limsupre - Mathbox for Glauco Siliprandi

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Theorem limsupre 43072

Description: If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)

**Hypotheses**
Ref	Expression
limsupre.1	⊢ (𝜑 → 𝐵 ⊆ ℝ)
limsupre.2	⊢ (𝜑 → sup(𝐵, ℝ^*, < ) = +∞)
limsupre.f	⊢ (𝜑 → 𝐹:𝐵⟶ℝ)
limsupre.bnd	⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))

**Assertion**
Ref	Expression
limsupre	⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Distinct variable groups: 𝐵,𝑗,𝑘 𝐹,𝑏,𝑗,𝑘 𝜑,𝑏,𝑗,𝑘 Allowed substitution hint: 𝐵(𝑏)

Proof of Theorem limsupre Dummy variables ℎ 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnfxr 10963	. . . . 5 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ^*)
3		renegcl 11214	. . . . . 6 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
4	3	rexrd 10956	. . . . 5 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ^*)
5	4	ad2antlr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -𝑏 ∈ ℝ^*)
6		limsupre.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ)
7		reex 10893	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9		limsupre.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10	8, 9	ssexd 5243	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
11	6, 10	fexd 7085	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
12		limsupcl 15110	. . . . . 6 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
14	13	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ^*)
15	3	mnfltd 12789	. . . . 5 ⊢ (𝑏 ∈ ℝ → -∞ < -𝑏)
16	15	ad2antlr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
17	9	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
18		ressxr 10950	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
20	6, 19	fssd 6602	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ^*)
21	20	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝐹:𝐵⟶ℝ^*)
22		limsupre.2	. . . . . 6 ⊢ (𝜑 → sup(𝐵, ℝ^*, < ) = +∞)
23	22	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → sup(𝐵, ℝ^*, < ) = +∞)
24		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
25		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑏 ∈ ℝ)
26		nfre1 3234	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
27	25, 26	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
28		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑏 ∈ ℝ)
29		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ
30		nfra1 3142	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
31	28, 29, 30	nf3an 1905	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
32		simp13 1203	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
33		simp2 1135	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐵)
34		simp3 1136	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
35		rspa 3130	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ∧ 𝑗 ∈ 𝐵) → (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
36	35	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ∧ 𝑗 ∈ 𝐵) ∧ 𝑘 ≤ 𝑗) → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
37	32, 33, 34, 36	syl21anc 834	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
38		simp11l 1282	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝜑)
39	6	ffvelrnda 6943	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ)
40	38, 33, 39	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
41		simp11r 1283	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑏 ∈ ℝ)
42	40, 41	absled 15070	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (-𝑏 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ 𝑏)))
43	37, 42	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (-𝑏 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ 𝑏))
44	43	simpld 494	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → -𝑏 ≤ (𝐹‘𝑗))
45	44	3exp 1117	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (𝑗 ∈ 𝐵 → (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
46	31, 45	ralrimi 3139	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
47	46	3exp 1117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))))
48	47	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))))
49	27, 48	reximdai 3239	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
50	24, 49	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
51		breq2 5074	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℎ ≤ 𝑖 ↔ ℎ ≤ 𝑗))
52		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐹‘𝑖) = (𝐹‘𝑗))
53	52	breq2d 5082	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹‘𝑖) ↔ -𝑏 ≤ (𝐹‘𝑗)))
54	51, 53	imbi12d 344	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
55	54	cbvralvw 3372	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
56		breq1 5073	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ≤ 𝑗 ↔ 𝑘 ≤ 𝑗))
57	56	imbi1d 341	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
58	57	ralbidv 3120	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
59	55, 58	syl5bb 282	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
60	59	cbvrexvw 3373	. . . . . 6 ⊢ (∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
61	50, 60	sylibr 233	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)))
62	17, 21, 5, 23, 61	limsupbnd2 15120	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -𝑏 ≤ (lim sup‘𝐹))
63	2, 5, 14, 16, 62	xrltletrd 12824	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ < (lim sup‘𝐹))
64		limsupre.bnd	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
65	63, 64	r19.29a 3217	. 2 ⊢ (𝜑 → -∞ < (lim sup‘𝐹))
66		rexr 10952	. . . . 5 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℝ^*)
67	66	ad2antlr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ^*)
68		pnfxr 10960	. . . . 5 ⊢ +∞ ∈ ℝ^*
69	68	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ^*)
70	43	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑏)
71	70	3exp 1117	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (𝑗 ∈ 𝐵 → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
72	31, 71	ralrimi 3139	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
73	72	3exp 1117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))))
74	73	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))))
75	27, 74	reximdai 3239	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
76	24, 75	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
77	52	breq1d 5080	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑏 ↔ (𝐹‘𝑗) ≤ 𝑏))
78	51, 77	imbi12d 344	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
79	78	cbvralvw 3372	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
80	56	imbi1d 341	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
81	80	ralbidv 3120	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
82	79, 81	syl5bb 282	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
83	82	cbvrexvw 3373	. . . . . 6 ⊢ (∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
84	76, 83	sylibr 233	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏))
85	17, 21, 67, 84	limsupbnd1 15119	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
86		ltpnf 12785	. . . . 5 ⊢ (𝑏 ∈ ℝ → 𝑏 < +∞)
87	86	ad2antlr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
88	14, 67, 69, 85, 87	xrlelttrd 12823	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
89	88, 64	r19.29a 3217	. 2 ⊢ (𝜑 → (lim sup‘𝐹) < +∞)
90		xrrebnd 12831	. . 3 ⊢ ((lim sup‘𝐹) ∈ ℝ^* → ((lim sup‘𝐹) ∈ ℝ ↔ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) < +∞)))
91	13, 90	syl 17	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) < +∞)))
92	65, 89, 91	mpbir2and 709	1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ⊆ wss 3883 class class class wbr 5070 ⟶wf 6414 ‘cfv 6418 supcsup 9129 ℝcr 10801 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 -cneg 11136 abscabs 14873 lim supclsp 15107

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ico 13014 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108

This theorem is referenced by: limsupref 43116 ioodvbdlimc1lem2 43363 ioodvbdlimc2lem 43365

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