Theorem limsupmnf 42000

Description: The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupmnf.j	⊢ Ⅎ𝑗𝐹
limsupmnf.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupmnf.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupmnf	⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀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 limsupmnf

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

Step	Hyp	Ref	Expression
1		limsupmnf.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		limsupmnf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
3		eqid 2821	. . 3 ⊢ (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < ))
4	1, 2, 3	limsupmnflem 41999	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)))
5		breq2 5069	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
6	5	imbi2d 343	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
7	6	ralbidv 3197	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
8	7	rexbidv 3297	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
9		breq1 5068	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙))
10	9	imbi1d 344	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
11	10	ralbidv 3197	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
12		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
13		limsupmnf.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
14		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
15	13, 14	nffv 6679	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
16		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
17		nfcv 2977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑥
18	15, 16, 17	nfbr 5112	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
19	12, 18	nfim 1893	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)
20		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
21		breq2 5069	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
22		fveq2 6669	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
23	22	breq1d 5075	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
24	21, 23	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
25	19, 20, 24	cbvralw 3441	. . . . . . . . 9 ⊢ (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
27	11, 26	bitrd 281	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
28	27	cbvrexvw 3450	. . . . . 6 ⊢ (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
29	28	a1i 11	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
30	8, 29	bitrd 281	. . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
31	30	cbvralvw 3449	. . 3 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
32	31	a1i 11	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
33	4, 32	bitrd 281	1 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533  Ⅎwnfc 2961  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935   class class class wbr 5065   ↦ cmpt 5145   “ cima 5557  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  supcsup 8903  ℝcr 10535  +∞cpnf 10671  -∞cmnf 10672  ℝ^*cxr 10673   < clt 10674   ≤ cle 10675  [,)cico 12739  lim supclsp 14826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-ico 12743  df-limsup 14827

This theorem is referenced by:  limsupre2lem  42003  limsupmnfuzlem  42005