limsupgtlem - Mathbox for Glauco Siliprandi

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Theorem limsupgtlem 45782

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 1914	. . . 4 ⊢ Ⅎ𝑗𝜑
2		limsupgtlem.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		limsupgtlem.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	2, 3	uzn0d 45428	. . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅)
5		rnresss 5991	. . . . . . . 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 45388	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
9	8	frnd 6699	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ^*)
10	6, 9	sstrd 3960	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (ℤ_≥‘𝑗)) ⊆ ℝ^*)
11	10	supxrcld 45108	. . . . 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 2892	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
15	14, 2, 3, 7	limsupreuz 45742	. . . . . . 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 11227	. . . . . . . . . 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 12819	. . . . . . . . . . . . . 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 7060	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
24	23	rexrd 11231	. . . . . . . . . . 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 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) = (𝐹‘𝑘))
31	30	eqcomd 2736	. . . . . . . . . . . . . 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 6692	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑍)
34	33	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍)
35	22	ssd 45081	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ_≥‘𝑗) ⊆ 𝑍)
36		fnssres 6644	. . . . . . . . . . . . . . . 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 7057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹 ↾ (ℤ_≥‘𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
41	32, 40	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑗)))
42		eqid 2730	. . . . . . . . . . . 12 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )
43	29, 41, 42	supxrubd 45114	. . . . . . . . . . 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 13129	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 ≤ (𝐹‘𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < ))
47	46	rexlimdva2 3137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
48	47	ralimdva 3146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 ∃𝑘 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝐹‘𝑘) → ∀𝑗 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^*, < )))
49	48	reximdva 3147	. . . . 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 13014	. . . 4 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ⁺)
53	1, 4, 12, 50, 52	infrpgernmpt 45468	. . 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 45713	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ))
56	55	eqcomd 2736	. . . . . . . . 9 ⊢ (𝜑 → inf(ran (𝑗 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑗)), ℝ^, < )), ℝ^, < ) = (lim sup‘𝐹))
57	56	oveq1d 7405	. . . . . . . 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 5136	. . . . . 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 6875	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ V)
65	7, 64	fexd 7204	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
66	65	limsupcld 45695	. . . . . . . . . . . 12 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
67	51	rpred 13002	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
68	67	rehalfcld 12436	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ)
69	68	rexrd 11231	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / 2) ∈ ℝ^*)
70	66, 69	xaddcld 13268	. . . . . . . . . . 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 13129	. . . . . . . . 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 13201	. . . . . . . . . 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 5136	. . . . . . . 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 11782	. . . . . . . . 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 3126	. . . . . 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 3240	. . 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 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
90	67	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ∈ ℝ)
91	89, 90	resubcld 11613	. . . . . . . 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 11613	. . . . . . . 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 45458	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 / 2) < 𝑋)
98	97	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑋 / 2) < 𝑋)
99	93, 90, 89, 98	ltsub2dd 11798	. . . . . . . 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 11341	. . . . . 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 3146	. . 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 3147	. 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 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 class class class wbr 5110 ↦ cmpt 5191 ran crn 5642 ↾ cres 5643 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 supcsup 9398 infcinf 9399 ℝcr 11074 + caddc 11078 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 / cdiv 11842 2c2 12248 ℤcz 12536 ℤ_≥cuz 12800 ℝ⁺crp 12958 +_𝑒 cxad 13077 lim supclsp 15443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-xadd 13080 df-ico 13319 df-fz 13476 df-fzo 13623 df-fl 13761 df-ceil 13762 df-limsup 15444

This theorem is referenced by: limsupgt 45783

W3C validator