limsupre - Mathbox for Glauco Siliprandi

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Theorem limsupre 44655

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 11275	. . . . 5 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ^*)
3		renegcl 11527	. . . . . 6 ⊢ (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
4	3	rexrd 11268	. . . . 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 11203	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9		limsupre.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10	8, 9	ssexd 5323	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
11	6, 10	fexd 7230	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
12		limsupcl 15421	. . . . . 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 13108	. . . . 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 11262	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
20	6, 19	fssd 6734	. . . . . 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 483	. . . . . . 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 3280	. . . . . . . . 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 3279	. . . . . . . . . . . 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 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 3243	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ∧ 𝑗 ∈ 𝐵) → (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
36	35	imp 405	. . . . . . . . . . . . . . 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	ffvelcdmda 7085	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ)
40	38, 33, 39	syl2anc 582	. . . . . . . . . . . . . . 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 15381	. . . . . . . . . . . . . 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 493	. . . . . . . . . . . 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 3252	. . . . . . . . . 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 479	. . . . . . . 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 3256	. . . . . . 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 5151	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (ℎ ≤ 𝑖 ↔ ℎ ≤ 𝑗))
52		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐹‘𝑖) = (𝐹‘𝑗))
53	52	breq2d 5159	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹‘𝑖) ↔ -𝑏 ≤ (𝐹‘𝑗)))
54	51, 53	imbi12d 343	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
55	54	cbvralvw 3232	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)))
56		breq1 5150	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ≤ 𝑗 ↔ 𝑘 ≤ 𝑗))
57	56	imbi1d 340	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
58	57	ralbidv 3175	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
59	55, 58	bitrid 282	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → -𝑏 ≤ (𝐹‘𝑖)) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → -𝑏 ≤ (𝐹‘𝑗))))
60	59	cbvrexvw 3233	. . . . . 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 15431	. . . 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 13144	. . 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 3160	. 2 ⊢ (𝜑 → -∞ < (lim sup‘𝐹))
66		rexr 11264	. . . . 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 11272	. . . . 5 ⊢ +∞ ∈ ℝ^*
69	68	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ^*)
70	43	simprd 494	. . . . . . . . . . . 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 3252	. . . . . . . . . 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 479	. . . . . . . 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 3256	. . . . . . 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 5157	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑏 ↔ (𝐹‘𝑗) ≤ 𝑏))
78	51, 77	imbi12d 343	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
79	78	cbvralvw 3232	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏))
80	56	imbi1d 340	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
81	80	ralbidv 3175	. . . . . . . 8 ⊢ (ℎ = 𝑘 → (∀𝑗 ∈ 𝐵 (ℎ ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
82	79, 81	bitrid 282	. . . . . . 7 ⊢ (ℎ = 𝑘 → (∀𝑖 ∈ 𝐵 (ℎ ≤ 𝑖 → (𝐹‘𝑖) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑏)))
83	82	cbvrexvw 3233	. . . . . 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 15430	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
86		ltpnf 13104	. . . . 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 13143	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
89	88, 64	r19.29a 3160	. 2 ⊢ (𝜑 → (lim sup‘𝐹) < +∞)
90		xrrebnd 13151	. . 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 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ⊆ wss 3947 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 supcsup 9437 ℝcr 11111 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 -cneg 11449 abscabs 15185 lim supclsp 15418

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-ico 13334 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419

This theorem is referenced by: limsupref 44699 ioodvbdlimc1lem2 44946 ioodvbdlimc2lem 44948

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