Theorem limsupre3mpt 43600

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 5197	. . 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 43095	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*)
6	1, 2, 5	limsupre3 43599	. 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 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
9	8, 4	fvmpt2d 6938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
10	9	breq2d 5101	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑤 ≤ 𝐵))
11	10	anbi2d 629	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
12	3, 11	rexbida 3251	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
13	12	ralbidv 3170	. . . 4 ⊢ (𝜑 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
14	13	rexbidv 3171	. . 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 5099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤 ↔ 𝐵 ≤ 𝑤))
16	15	imbi2d 340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
17	3, 16	ralbida 3249	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
18	17	rexbidv 3171	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
19	18	rexbidv 3171	. . 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 631	. 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 5092	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝐵 ↔ 𝑦 ≤ 𝐵))
22	21	anbi2d 629	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
23	22	rexbidv 3171	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
24	23	ralbidv 3170	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
25		breq1 5092	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ≤ 𝑥 ↔ 𝑘 ≤ 𝑥))
26	25	anbi1d 630	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
27	26	rexbidv 3171	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
28	27	cbvralvw 3221	. . . . . . 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 278	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
31	30	cbvrexvw 3222	. . . 4 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵))
32		breq2 5093	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
33	32	imbi2d 340	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
34	33	ralbidv 3170	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
35	34	rexbidv 3171	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
36	25	imbi1d 341	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
37	36	ralbidv 3170	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
38	37	cbvrexvw 3222	. . . . . . 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 278	. . . . 5 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
41	40	cbvrexvw 3222	. . . 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 627	. . 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 304	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 205   ∧ wa 396   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  ∀wral 3061  ∃wrex 3070   ⊆ wss 3897   class class class wbr 5089   ↦ cmpt 5172  ‘cfv 6473  ℝcr 10963  ℝ^*cxr 11101   ≤ cle 11103  lim supclsp 15270

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041  ax-pre-sup 11042

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-po 5526  df-so 5527  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-sup 9291  df-inf 9292  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-ico 13178  df-limsup 15271

This theorem is referenced by: (None)