Theorem limsupre3lem 46168

Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupre3lem.1	⊢ Ⅎ𝑗𝐹
limsupre3lem.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre3lem.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupre3lem	⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))

Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥   𝜑,𝑗,𝑘,𝑥

Allowed substitution hint:   𝐹(𝑗)

Proof of Theorem limsupre3lem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupre3lem.1	. . 3 ⊢ Ⅎ𝑗𝐹
2		limsupre3lem.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsupre3lem.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsupre2 46161	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦))))
5		simp2 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ∈ ℝ)
6		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
7		simp3l 1203	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑘 ≤ 𝑗)
8		simp1r 1200	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ)
9	8	rexrd 11184	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ*)
10	3	ffvelcdmda 7028	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
11	10	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
12	11	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ*)
13		simp3 1139	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 < (𝐹‘𝑗))
14	9, 12, 13	xrltled 13090	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ≤ (𝐹‘𝑗))
15	14	3adant3l 1182	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ≤ (𝐹‘𝑗))
16	7, 15	jca 511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))
17	16	3exp 1120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))))
18	6, 17	reximdai 3240	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
19	18	ralimdv 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
20	19	3impia 1118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))
21		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ 𝑦 ≤ (𝐹‘𝑗)))
22	21	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
23	22	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
24	23	ralbidv 3161	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
25	24	rspcev 3565	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
26	5, 20, 25	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
27	26	3exp 1120	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))))
28	27	rexlimdv 3137	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
29		peano2rem 11450	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
30	29	ad2antlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈ ℝ)
31		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
32		simp3l 1203	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑘 ≤ 𝑗)
33		simp1r 1200	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ)
34	29	rexrd 11184	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ*)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈ ℝ*)
36	33	rexrd 11184	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ*)
37	10	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
38	37	3adant3 1133	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝐹‘𝑗) ∈ ℝ*)
39	33	ltm1d 12077	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < 𝑥)
40		simp3r 1204	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ≤ (𝐹‘𝑗))
41	35, 36, 38, 39, 40	xrltletrd 13101	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < (𝐹‘𝑗))
42	32, 41	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))
43	42	3exp 1120	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))))
44	31, 43	reximdai 3240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
45	44	ralimdv 3152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
46	45	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))
47		breq1 5089	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 − 1) → (𝑦 < (𝐹‘𝑗) ↔ (𝑥 − 1) < (𝐹‘𝑗)))
48	47	anbi2d 631	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 1) → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
49	48	rexbidv 3162	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 1) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
50	49	ralbidv 3161	. . . . . . 7 ⊢ (𝑦 = (𝑥 − 1) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
51	50	rspcev 3565	. . . . . 6 ⊢ (((𝑥 − 1) ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)))
52	30, 46, 51	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)))
53	52	rexlimdva2 3141	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))))
54	28, 53	impbid 212	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
55		simplr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → 𝑦 ∈ ℝ)
56	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) ∈ ℝ*)
57		rexr 11180	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
58	57	ad3antlr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → 𝑦 ∈ ℝ*)
59		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) < 𝑦)
60	56, 58, 59	xrltled 13090	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) < 𝑦) → (𝐹‘𝑗) ≤ 𝑦)
61	60	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) < 𝑦 → (𝐹‘𝑗) ≤ 𝑦))
62	61	imim2d 57	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
63	62	ralimdva 3150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
64	63	reximdv 3153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
65	64	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦))
66		breq2 5090	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑦))
67	66	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
68	67	ralbidv 3161	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
69	68	rexbidv 3162	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
70	69	rspcev 3565	. . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
71	55, 65, 70	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
72	71	rexlimdva2 3141	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
73		peano2re 11308	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
74	73	ad2antlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑥 + 1) ∈ ℝ)
75	37	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ∈ ℝ*)
76		rexr 11180	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
77	76	ad3antlr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*)
78	73	rexrd 11184	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ*)
79	78	ad3antlr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝑥 + 1) ∈ ℝ*)
80		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥)
81		ltp1 11984	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
82	81	ad3antlr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < (𝑥 + 1))
83	75, 77, 79, 80, 82	xrlelttrd 13100	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) < (𝑥 + 1))
84	83	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) ≤ 𝑥 → (𝐹‘𝑗) < (𝑥 + 1)))
85	84	imim2d 57	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
86	85	ralimdva 3150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
87	86	reximdv 3153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
88	87	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1)))
89		breq2 5090	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 + 1) → ((𝐹‘𝑗) < 𝑦 ↔ (𝐹‘𝑗) < (𝑥 + 1)))
90	89	imbi2d 340	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 + 1) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
91	90	ralbidv 3161	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
92	91	rexbidv 3162	. . . . . . 7 ⊢ (𝑦 = (𝑥 + 1) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
93	92	rspcev 3565	. . . . . 6 ⊢ (((𝑥 + 1) ∈ ℝ ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦))
94	74, 88, 93	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦))
95	94	rexlimdva2 3141	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)))
96	72, 95	impbid 212	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
97	54, 96	anbi12d 633	. 2 ⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))
98	4, 97	bitrd 279	1 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890   class class class wbr 5086  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  ℝcr 11026  1c1 11028   + caddc 11030  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169   − cmin 11366  lim supclsp 15421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-po 5530  df-so 5531  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9346  df-inf 9347  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-ico 13293  df-limsup 15422

This theorem is referenced by:  limsupre3  46169