limsupre - Mathbox for Glauco Siliprandi

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Theorem limsupre 46084

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 11193	. . . . 5 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ^*)
3		renegcl 11448	. . . . . 6 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
4	3	rexrd 11186	. . . . 5 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ^*)
5	4	ad2antlr 733	. . . 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 11120	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9		limsupre.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10	8, 9	ssexd 5252	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
11	6, 10	fexd 7171	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
12		limsupcl 15426	. . . . . 6 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
14	13	ad2antrr 732	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ^*)
15	3	mnfltd 13066	. . . . 5 ⊢ (𝑏 ∈ ℝ → -∞ < -𝑏)
16	15	ad2antlr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
17	9	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
18		ressxr 11180	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
20	6, 19	fssd 6672	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ^*)
21	20	ad2antrr 732	. . . . 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 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → sup(𝐵, ℝ^*, < ) = +∞)
24		simpr 485	. . . . . . 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 1921	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑏 ∈ ℝ)
26		nfre1 3264	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
27	25, 26	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
28		nfv 1921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑏 ∈ ℝ)
29		nfv 1921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ
30		nfra1 3263	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
31	28, 29, 30	nf3an 1908	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
32		simp13 1212	. . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐵)
34		simp3 1144	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
35		rspa 3228	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ∧ 𝑗 ∈ 𝐵) → (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
36	35	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ∧ 𝑗 ∈ 𝐵) ∧ 𝑘 ≤ 𝑗) → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
37	32, 33, 34, 36	syl21anc 843	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (abs‘(𝐹‘𝑗)) ≤ 𝑏)
38		simp11l 1291	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝜑)
39	6	ffvelcdmda 7025	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ)
40	38, 33, 39	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
41		simp11r 1292	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → 𝑏 ∈ ℝ)
42	40, 41	absled 15386	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → -𝑏 ≤ (𝐹‘𝑗))
45	44	3exp 1125	. . . . . . . . . . 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 3237	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
47	46	3exp 1125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))))
48	47	adantr 481	. . . . . . . 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 3241	. . . . . . 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 5076	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℎ ≤ 𝑖 ↔ ℎ ≤ 𝑗))
52		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐹‘𝑖) = (𝐹‘𝑗))
53	52	breq2d 5084	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹‘𝑖) ↔ -𝑏 ≤ (𝐹‘𝑗)))
54	51, 53	imbi12d 345	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
55	54	cbvralvw 3217	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
56		breq1 5075	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ≤ 𝑗 ↔ 𝑘 ≤ 𝑗))
57	56	imbi1d 342	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
58	57	ralbidv 3162	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
59	55, 58	bitrid 284	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
60	59	cbvrexvw 3218	. . . . . 6 ⊢ (∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
61	50, 60	sylibr 235	. . . . 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 15436	. . . 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 13103	. . 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 3147	. 2 ⊢ (𝜑 → -∞ < (lim sup‘𝐹))
66		rexr 11182	. . . . 5 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℝ^*)
67	66	ad2antlr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ^*)
68		pnfxr 11190	. . . . 5 ⊢ +∞ ∈ ℝ^*
69	68	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ^*)
70	43	simprd 496	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ∧ 𝑗 ∈ 𝐵 ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑏)
71	70	3exp 1125	. . . . . . . . . . 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 3237	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
73	72	3exp 1125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) → ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))))
74	73	adantr 481	. . . . . . . 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 3241	. . . . . . 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 5082	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑏 ↔ (𝐹‘𝑗) ≤ 𝑏))
78	51, 77	imbi12d 345	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
79	78	cbvralvw 3217	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
80	56	imbi1d 342	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
81	80	ralbidv 3162	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
82	79, 81	bitrid 284	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
83	82	cbvrexvw 3218	. . . . . 6 ⊢ (∃ℎ ∈ ℝ ∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
84	76, 83	sylibr 235	. . . . 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 15435	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
86		ltpnf 13062	. . . . 5 ⊢ (𝑏 ∈ ℝ → 𝑏 < +∞)
87	86	ad2antlr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
88	14, 67, 69, 85, 87	xrlelttrd 13102	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
89	88, 64	r19.29a 3147	. 2 ⊢ (𝜑 → (lim sup‘𝐹) < +∞)
90		xrrebnd 13111	. . 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 719	1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ⊆ wss 3883 class class class wbr 5072 ⟶wf 6481 ‘cfv 6485 supcsup 9343 ℝcr 11028 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 -cneg 11369 abscabs 15187 lim supclsp 15423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-ico 13295 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424

This theorem is referenced by: limsupref 46128 ioodvbdlimc1lem2 46375 ioodvbdlimc2lem 46377

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