Theorem liminfreuz 45787

Description: Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfreuz.1	⊢ Ⅎ𝑗𝐹
liminfreuz.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
liminfreuz.3	⊢ 𝑍 = (ℤ≥‘𝑀)
liminfreuz.4	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)

Assertion

Ref	Expression
liminfreuz	⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))

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

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

Proof of Theorem liminfreuz

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

Step	Hyp	Ref	Expression
1		nfcv 2905	. . 3 ⊢ Ⅎ𝑙𝐹
2		liminfreuz.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		liminfreuz.3	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		liminfreuz.4	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
5	1, 2, 3, 4	liminfreuzlem 45786	. 2 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙))))
6		breq2 5155	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
7	6	rexbidv 3179	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
8	7	ralbidv 3178	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
9		fveq2 6914	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
10	9	rexeqdv 3327	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥))
11		liminfreuz.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
12		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
13	11, 12	nffv 6924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
14		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
15		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑥
16	13, 14, 15	nfbr 5198	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
17		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
18		fveq2 6914	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
19	18	breq1d 5161	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
20	16, 17, 19	cbvrexw 3307	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
22	10, 21	bitrd 279	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
23	22	cbvralvw 3237	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
24	23	a1i 11	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
25	8, 24	bitrd 279	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
26	25	cbvrexvw 3238	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
27		breq1 5154	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙)))
28	27	ralbidv 3178	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙)))
29	15, 14, 13	nfbr 5198	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙)
30		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗)
31	18	breq2d 5163	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗)))
32	29, 30, 31	cbvralw 3306	. . . . . . 7 ⊢ (∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
33	32	a1i 11	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
34	28, 33	bitrd 279	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
35	34	cbvrexvw 3238	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
36	26, 35	anbi12i 628	. . 3 ⊢ ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
37	36	a1i 11	. 2 ⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))
38	5, 37	bitrd 279	1 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  ∃wrex 3070   class class class wbr 5151  ⟶wf 6565  ‘cfv 6569  ℝcr 11161   ≤ cle 11303  ℤcz 12620  ℤ_≥cuz 12885  lim infclsi 45735

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761  ax-cnex 11218  ax-resscn 11219  ax-1cn 11220  ax-icn 11221  ax-addcl 11222  ax-addrcl 11223  ax-mulcl 11224  ax-mulrcl 11225  ax-mulcom 11226  ax-addass 11227  ax-mulass 11228  ax-distr 11229  ax-i2m1 11230  ax-1ne0 11231  ax-1rid 11232  ax-rnegex 11233  ax-rrecex 11234  ax-cnre 11235  ax-pre-lttri 11236  ax-pre-lttrn 11237  ax-pre-ltadd 11238  ax-pre-mulgt0 11239  ax-pre-sup 11240

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-isom 6578  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-om 7895  df-1st 8022  df-2nd 8023  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-1o 8514  df-er 8753  df-en 8994  df-dom 8995  df-sdom 8996  df-fin 8997  df-sup 9489  df-inf 9490  df-pnf 11304  df-mnf 11305  df-xr 11306  df-ltxr 11307  df-le 11308  df-sub 11501  df-neg 11502  df-div 11928  df-nn 12274  df-n0 12534  df-z 12621  df-uz 12886  df-q 12998  df-xneg 13161  df-ico 13399  df-fz 13554  df-fzo 13701  df-fl 13838  df-ceil 13839  df-limsup 15513  df-liminf 45736

This theorem is referenced by: (None)