Theorem limsupre3 45731

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
limsupre3.1	⊢ Ⅎ𝑗𝐹
limsupre3.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre3.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupre3	⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))

Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)

Proof of Theorem limsupre3

Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2891	. . 3 ⊢ Ⅎ𝑙𝐹
2		limsupre3.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsupre3.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsupre3lem 45730	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))))
5		breq1 5110	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙)))
6	5	anbi2d 630	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙))))
7	6	rexbidv 3157	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙))))
8	7	ralbidv 3156	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙))))
9		breq1 5110	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙))
10	9	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙))))
11	10	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙))))
12		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
13		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑥
14		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 ≤
15		limsupre3.1	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
16		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑙
17	15, 16	nffv 6868	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑙)
18	13, 14, 17	nfbr 5154	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙)
19	12, 18	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙))
20		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))
21		breq2 5111	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
22		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
23	22	breq2d 5119	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗)))
24	21, 23	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
25	19, 20, 24	cbvrexw 3281	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
26	25	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
27	11, 26	bitrd 279	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
28	27	cbvralvw 3215	. . . . . . 7 ⊢ (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
29	28	a1i 11	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
30	8, 29	bitrd 279	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
31	30	cbvrexvw 3216	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
32		breq2 5111	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
33	32	imbi2d 340	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
34	33	ralbidv 3156	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
35	34	rexbidv 3157	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
36	9	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
37	36	ralbidv 3156	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
38	17, 14, 13	nfbr 5154	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
39	12, 38	nfim 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)
40		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
41	22	breq1d 5117	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
42	21, 41	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
43	39, 40, 42	cbvralw 3280	. . . . . . . . . 10 ⊢ (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
44	43	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
45	37, 44	bitrd 279	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
46	45	cbvrexvw 3216	. . . . . . 7 ⊢ (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
47	46	a1i 11	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
48	35, 47	bitrd 279	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
49	48	cbvrexvw 3216	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
50	31, 49	anbi12i 628	. . 3 ⊢ ((∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
51	50	a1i 11	. 2 ⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))
52	4, 51	bitrd 279	1 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914   class class class wbr 5107  ⟶wf 6507  ‘cfv 6511  ℝcr 11067  ℝ^*cxr 11207   ≤ cle 11209  lim supclsp 15436

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-ico 13312  df-limsup 15437

This theorem is referenced by:  limsupre3mpt  45732  limsupre3uzlem  45733