Theorem limsupre2mpt 45712

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 smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupre2mpt.p	⊢ Ⅎ𝑥𝜑
limsupre2mpt.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre2mpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)

Assertion

Ref	Expression
limsupre2mpt	⊢ (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 limsupre2mpt

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

Step	Hyp	Ref	Expression
1		nfmpt1 5194	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
2		limsupre2mpt.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsupre2mpt.p	. . . 4 ⊢ Ⅎ𝑥𝜑
4		limsupre2mpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	3, 4	fmptd2f 45213	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*)
6	1, 2, 5	limsupre2 45707	. 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 2729	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
9	8, 4	fvmpt2d 6947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
10	9	breq2d 5107	. . . . . . 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 3241	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < 𝐵)))
13	12	ralbidv 3152	. . . 4 ⊢ (𝜑 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < 𝐵)))
14	13	rexbidv 3153	. . 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 5105	. . . . . . 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 3240	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 < 𝑤)))
18	17	rexbidv 3153	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑤) ↔ ∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 < 𝑤)))
19	18	rexbidv 3153	. . 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 5098	. . . . . . . . 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 3153	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 < 𝐵)))
24	23	ralbidv 3152	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < 𝐵) ↔ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 < 𝐵)))
25		breq1 5098	. . . . . . . . . 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 3153	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑦 < 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 < 𝐵)))
28	27	cbvralvw 3207	. . . . . . 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 279	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < 𝐵) ↔ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 < 𝐵)))
31	30	cbvrexvw 3208	. . . 4 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 ∧ 𝑤 < 𝐵) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 < 𝐵))
32		breq2 5099	. . . . . . . . 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 3152	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 < 𝑤) ↔ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 < 𝑦)))
35	34	rexbidv 3153	. . . . . 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 3152	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 < 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 < 𝑦)))
38	37	cbvrexvw 3208	. . . . . . 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 279	. . . . 5 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑗 ≤ 𝑥 → 𝐵 < 𝑤) ↔ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 < 𝑦)))
41	40	cbvrexvw 3208	. . . 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 305	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 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486  ℝcr 11027  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169  lim supclsp 15395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-ico 13272  df-limsup 15396

This theorem is referenced by: (None)