Theorem limsupubuz 45709

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 1914	. . . . . 6 ⊢ Ⅎ𝑙𝜑
2		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑙𝐹
3		limsupubuz.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		uzssre 12879	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
5	3, 4	eqsstri 4010	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
7		limsupubuz.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
8	7	frexr 45379	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
9		limsupubuz.n	. . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
10	1, 2, 6, 8, 9	limsupub 45700	. . . . 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 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑙 𝑀 ∈ ℤ
13	1, 12	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝜑 ∧ 𝑀 ∈ ℤ)
14		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑙 𝑦 ∈ ℝ
15	13, 14	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑙((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ)
16		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑙 𝑘 ∈ ℝ
17	15, 16	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑙(((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ)
18		nfra1 3270	. . . . . . . 8 ⊢ Ⅎ𝑙∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)
19	17, 18	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑙((((𝜑 ∧ 𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙 ∈ 𝑍 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))
20		nfmpt1 5225	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
21	20	nfrn 5937	. . . . . . . . . 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 9468	. . . . . . . . 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 5171	. . . . . . . 8 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦
28	27, 26, 24	nfif 4536	. . . . . . 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 5128	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑖))
30		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → (𝐹‘𝑙) = (𝐹‘𝑖))
31	30	breq1d 5134	. . . . . . . . . . . 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 3224	. . . . . . . . . 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 2736	. . . . . . 7 ⊢ if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47		eqid 2736	. . . . . . 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 2736	. . . . . . 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 45708	. . . . . 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 3144	. . . . 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 3142	. . . 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 45406	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
55	53, 54	eqtrd 2771	. . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → 𝑍 = ∅)
56		0red 11243	. . . . . 6 ⊢ (𝑍 = ∅ → 0 ∈ ℝ)
57		rzal 4489	. . . . . 6 ⊢ (𝑍 = ∅ → ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 0)
58		brralrspcev 5184	. . . . . 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 2899	. . . . . 6 ⊢ Ⅎ𝑗𝑙
65	63, 64	nffv 6891	. . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑙)
66		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗 ≤
67		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗𝑥
68	65, 66, 67	nfbr 5171	. . . 4 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
69		nfv 1914	. . . 4 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
70		fveq2 6881	. . . . 5 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
71	70	breq1d 5134	. . . 4 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
72	68, 69, 71	cbvralw 3290	. . 3 ⊢ (∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
73	72	rexbii 3084	. 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 1540   ∈ wcel 2109  Ⅎwnfc 2884   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ⊆ wss 3931  ∅c0 4313  ifcif 4505   class class class wbr 5124   ↦ cmpt 5206  ran crn 5660  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  supcsup 9457  ℝcr 11133  0cc0 11134  +∞cpnf 11271   < clt 11274   ≤ cle 11275  ℤcz 12593  ℤ_≥cuz 12857  ...cfz 13529  ⌈cceil 13813  lim supclsp 15491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-ico 13373  df-fz 13530  df-fl 13814  df-ceil 13815  df-limsup 15492

This theorem is referenced by:  limsupubuzmpt  45715  limsupvaluz2  45734  supcnvlimsup  45736