Theorem limsupre3lem 43163

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 43156	. 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 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑦 ∈ ℝ)
6		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
7		simp3l 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → 𝑘 ≤ 𝑗)
8		simp1r 1196	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ)
9	8	rexrd 10956	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ∈ ℝ*)
10	3	ffvelrnda 6943	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
11	10	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
12	11	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ*)
13		simp3 1136	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 < (𝐹‘𝑗))
14	9, 12, 13	xrltled 12813	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ 𝑦 < (𝐹‘𝑗)) → 𝑦 ≤ (𝐹‘𝑗))
15	14	3adant3l 1178	. . . . . . . . . . . 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 1117	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))))
18	6, 17	reximdai 3239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
19	18	ralimdv 3103	. . . . . . . 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 1115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))
21		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ 𝑦 ≤ (𝐹‘𝑗)))
22	21	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
23	22	rexbidv 3225	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
24	23	ralbidv 3120	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
25	24	rspcev 3552	. . . . . . 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 583	. . . . . 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 1117	. . . . 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 3211	. . . 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 11218	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
30	29	ad2antlr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) ∈ ℝ)
31		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
32		simp3l 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑘 ≤ 𝑗)
33		simp1r 1196	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ)
34	29	rexrd 10956	. . . . . . . . . . . . 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 10956	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ∈ ℝ*)
37	10	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
38	37	3adant3 1130	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝐹‘𝑗) ∈ ℝ*)
39	33	ltm1d 11837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (𝑥 − 1) < 𝑥)
40		simp3r 1200	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴 ∧ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝑥 ≤ (𝐹‘𝑗))
41	35, 36, 38, 39, 40	xrltletrd 12824	. . . . . . . . . . 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 1117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝐴 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗)))))
44	31, 43	reximdai 3239	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
45	44	ralimdv 3103	. . . . . . 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 5073	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 − 1) → (𝑦 < (𝐹‘𝑗) ↔ (𝑥 − 1) < (𝐹‘𝑗)))
48	47	anbi2d 628	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 1) → ((𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
49	48	rexbidv 3225	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 1) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 < (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ (𝑥 − 1) < (𝐹‘𝑗))))
50	49	ralbidv 3120	. . . . . . 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 3552	. . . . . 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 583	. . . . 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 3215	. . . 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 211	. . 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 765	. . . . . 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 10952	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
58	57	ad3antlr 727	. . . . . . . . . . . 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 12813	. . . . . . . . . . 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 3102	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
64	63	reximdv 3201	. . . . . . 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 5074	. . . . . . . . . 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 3120	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
69	68	rexbidv 3225	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑦)))
70	69	rspcev 3552	. . . . . 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 583	. . . . 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 3215	. . . 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 11078	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
74	73	ad2antlr 723	. . . . . 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 10952	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
77	76	ad3antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*)
78	73	rexrd 10956	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ*)
79	78	ad3antlr 727	. . . . . . . . . . . 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 11745	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
82	81	ad3antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < (𝑥 + 1))
83	75, 77, 79, 80, 82	xrlelttrd 12823	. . . . . . . . . . 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 3102	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
87	86	reximdv 3201	. . . . . . 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 5074	. . . . . . . . . 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 3120	. . . . . . . 8 ⊢ (𝑦 = (𝑥 + 1) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < (𝑥 + 1))))
92	91	rexbidv 3225	. . . . . . 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 3552	. . . . . 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 583	. . . . 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 3215	. . . 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 211	. . 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 630	. 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 278	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  1c1 10803   + caddc 10805  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   − cmin 11135  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-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  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-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-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-ico 13014  df-limsup 15108

This theorem is referenced by:  limsupre3  43164