Theorem limsupre3mpt 41557

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
limsupre3mpt.p	⊢ Ⅎ𝑥𝜑
limsupre3mpt.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre3mpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)

Assertion

Ref	Expression
limsupre3mpt	⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦))))

Distinct variable groups:   𝐴,𝑘,𝑥,𝑦   𝐵,𝑘,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑘)   𝐵(𝑥)

Proof of Theorem limsupre3mpt

Dummy variables 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfmpt1 5058	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
2		limsupre3mpt.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsupre3mpt.p	. . . 4 ⊢ Ⅎ𝑥𝜑
4		limsupre3mpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	3, 4	fmptd2f 41046	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*)
6	1, 2, 5	limsupre3 41556	. 2 ⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∧ ∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤))))
7		eqid 2795	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
9	8, 4	fvmpt2d 6647	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
10	9	breq2d 4974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑤 ≤ 𝐵))
11	10	anbi2d 628	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
12	3, 11	rexbida 3279	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
13	12	ralbidv 3164	. . . 4 ⊢ (𝜑 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
14	13	rexbidv 3260	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
15	9	breq1d 4972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤 ↔ 𝐵 ≤ 𝑤))
16	15	imbi2d 342	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
17	3, 16	ralbida 3194	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
18	17	rexbidv 3260	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
19	18	rexbidv 3260	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
20	14, 19	anbi12d 630	. 2 ⊢ (𝜑 → ((∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∧ ∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤)) ↔ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ∧ ∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤))))
21		breq1 4965	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝐵 ↔ 𝑦 ≤ 𝐵))
22	21	anbi2d 628	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
23	22	rexbidv 3260	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
24	23	ralbidv 3164	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
25		breq1 4965	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ≤ 𝑥 ↔ 𝑘 ≤ 𝑥))
26	25	anbi1d 629	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
27	26	rexbidv 3260	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
28	27	cbvralv 3403	. . . . . . 7 ⊢ (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵))
29	28	a1i 11	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
30	24, 29	bitrd 280	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
31	30	cbvrexv 3404	. . . 4 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵))
32		breq2 4966	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
33	32	imbi2d 342	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
34	33	ralbidv 3164	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
35	34	rexbidv 3260	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
36	25	imbi1d 343	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
37	36	ralbidv 3164	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
38	37	cbvrexv 3404	. . . . . . 7 ⊢ (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦))
39	38	a1i 11	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
40	35, 39	bitrd 280	. . . . 5 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
41	40	cbvrexv 3404	. . . 4 ⊢ (∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦))
42	31, 41	anbi12i 626	. . 3 ⊢ ((∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ∧ ∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)) ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
43	42	a1i 11	. 2 ⊢ (𝜑 → ((∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ∧ ∃𝑤 ∈ ℝ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)) ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦))))
44	6, 20, 43	3bitrd 306	1 ⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  Ⅎwnf 1765   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106   ⊆ wss 3859   class class class wbr 4962   ↦ cmpt 5041  ‘cfv 6225  ℝcr 10382  ℝ^*cxr 10520   ≤ cle 10522  lim supclsp 14661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-sup 8752  df-inf 8753  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-ico 12594  df-limsup 14662

This theorem is referenced by: (None)