Theorem limsupubuz 41987

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 1911	. . . . . 6 ⊢ Ⅎ𝑙𝜑
2		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑙𝐹
3		limsupubuz.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		uzssre 41662	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
5	3, 4	eqsstri 4000	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
7		limsupubuz.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
8	7	frexr 41648	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
9		limsupubuz.n	. . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
10	1, 2, 6, 8, 9	limsupub 41978	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
11	10	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
12		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑙 𝑀 ∈ ℤ
13	1, 12	nfan 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝜑 ∧ 𝑀 ∈ ℤ)
14		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑙 𝑦 ∈ ℝ
15	13, 14	nfan 1896	. . . . . . . . 9 ⊢ Ⅎ𝑙((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ)
16		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑙 𝑘 ∈ ℝ
17	15, 16	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑙(((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ)
18		nfra1 3219	. . . . . . . 8 ⊢ Ⅎ𝑙∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)
19	17, 18	nfan 1896	. . . . . . 7 ⊢ Ⅎ𝑙((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
20		nfmpt1 5156	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
21	20	nfrn 5818	. . . . . . . . . 10 ⊢ Ⅎ𝑙ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
22		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑙ℝ
23		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑙 <
24	21, 22, 23	nfsup 8909	. . . . . . . . 9 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < )
25		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑙 ≤
26		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑙𝑦
27	24, 25, 26	nfbr 5105	. . . . . . . 8 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦
28	27, 26, 24	nfif 4495	. . . . . . 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 5062	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑖))
30		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → (𝐹‘𝑙) = (𝐹‘𝑖))
31	30	breq1d 5068	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑖) ≤ 𝑦))
32	29, 31	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)))
33	32	cbvralvw 3449	. . . . . . . . . 10 ⊢ (∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦))
34	33	biimpi 218	. . . . . . . . 9 ⊢ (∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) → ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦))
35	34	adantl 484	. . . . . . . 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 782	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑀 ∈ ℤ)
37	35, 36	syldan 593	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑀 ∈ ℤ)
38	7	ad4antr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
39	35, 38	syldan 593	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
40		simpllr 774	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑦 ∈ ℝ)
41	35, 40	syldan 593	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑦 ∈ ℝ)
42		simplr 767	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖 ∈ 𝑍 (𝑘 ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑦)) → 𝑘 ∈ ℝ)
43	35, 42	syldan 593	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) → 𝑘 ∈ ℝ)
44	33	biimpri 230	. . . . . . . 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 2821	. . . . . . 7 ⊢ if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47		eqid 2821	. . . . . . 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 2821	. . . . . . 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 41986	. . . . . 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 3287	. . . . 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 3284	. . . 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 41679	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
55	53, 54	eqtrd 2856	. . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → 𝑍 = ∅)
56		0red 10638	. . . . . 6 ⊢ (𝑍 = ∅ → 0 ∈ ℝ)
57		rzal 4452	. . . . . 6 ⊢ (𝑍 = ∅ → ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 0)
58		brralrspcev 5118	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 0) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
59	56, 57, 58	syl2anc 586	. . . . 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 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
62	52, 61	pm2.61dan 811	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥)
63		limsupubuz.j	. . . . . 6 ⊢ Ⅎ𝑗𝐹
64		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑗𝑙
65	63, 64	nffv 6674	. . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑙)
66		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑗 ≤
67		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑗𝑥
68	65, 66, 67	nfbr 5105	. . . 4 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
69		nfv 1911	. . . 4 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
70		fveq2 6664	. . . . 5 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
71	70	breq1d 5068	. . . 4 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
72	68, 69, 71	cbvralw 3441	. . 3 ⊢ (∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
73	72	rexbii 3247	. 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
74	62, 73	sylib 220	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Ⅎwnfc 2961   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  ∅c0 4290  ifcif 4466   class class class wbr 5058   ↦ cmpt 5138  ran crn 5550  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  supcsup 8898  ℝcr 10530  0cc0 10531  +∞cpnf 10666   < clt 10669   ≤ cle 10670  ℤcz 11975  ℤ_≥cuz 12237  ...cfz 12886  ⌈cceil 13155  lim supclsp 14821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-ico 12738  df-fz 12887  df-fl 13156  df-ceil 13157  df-limsup 14822

This theorem is referenced by:  limsupubuzmpt  41993  limsupvaluz2  42012  supcnvlimsup  42014