Theorem limsupre3mpt 46177

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 5171	. . 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 45679	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*)
6	1, 2, 5	limsupre3 46176	. 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 2739	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
9	8, 4	fvmpt2d 6949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
10	9	breq2d 5084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝑤 ≤ 𝐵))
11	10	anbi2d 636	. . . . . 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 3162	. . . 4 ⊢ (𝜑 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵)))
14	13	rexbidv 3163	. . 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 5082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤 ↔ 𝐵 ≤ 𝑤))
16	15	imbi2d 341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
17	3, 16	ralbida 3250	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
18	17	rexbidv 3163	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤)))
19	18	rexbidv 3163	. . 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 638	. 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 5075	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝐵 ↔ 𝑦 ≤ 𝐵))
22	21	anbi2d 636	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
23	22	rexbidv 3163	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
24	23	ralbidv 3162	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
25		breq1 5075	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 ≤ 𝑥 ↔ 𝑘 ≤ 𝑥))
26	25	anbi1d 637	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
27	26	rexbidv 3163	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵)))
28	27	cbvralvw 3217	. . . . . . 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	cbvrexvw 3218	. . . 4 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 ≤ 𝐵) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵))
32		breq2 5076	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
33	32	imbi2d 341	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
34	33	ralbidv 3162	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
35	34	rexbidv 3163	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
36	25	imbi1d 342	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
37	36	ralbidv 3162	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))
38	37	cbvrexvw 3218	. . . . . . 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	cbvrexvw 3218	. . . 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 634	. . 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 1547  Ⅎwnf 1790   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883   class class class wbr 5072   ↦ cmpt 5153  ‘cfv 6485  ℝcr 11028  ℝ^*cxr 11169   ≤ cle 11171  lim supclsp 15423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-ico 13295  df-limsup 15424

This theorem is referenced by: (None)