limsupre3uzlem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupre3uzlem		Structured version Visualization version GIF version

Theorem limsupre3uzlem 44437

Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
limsupre3uzlem.1	⊢ Ⅎ𝑗𝐹
limsupre3uzlem.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupre3uzlem.3	⊢ 𝑍 = (ℤ_≥‘𝑀)
limsupre3uzlem.4	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)

**Assertion**
Ref	Expression
limsupre3uzlem	⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))

Distinct variable groups: 𝑘,𝐹,𝑥 𝑗,𝑀,𝑘 𝑗,𝑍,𝑘,𝑥 𝜑,𝑗,𝑘,𝑥 Allowed substitution hints: 𝐹(𝑗) 𝑀(𝑥)

Proof of Theorem limsupre3uzlem Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupre3uzlem.1	. . 3 ⊢ Ⅎ𝑗𝐹
2		limsupre3uzlem.3	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3		uzssre 12840	. . . . 5 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ
4	2, 3	eqsstri 4015	. . . 4 ⊢ 𝑍 ⊆ ℝ
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
6		limsupre3uzlem.4	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7	1, 5, 6	limsupre3 44435	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))
8		breq1 5150	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑘 → (𝑦 ≤ 𝑗 ↔ 𝑘 ≤ 𝑗))
9	8	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑦 = 𝑘 → ((𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
10	9	rexbidv 3178	. . . . . . . . 9 ⊢ (𝑦 = 𝑘 → (∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
11	10	cbvralvw 3234	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
12	11	biimpi 215	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
13		nfra1 3281	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))
14		simpr 485	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
15	4, 14	sselid 3979	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℝ)
16		rspa 3245	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
17	15, 16	syldan 591	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
18		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ∈ 𝑍
19		nfre1 3282	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)
20		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑘) = (ℤ_≥‘𝑘)
21	2	eluzelz2 44099	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
22	21	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℤ)
23	2	eluzelz2 44099	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
24	23	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ ℤ)
25		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
26	20, 22, 24, 25	eluzd 44105	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (ℤ_≥‘𝑘))
27	26	3adant3r 1181	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑗 ∈ (ℤ_≥‘𝑘))
28		simp3r 1202	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ≤ (𝐹‘𝑗))
29		rspe 3246	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑘) ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
30	27, 28, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
31	30	3exp 1119	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝑗 ∈ 𝑍 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))))
32	18, 19, 31	rexlimd 3263	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → (∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
33	32	imp 407	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑍 ∧ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
34	14, 17, 33	syl2anc 584	. . . . . . . 8 ⊢ ((∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ∈ 𝑍) → ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
35	13, 34	ralrimia 3255	. . . . . . 7 ⊢ (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
36	12, 35	syl 17	. . . . . 6 ⊢ (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
38		iftrue 4533	. . . . . . . . . . . . 13 ⊢ (𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦))
39	38	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦))
40		limsupre3uzlem.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
41	40	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ ℤ)
42		ceilcl 13803	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℤ)
43	42	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ ℤ)
44		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ≤ (⌈‘𝑦))
45	2, 41, 43, 44	eluzd 44105	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ 𝑍)
46	39, 45	eqeltrd 2833	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
47		iffalse 4536	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
48	47	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
49	40, 2	uzidd2 44112	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
50	49	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ 𝑍)
51	48, 50	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
52	51	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
53	46, 52	pm2.61dan 811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
54	53	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
55		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
56		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (ℤ_≥‘𝑘) = (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)))
57	56	rexeqdv 3326	. . . . . . . . . 10 ⊢ (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗)))
58	57	rspcva 3610	. . . . . . . . 9 ⊢ ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗))
59	54, 55, 58	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗))
60		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝜑
61	18	nfci 2886	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑍
62	61, 19	nfralw 3308	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)
63	60, 62	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
64		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑦 ∈ ℝ
65	63, 64	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ)
66		nfre1 3282	. . . . . . . . 9 ⊢ Ⅎ𝑗∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))
67	40	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℤ)
68		eluzelz 12828	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → 𝑗 ∈ ℤ)
69	68	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℤ)
70	67	zred 12662	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℝ)
