limsupgtlem - Mathbox for Glauco Siliprandi

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

Theorem limsupgtlem 42419

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 1915	. . . 4 ⊢ Ⅎ𝑗𝜑
2		limsupgtlem.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		limsupgtlem.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	2, 3	uzn0d 42062	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
5		rnresss 5854	. . . . . . . 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 42019	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
9	8	frnd 6494	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ^*)
10	6, 9	sstrd 3925	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
11	10	supxrcld 41743	. . . . 5 ⊢ (𝜑 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
12	11	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
13		limsupgtlem.r	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
15	14, 2, 3, 7	limsupreuz 42379	. . . . . . 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 498	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))
18		rexr 10676	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
19	18	ad4antlr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ∈ ℝ^*)
20	7	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ)
21	3	uztrn2 12250	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
22	21	adantll 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
23	20, 22	ffvelrnd 6829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
24	23	rexrd 10680	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
25	24	3impa 1107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
26	25	ad5ant134 1364	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ ℝ^*)
27	11	ad4antr 731	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
28		simpr 488	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ (𝐹‘𝑘))
29	10	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
30		fvres 6664	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) = (𝐹‘𝑘))
31	30	eqcomd 2804	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
32	31	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
33	7	ffnd 6488	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑍)
34	33	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍)
35	22	ssd 41716	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ_≥‘𝑗) ⊆ 𝑍)
36		fnssres 6442	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
37	34, 35, 36	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
38	37	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
39		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
40	38, 39	fnfvelrnd 41900	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
41	32, 40	eqeltrd 2890	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
42		eqid 2798	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )
43	29, 41, 42	supxrubd 41749	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
44	43	3impa 1107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
45	44	ad5ant134 1364	. . . . . . . . 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 12543	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
47	46	rexlimdva2 3246	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
48	47	ralimdva 3144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
49	48	reximdva 3233	. . . . 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 15	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
51		limsupgtlem.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
52	51	rphalfcld 12431	. . . 4 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ⁺)
53	1, 4, 12, 50, 52	infrpgernmpt 42104	. . 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 1135	. . . . . . 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 42350	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ))
56	55	eqcomd 2804	. . . . . . . . 9 ⊢ (𝜑 → inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) = (lim sup‘𝐹))
57	56	oveq1d 7150	. . . . . . . 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 1130	. . . . . . 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 5056	. . . . . 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 1165	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^)
61		simpl1 1188	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
62	61, 11	syl 17	. . . . . . . . . 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 6659	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ V)
65	7, 64	fexd 6967	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
66	65	limsupcld 42332	. . . . . . . . . . . 12 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
67	51	rpred 12419	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
68	67	rehalfcld 11872	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ)
69	68	rexrd 10680	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ^*)
70	66, 69	xaddcld 12682	. . . . . . . . . . 11 ⊢ (𝜑 → ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)) ∈ ℝ^*)
71	61, 70	syl 17	. . . . . . . . . 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 1165	. . . . . . . . . 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 1190	. . . . . . . . . 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 12543	. . . . . . . . 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 12615	. . . . . . . . . 10 ⊢ (𝜑 → ((lim sup‘𝐹) +_𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) + (𝑋 / 2)))
76	61, 75	syl 17	. . . . . . . . 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 5056	. . . . . . . 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 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑋 / 2) ∈ ℝ)
79	13	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim sup‘𝐹) ∈ ℝ)
80	23, 78, 79	lesubaddd 11226	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹‘𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81	80	3adantl3 1165	. . . . . . . 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 3149	. . . . . 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 1406	. . . . 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 1116	. . . 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 3270	. . 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 15	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
88		simpll 766	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
89	7	ffvelrnda 6828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
90	67	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ∈ ℝ)
91	89, 90	resubcld 11057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
92	91	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
93	68	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) ∈ ℝ)
94	89, 93	resubcld 11057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
95	94	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
96	13	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
97	51	rphalfltd 42094	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 / 2) < 𝑋)
98	97	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) < 𝑋)
99	93, 90, 89, 98	ltsub2dd 11242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
100	99	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
101		simpr 488	. . . . . . 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 10789	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))
103	102	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
104	88, 22, 103	syl2anc 587	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
105	104	ralimdva 3144	. . 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 3233	. 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 15	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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 infcinf 8889 ℝcr 10525 + caddc 10529 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 +_𝑒 cxad 12493 lim supclsp 14819

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-ceil 13158 df-limsup 14820

This theorem is referenced by: limsupgt 42420

W3C validator