Theorem liminfreuz 46955

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 2926	. . 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 46954	. 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 5110	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
7	6	rexbidv 3190	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
8	7	ralbidv 3189	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
9		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
10	9	rexeqdv 3324	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥))
11		liminfreuz.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
12		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
13	11, 12	nffv 6906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
14		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
15		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑥
16	13, 14, 15	nfbr 5155	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
17		nfv 1948	. . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
18		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
19	18	breq1d 5116	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
20	16, 17, 19	cbvrexw 3309	. . . . . . . . . 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 282	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
23	22	cbvralvw 3244	. . . . . . 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 282	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
26	25	cbvrexvw 3245	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
27		breq1 5109	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙)))
28	27	ralbidv 3189	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙)))
29	15, 14, 13	nfbr 5155	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙)
30		nfv 1948	. . . . . . . 8 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗)
31	18	breq2d 5118	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗)))
32	29, 30, 31	cbvralw 3308	. . . . . . 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 282	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
35	34	cbvrexvw 3245	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
36	26, 35	anbi12i 640	. . 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 282	1 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Ⅎwnfc 2911  ∀wral 3080  ∃wrex 3090   class class class wbr 5106  ⟶wf 6543  ‘cfv 6547  ℝcr 11212   ≤ cle 11357  ℤcz 12706  ℤ_≥cuz 12978  lim infclsi 46903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-sup 9447  df-inf 9448  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-div 11987  df-nn 12349  df-n0 12620  df-z 12707  df-uz 12979  df-q 13089  df-xneg 13254  df-ico 13495  df-fz 13653  df-fzo 13802  df-fl 13945  df-ceil 13946  df-limsup 15653  df-liminf 46904

This theorem is used by: (None)