Theorem limsupre2 45745

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
limsupre2.1	⊢ Ⅎ𝑗𝐹
limsupre2.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre2.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupre2	⊢ (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 limsupre2

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

Step	Hyp	Ref	Expression
1		nfcv 2904	. . 3 ⊢ Ⅎ𝑙𝐹
2		limsupre2.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsupre2.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsupre2lem 45744	. 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 5145	. . . . . . . . 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 3178	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 < (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 < (𝐹‘𝑙))))
8	7	ralbidv 3177	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 < (𝐹‘𝑙)) ↔ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 < (𝐹‘𝑙))))
9		breq1 5145	. . . . . . . . . . 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 3178	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑥 < (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑥 < (𝐹‘𝑙))))
12		nfv 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
13		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑥
14		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 <
15		limsupre2.1	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
16		nfcv 2904	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑙
17	15, 16	nffv 6915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑙)
18	13, 14, 17	nfbr 5189	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑥 < (𝐹‘𝑙)
19	12, 18	nfan 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 ∧ 𝑥 < (𝐹‘𝑙))
20		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))
21		breq2 5146	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
22		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
23	22	breq2d 5154	. . . . . . . . . . . 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 3306	. . . . . . . . . 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 3236	. . . . . . 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 3237	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 < (𝐹‘𝑙)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)))
32	31	a1i 11	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 < (𝐹‘𝑙)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
33		breq2 5146	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) < 𝑦 ↔ (𝐹‘𝑙) < 𝑥))
34	33	imbi2d 340	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑦) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥)))
35	34	ralbidv 3177	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑦) ↔ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥)))
36	35	rexbidv 3178	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥)))
37	9	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ (𝑘 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥)))
38	37	ralbidv 3177	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ ∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥)))
39	17, 14, 13	nfbr 5189	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) < 𝑥
40	12, 39	nfim 1895	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥)
41		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)
42	22	breq1d 5152	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) < 𝑥 ↔ (𝐹‘𝑗) < 𝑥))
43	21, 42	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
44	40, 41, 43	cbvralw 3305	. . . . . . . . . 10 ⊢ (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))
45	44	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
46	38, 45	bitrd 279	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
47	46	cbvrexvw 3237	. . . . . . 7 ⊢ (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))
48	47	a1i 11	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
49	36, 48	bitrd 279	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
50	49	cbvrexvw 3237	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))
51	50	a1i 11	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
52	32, 51	anbi12d 632	. 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))))
53	4, 52	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 1539   ∈ wcel 2107  Ⅎwnfc 2889  ∀wral 3060  ∃wrex 3069   ⊆ wss 3950   class class class wbr 5142  ⟶wf 6556  ‘cfv 6560  ℝcr 11155  ℝ^*cxr 11295   < clt 11296   ≤ cle 11297  lim supclsp 15507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-ico 13394  df-limsup 15508

This theorem is referenced by:  limsupre2mpt  45750  limsupre3lem  45752