limsupgtlem - Mathbox for Glauco Siliprandi

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Theorem limsupgtlem 46234

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 1922	. . . 4 ⊢ Ⅎ𝑗𝜑
2		limsupgtlem.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		limsupgtlem.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	2, 3	uzn0d 45882	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
5		rnresss 5976	. . . . . . . 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 45843	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
9	8	frnd 6667	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ^*)
10	6, 9	sstrd 3927	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
11	10	supxrcld 45568	. . . . 5 ⊢ (𝜑 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
12	11	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
13		limsupgtlem.r	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
15	14, 2, 3, 7	limsupreuz 46194	. . . . . . 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 234	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
17	16	simpld 496	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))
18		rexr 11186	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
19	18	ad4antlr 740	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ∈ ℝ^*)
20	7	ad2antrr 733	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ)
21	3	uztrn2 12802	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
22	21	adantll 721	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
23	20, 22	ffvelcdmd 7030	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
24	23	rexrd 11190	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
25	24	3impa 1116	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
26	25	ad5ant134 1376	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ ℝ^*)
27	11	ad4antr 739	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ∈ ℝ^)
28		simpr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ (𝐹‘𝑘))
29	10	ad2antrr 733	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
30		fvres 6850	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) = (𝐹‘𝑘))
31	30	eqcomd 2747	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
32	31	adantl 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘))
33	7	ffnd 6660	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑍)
34	33	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍)
35	22	ssd 45543	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ_≥‘𝑗) ⊆ 𝑍)
36		fnssres 6612	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
37	34, 35, 36	syl2anc 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
38	37	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹 ↾ (ℤ_≥‘𝑗)) Fn (ℤ_≥‘𝑗))
39		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
40	38, 39	fnfvelrnd 7027	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
41	32, 40	eqeltrd 2841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
42		eqid 2741	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )
43	29, 41, 42	supxrubd 45574	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
44	43	3impa 1116	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
45	44	ad5ant134 1376	. . . . . . . . 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 13108	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
47	46	rexlimdva2 3144	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
48	47	ralimdva 3153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
49	48	reximdva 3154	. . . . 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 12993	. . . 4 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ⁺)
53	1, 4, 12, 50, 52	infrpgernmpt 45922	. . 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 1145	. . . . . . 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 46165	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ))
56	55	eqcomd 2747	. . . . . . . . 9 ⊢ (𝜑 → inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) = (lim sup‘𝐹))
57	56	oveq1d 7375	. . . . . . . 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 1140	. . . . . . 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 5101	. . . . . 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 1176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) ≤ ((lim sup‘𝐹) +_𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^)
61		simpl1 1199	. . . . . . . . . . 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 6845	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ V)
65	7, 64	fexd 7175	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
66	65	limsupcld 46147	. . . . . . . . . . . 12 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
67	51	rpred 12981	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
68	67	rehalfcld 12419	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ)
69	68	rexrd 11190	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ^*)
70	66, 69	xaddcld 13248	. . . . . . . . . . 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 1176	. . . . . . . . . 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 1201	. . . . . . . . . 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 13108	. . . . . . . . 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 13181	. . . . . . . . . 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 5101	. . . . . . . 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 733	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑋 / 2) ∈ ℝ)
79	13	ad2antrr 733	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim sup‘𝐹) ∈ ℝ)
80	23, 78, 79	lesubaddd 11742	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹‘𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81	80	3adantl3 1176	. . . . . . . 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 259	. . . . . . 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 3133	. . . . . 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 1418	. . . . 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 1126	. . . 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 3243	. . 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 773	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
89	7	ffvelcdmda 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
90	67	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ∈ ℝ)
91	89, 90	resubcld 11573	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
92	91	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) ∈ ℝ)
93	68	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) ∈ ℝ)
94	89, 93	resubcld 11573	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
95	94	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − (𝑋 / 2)) ∈ ℝ)
96	13	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
97	51	rphalfltd 45912	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 / 2) < 𝑋)
98	97	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) < 𝑋)
99	93, 90, 89, 98	ltsub2dd 11758	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
100	99	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < ((𝐹‘𝑘) − (𝑋 / 2)))
101		simpr 486	. . . . . . 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 11301	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))
103	102	ex 414	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
104	88, 22, 103	syl2anc 591	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)))
105	104	ralimdva 3153	. . 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 3154	. 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 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∀wral 3055 ∃wrex 3065 Vcvv 3433 ⊆ wss 3885 class class class wbr 5075 ↦ cmpt 5156 ran crn 5622 ↾ cres 5623 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 supcsup 9347 infcinf 9348 ℝcr 11032 + caddc 11036 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 − cmin 11372 / cdiv 11802 2c2 12231 ℤcz 12519 ℤ_≥cuz 12783 ℝ⁺crp 12937 +_𝑒 cxad 13056 lim supclsp 15427

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-xadd 13059 df-ico 13299 df-fz 13457 df-fzo 13604 df-fl 13746 df-ceil 13747 df-limsup 15428

This theorem is referenced by: limsupgt 46235

W3C validator