z 12464  df-uz 12728  df-ico 13246  df-fz 13403  df-fl 13691  df-ceil 13692  df-limsup 15373

This theorem is referenced by:  limsupubuzmpt  45757  limsupvaluz2  45776  supcnvlimsup  45778