Theorem limsupubuz 45873

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 2895	. . . . . 6 ⊢ Ⅎ𝑙𝐹
3		limsupubuz.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		uzssre 12764	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
5	3, 4	eqsstri 3977	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
7		limsupubuz.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
8	7	frexr 45545	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
9		limsupubuz.n	. . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
10	1, 2, 6, 8, 9	limsupub 45864	. . . . 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 3257	. . . . . . . 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 5194	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
21	20	nfrn 5898	. . . . . . . . . 10 ⊢ Ⅎ𝑙ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙))
22		nfcv 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑙ℝ
23		nfcv 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑙 <
24	21, 22, 23	nfsup 9346	. . . . . . . . 9 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < )
25		nfcv 2895	. . . . . . . . 9 ⊢ Ⅎ𝑙 ≤
26		nfcv 2895	. . . . . . . . 9 ⊢ Ⅎ𝑙𝑦
27	24, 25, 26	nfbr 5142	. . . . . . . 8 ⊢ Ⅎ𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹‘𝑙)), ℝ, < ) ≤ 𝑦
28	27, 26, 24	nfif 4507	. . . . . . 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 5099	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑖))
30		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → (𝐹‘𝑙) = (𝐹‘𝑖))
31	30	breq1d 5105	. . . . . . . . . . . 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 3211	. . . . . . . . . 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 2733	. . . . . . 7 ⊢ if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47		eqid 2733	. . . . . . 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 2733	. . . . . . 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 45872	. . . . . 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 3136	. . . . 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 3134	. . . 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 45572	. . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅)
55	53, 54	eqtrd 2768	. . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → 𝑍 = ∅)
56		0red 11126	. . . . . 6 ⊢ (𝑍 = ∅ → 0 ∈ ℝ)
57		rzal 4444	. . . . . 6 ⊢ (𝑍 = ∅ → ∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 0)
58		brralrspcev 5155	. . . . . 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 2895	. . . . . 6 ⊢ Ⅎ𝑗𝑙
65	63, 64	nffv 6841	. . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑙)
66		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑗 ≤
67		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑗𝑥
68	65, 66, 67	nfbr 5142	. . . 4 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
69		nfv 1915	. . . 4 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
70		fveq2 6831	. . . . 5 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
71	70	breq1d 5105	. . . 4 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
72	68, 69, 71	cbvralw 3275	. . 3 ⊢ (∀𝑙 ∈ 𝑍 (𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
73	72	rexbii 3080	. 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 2113  Ⅎwnfc 2880   ≠ wne 2929  ∀wral 3048  ∃wrex 3057   ⊆ wss 3898  ∅c0 4282  ifcif 4476   class class class wbr 5095   ↦ cmpt 5176  ran crn 5622  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  supcsup 9335  ℝcr 11016  0cc0 11017  +∞cpnf 11154   < clt 11157   ≤ cle 11158  ℤcz 12479  ℤ_≥cuz 12742  ...cfz 13414  ⌈cceil 13702  lim supclsp 15384

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094  ax-pre-sup 11095

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9337  df-inf 9338  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-n0 12393  df-z 12480  df-uz 12743  df-ico 13258  df-fz 13415  df-fl 13703  df-ceil 13704  df-limsup 15385

This theorem is referenced by:  limsupubuzmpt  45879  limsupvaluz2  45898  supcnvlimsup  45900