limsupgtlem - Mathbox for Glauco Siliprandi

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Theorem limsupgtlem 45815

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 45463	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
5		rnresss 5961	. . . . . . . 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 45423	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
9	8	frnd 6654	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ^*)
10	6, 9	sstrd 3940	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
11	10	supxrcld 45144	. . . . 5 ⊢ (𝜑 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
13		limsupgtlem.r	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
15	14, 2, 3, 7	limsupreuz 45775	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
17	16	simpld 494	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))
18		rexr 11153	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
19	18	ad4antlr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ∈ ℝ^*)
20	7	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ)
21	3	uztrn2 12746	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
22	21	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
23	20, 22	ffvelcdmd 7013	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
24	23	rexrd 11157	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
25	24	3impa 1109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
26	25	ad5ant134 1369	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ ℝ^*)
27	11	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
28		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ (𝐹‘𝑘))
29	10	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
30		fvres 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) = (𝐹‘𝑘))
31	30	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
32	31	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
33	7	ffnd 6647	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑍)
34	33	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍)
35	22	ssd 45117	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ_≥‘𝑗) ⊆ 𝑍)
36		fnssres 6599	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
37	34, 35, 36	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
38	37	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
39		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
40	38, 39	fnfvelrnd 7010	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
41	32, 40	eqeltrd 2831	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
42		eqid 2731	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )
43	29, 41, 42	supxrubd 45150	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
44	43	3impa 1109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
45	44	ad5ant134 1369	. . . . . . . . 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 13056	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
47	46	rexlimdva2 3135	. . . . . . 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 3145	. . . . 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 12941	. . . 4 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ⁺)
53	1, 4, 12, 50, 52	infrpgernmpt 45503	. . 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 1138	. . . . . . 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 45746	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ))
56	55	eqcomd 2737	. . . . . . . . 9 ⊢ (𝜑 → inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) = (lim sup‘𝐹))
57	56	oveq1d 7356	. . . . . . . 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 1133	. . . . . . 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 5112	. . . . . 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 1169	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^)
61		simpl1 1192	. . . . . . . . . . 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 6831	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ V)
65	7, 64	fexd 7156	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
66	65	limsupcld 45728	. . . . . . . . . . . 12 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
67	51	rpred 12929	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
68	67	rehalfcld 12363	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ)
69	68	rexrd 11157	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ^*)
70	66, 69	xaddcld 13195	. . . . . . . . . . 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 1169	. . . . . . . . . 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 1194	. . . . . . . . . 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 13056	. . . . . . . . 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 13128	. . . . . . . . . 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 5112	. . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑋 / 2) ∈ ℝ)
79	13	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim sup‘𝐹) ∈ ℝ)
80	23, 78, 79	lesubaddd 11709	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹‘𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81	80	3adantl3 1169	. . . . . . . 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 257	. . . . . . 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 3124	. . . . . 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 1411	. . . . 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 1119	. . . 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 3234	. . 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	ffvelcdmda 7012	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
90	67	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ∈ ℝ)
91	89, 90	resubcld 11540	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
92	91	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
93	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) ∈ ℝ)
94	89, 93	resubcld 11540	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
95	94	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
96	13	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
97	51	rphalfltd 45493	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 / 2) < 𝑋)
98	97	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) < 𝑋)
99	93, 90, 89, 98	ltsub2dd 11725	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
100	99	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
101		simpr 484	. . . . . . 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 11268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))
103	102	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
104	88, 22, 103	syl2anc 584	. . . 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 3145	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 class class class wbr 5086 ↦ cmpt 5167 ran crn 5612 ↾ cres 5613 Fn wfn 6471 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 supcsup 9319 infcinf 9320 ℝcr 11000 + caddc 11004 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 2c2 12175 ℤcz 12463 ℤ_≥cuz 12727 ℝ⁺crp 12885 +_𝑒 cxad 13004 lim supclsp 15372

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-n0 12377 df-z 12464 df-uz 12728 df-rp 12886 df-xadd 13007 df-ico 13246 df-fz 13403 df-fzo 13550 df-fl 13691 df-ceil 13692 df-limsup 15373

This theorem is referenced by: limsupgt 45816

W3C validator