71	4, 53	sselid 3979	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ)
72	71	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ)
73	69	zred 12662	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℝ)
74	4, 49	sselid 3979	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℝ)
75	74	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑀 ∈ ℝ)
76	42	zred 12662	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℝ)
77	76	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (⌈‘𝑦) ∈ ℝ)
78		max1 13160	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
79	75, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
80	79	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
81		eluzle 12831	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗)
82	81	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗)
83	70, 72, 73, 80, 82	letrd 11367	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ≤ 𝑗)
84	2, 67, 69, 83	eluzd 44105	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ 𝑍)
85	84	3adant3 1132	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑗 ∈ 𝑍)
86		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ∈ ℝ)
87		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
88		ceilge 13806	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ → 𝑦 ≤ (⌈‘𝑦))
89	88	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ (⌈‘𝑦))
90		max2 13162	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
91	75, 77, 90	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
92	87, 77, 71, 89, 91	letrd 11367	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
93	92	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
94	86, 72, 73, 93, 82	letrd 11367	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ 𝑗)
95	94	3adant3 1132	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑦 ≤ 𝑗)
96		simp3 1138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
97	95, 96	jca 512	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
98		rspe 3246	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
99	85, 97, 98	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
100	99	3exp 1119	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))))
101	100	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))))
102	65, 66, 101	rexlimd 3263	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
103	59, 102	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
104	103	ralrimiva 3146	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)) → ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
105	104	ex 413	. . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) → ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
106	37, 105	impbid 211	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
107	106	rexbidv 3178	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
108	53	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
109	60, 64	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
110		nfra1 3281	. . . . . . . . 9 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
111	109, 110	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
112	94	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ 𝑗)
113		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
114	84	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ 𝑍)
115		rspa 3245	. . . . . . . . . . 11 ⊢ ((∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍) → (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
116	113, 114, 115	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
117	112, 116	mpd 15	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝐹‘𝑗) ≤ 𝑥)
118	117	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝐹‘𝑗) ≤ 𝑥))
119	111, 118	ralrimi 3254	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹‘𝑗) ≤ 𝑥)
120	56	raleqdv 3325	. . . . . . . 8 ⊢ (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹‘𝑗) ≤ 𝑥))
121	120	rspcev 3612	. . . . . . 7 ⊢ ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
122	108, 119, 121	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
123	122	rexlimdva2 3157	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
124	4	sseli 3977	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℝ)
125	124	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → 𝑘 ∈ ℝ)
126		nfra1 3281	. . . . . . . . . 10 ⊢ Ⅎ𝑗∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥
127	18, 126	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
128		simp1r 1198	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
129	26	3adant1r 1177	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (ℤ_≥‘𝑘))
130		rspa 3245	. . . . . . . . . . 11 ⊢ ((∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ 𝑗 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗) ≤ 𝑥)
131	128, 129, 130	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
132	131	3exp 1119	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → (𝑗 ∈ 𝑍 → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
133	127, 132	ralrimi 3254	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
134	133	adantll 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
135	8	rspceaimv 3616	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
136	125, 134, 135	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
137	136	rexlimdva2 3157	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
138	123, 137	impbid 211	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
139	138	rexbidv 3178	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
140	107, 139	anbi12d 631	. 2 ⊢ (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝑦 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))
141	7, 140	bitrd 278	1 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2883 ∀wral 3061 ∃wrex 3070 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ⟶wf 6536 ‘cfv 6540 ℝcr 11105 ℝ^*cxr 11243 ≤ cle 11245 ℤcz 12554 ℤ_≥cuz 12818 ⌈cceil 13752 lim supclsp 15410

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-ico 13326 df-fl 13753 df-ceil 13754 df-limsup 15411

This theorem is referenced by: limsupre3uz 44438 limsupreuz 44439

W3C validator