Theorem limsupre3mpt 42022

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 5166	. . 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 41512	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*)
6	1, 2, 5	limsupre3 42021	. 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 2823	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
9	8, 4	fvmpt2d 6783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
10	9	breq2d 5080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑤 ≤ 𝐵))
11	10	anbi2d 630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
12	3, 11	rexbida 3320	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
13	12	ralbidv 3199	. . . 4 ⊢ (𝜑 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
14	13	rexbidv 3299	. . 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 5078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤 ↔ 𝐵 ≤ 𝑤))
16	15	imbi2d 343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
17	3, 16	ralbida 3232	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
18	17	rexbidv 3299	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
19	18	rexbidv 3299	. . 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 632	. 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 5071	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝐵 ↔ 𝑦 ≤ 𝐵))
22	21	anbi2d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
23	22	rexbidv 3299	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
24	23	ralbidv 3199	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
25		breq1 5071	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ≤ 𝑥 ↔ 𝑘 ≤ 𝑥))
26	25	anbi1d 631	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
27	26	rexbidv 3299	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
28	27	cbvralvw 3451	. . . . . . 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 281	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
31	30	cbvrexvw 3452	. . . 4 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵))
32		breq2 5072	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
33	32	imbi2d 343	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
34	33	ralbidv 3199	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
35	34	rexbidv 3299	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
36	25	imbi1d 344	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
37	36	ralbidv 3199	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
38	37	cbvrexvw 3452	. . . . . . 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 281	. . . . 5 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
41	40	cbvrexvw 3452	. . . 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 628	. . 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 307	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 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938   class class class wbr 5068   ↦ cmpt 5148  ‘cfv 6357  ℝcr 10538  ℝ^*cxr 10676   ≤ cle 10678  lim supclsp 14829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-sup 8908  df-inf 8909  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-ico 12747  df-limsup 14830

This theorem is referenced by: (None)