limsupre - Mathbox for Glauco Siliprandi

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Theorem limsupre 45749

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 11169	. . . . 5 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ^*)
3		renegcl 11424	. . . . . 6 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
4	3	rexrd 11162	. . . . 5 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ^*)
5	4	ad2antlr 727	. . . 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 11097	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9		limsupre.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10	8, 9	ssexd 5260	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
11	6, 10	fexd 7161	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
12		limsupcl 15380	. . . . . 6 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
14	13	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ^*)
15	3	mnfltd 13023	. . . . 5 ⊢ (𝑏 ∈ ℝ → -∞ < -𝑏)
16	15	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
17	9	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
18		ressxr 11156	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
20	6, 19	fssd 6668	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ^*)
21	20	ad2antrr 726	. . . . 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 726	. . . . 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 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑏 ∈ ℝ)
26		nfre1 3257	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
27	25, 26	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
28		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑏 ∈ ℝ)
29		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ
30		nfra1 3256	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
31	28, 29, 30	nf3an 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
32		simp13 1206	. . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐵)
34		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
35		rspa 3221	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
38		simp11l 1285	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝜑)
39	6	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ)
40	38, 33, 39	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
41		simp11r 1286	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑏 ∈ ℝ)
42	40, 41	absled 15340	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 1119	. . . . . . . . . . 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 3230	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
47	46	3exp 1119	. . . . . . . . 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 3234	. . . . . . 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 5093	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℎ ≤ 𝑖 ↔ ℎ ≤ 𝑗))
52		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐹‘𝑖) = (𝐹‘𝑗))
53	52	breq2d 5101	. . . . . . . . . 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 3210	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
56		breq1 5092	. . . . . . . . . 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 3155	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
59	55, 58	bitrid 283	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
60	59	cbvrexvw 3211	. . . . . 6 ⊢ (∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
61	50, 60	sylibr 234	. . . . 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 15390	. . . 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 13060	. . 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 3140	. 2 ⊢ (𝜑 → -∞ < (lim sup‘𝐹))
66		rexr 11158	. . . . 5 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℝ^*)
67	66	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ^*)
68		pnfxr 11166	. . . . 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 1119	. . . . . . . . . . 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 3230	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
73	72	3exp 1119	. . . . . . . . 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 3234	. . . . . . 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 5099	. . . . . . . . . 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 3210	. . . . . . . 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 3155	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
82	79, 81	bitrid 283	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
83	82	cbvrexvw 3211	. . . . . 6 ⊢ (∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
84	76, 83	sylibr 234	. . . . 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 15389	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
86		ltpnf 13019	. . . . 5 ⊢ (𝑏 ∈ ℝ → 𝑏 < +∞)
87	86	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
88	14, 67, 69, 85, 87	xrlelttrd 13059	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
89	88, 64	r19.29a 3140	. 2 ⊢ (𝜑 → (lim sup‘𝐹) < +∞)
90		xrrebnd 13067	. . 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 713	1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

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 5089 ⟶wf 6477 ‘cfv 6481 supcsup 9324 ℝcr 11005 +∞cpnf 11143 -∞cmnf 11144 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 -cneg 11345 abscabs 15141 lim supclsp 15377

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 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

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 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-rp 12891 df-ico 13251 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378

This theorem is referenced by: limsupref 45793 ioodvbdlimc1lem2 46040 ioodvbdlimc2lem 46042

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