limsupgtlem - Mathbox for Glauco Siliprandi

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Theorem limsupgtlem 46376

Description: For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
limsupgtlem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupgtlem.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
limsupgtlem.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
limsupgtlem.r	⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
limsupgtlem.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)

**Assertion**
Ref	Expression
limsupgtlem	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))

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

Proof of Theorem limsupgtlem Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1941	. . . 4 ⊢ Ⅎ𝑗𝜑
2		limsupgtlem.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		limsupgtlem.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	2, 3	uzn0d 46024	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
5		rnresss 6014	. . . . . . . 8 ⊢ ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ran 𝐹
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ran 𝐹)
7		limsupgtlem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
8	7	frexr 45985	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
9	8	frnd 6712	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ^*)
10	6, 9	sstrd 3955	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
11	10	supxrcld 45710	. . . . 5 ⊢ (𝜑 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
12	11	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
13		limsupgtlem.r	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
15	14, 2, 3, 7	limsupreuz 46336	. . . . . . 7 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)))
16	13, 15	mpbid 235	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
17	16	simpld 499	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))
18		rexr 11251	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
19	18	ad4antlr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ∈ ℝ^*)
20	7	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ)
21	3	uztrn2 12877	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
22	21	adantll 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
23	20, 22	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
24	23	rexrd 11255	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
25	24	3impa 1125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
26	25	ad5ant134 1390	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ ℝ^*)
27	11	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
28		simpr 489	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ (𝐹‘𝑘))
29	10	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
30		fvres 6898	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) = (𝐹‘𝑘))
31	30	eqcomd 2775	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
32	31	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
33	7	ffnd 6704	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑍)
34	33	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍)
35	22	ssd 45685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ_≥‘𝑗) ⊆ 𝑍)
36		fnssres 6656	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
37	34, 35, 36	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
38	37	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
39		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
40	38, 39	fnfvelrnd 7075	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
41	32, 40	eqeltrd 2869	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
42		eqid 2769	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )
43	29, 41, 42	supxrubd 45716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
44	43	3impa 1125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
45	44	ad5ant134 1390	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
46	19, 26, 27, 28, 45	xrletrd 13183	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
47	46	rexlimdva2 3174	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
48	47	ralimdva 3183	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
49	48	reximdva 3184	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
50	17, 49	mpd 16	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
51		limsupgtlem.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
52	51	rphalfcld 13068	. . . 4 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ⁺)
53	1, 4, 12, 50, 52	infrpgernmpt 46064	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^*, < ) +_𝑒 (𝑋 / 2)))
54		simp3 1154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) +_𝑒 (𝑋 / 2))) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) +_𝑒 (𝑋 / 2)))
55	2, 3, 8	limsupvaluz 46307	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ))
56	55	eqcomd 2775	. . . . . . . . 9 ⊢ (𝜑 → inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) = (lim sup‘𝐹))
57	56	oveq1d 7423	. . . . . . . 8 ⊢ (𝜑 → (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) +_𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)))
58	57	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) +_𝑒 (𝑋 / 2))) → (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^*, < ) +_𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)))
59	54, 58	breqtrd 5138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) +_𝑒 (𝑋 / 2))) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)))
60	24	3adantl3 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^)
61		simpl1 1208	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
62	61, 11	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^*)
63	3	fvexi 6893	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ V)
65	7, 64	fexd 7223	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
66	65	limsupcld 46289	. . . . . . . . . . . 12 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
67	51	rpred 13056	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
68	67	rehalfcld 12487	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ)
69	68	rexrd 11255	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ^*)
70	66, 69	xaddcld 13323	. . . . . . . . . . 11 ⊢ (𝜑 → ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)) ∈ ℝ^*)
71	61, 70	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)) ∈ ℝ^)
72	43	3adantl3 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ))
73		simpl3 1210	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)))
74	60, 62, 71, 72, 73	xrletrd 13183	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)))
75	13, 68	rexaddd 13256	. . . . . . . . . 10 ⊢ (𝜑 → ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) + (𝑋 / 2)))
76	61, 75	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) + (𝑋 / 2)))
77	74, 76	breqtrd 5138	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2)))
78	68	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑋 / 2) ∈ ℝ)
79	13	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim sup‘𝐹) ∈ ℝ)
80	23, 78, 79	lesubaddd 11807	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹‘𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81	80	3adantl3 1185	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹‘𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
82	77, 81	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
83	82	ralrimiva 3163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
84	59, 83	syld3an3 1434	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^*, < ) +_𝑒 (𝑋 / 2))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
85	84	3exp 1135	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^*, < ) +_𝑒 (𝑋 / 2)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))))
86	1, 85	reximdai 3273	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ (inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^*, < ) +_𝑒 (𝑋 / 2)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)))
87	53, 86	mpd 16	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
88		simpll 778	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
89	7	ffvelcdmda 7077	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
90	67	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ∈ ℝ)
91	89, 90	resubcld 11638	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
92	91	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
93	68	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) ∈ ℝ)
94	89, 93	resubcld 11638	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
95	94	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
96	13	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
97	51	rphalfltd 46054	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 / 2) < 𝑋)
98	97	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) < 𝑋)
99	93, 90, 89, 98	ltsub2dd 11823	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
100	99	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
101		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
102	92, 95, 96, 100, 101	ltletrd 11366	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))
103	102	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
104	88, 22, 103	syl2anc 595	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
105	104	ralimdva 3183	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
106	105	reximdva 3184	. 2 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
107	87, 106	mpd 16	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 Vcvv 3463 ⊆ wss 3913 class class class wbr 5110 ↦ cmpt 5193 ran crn 5660 ↾ cres 5661 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7408 supcsup 9396 infcinf 9397 ℝcr 11095 + caddc 11099 ℝ^*cxr 11238 < clt 11239 ≤ cle 11240 − cmin 11437 / cdiv 11867 2c2 12291 ℤcz 12587 ℤ_≥cuz 12858 ℝ⁺crp 13012 +_𝑒 cxad 13131 lim supclsp 15517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-n0 12501 df-z 12588 df-uz 12859 df-rp 13013 df-xadd 13134 df-ico 13374 df-fz 13532 df-fzo 13679 df-fl 13821 df-ceil 13822 df-limsup 15518

This theorem is referenced by: limsupgt 46377

W3C validator