Theorem limsuppnf 46154

Description: If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem limsuppnf

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

Step	Hyp	Ref	Expression
1		nfcv 2901	. . 3 ⊢ Ⅎ𝑙𝐹
2		limsuppnf.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsuppnf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsuppnflem 46153	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
5		breq1 5075	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙))
6	5	anbi1d 637	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
7	6	rexbidv 3163	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
8		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
9		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
10		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
11		limsuppnf.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
12		nfcv 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
13	11, 12	nffv 6837	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
14	9, 10, 13	nfbr 5119	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑦 ≤ (𝐹‘𝑙)
15	8, 14	nfan 1906	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))
16		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))
17		breq2 5076	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
18		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
19	18	breq2d 5084	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑦 ≤ (𝐹‘𝑗)))
20	17, 19	anbi12d 638	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
21	15, 16, 20	cbvrexw 3282	. . . . . . . . 9 ⊢ (∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))
22	21	a1i 11	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
23	7, 22	bitrd 280	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
24	23	cbvralvw 3217	. . . . . 6 ⊢ (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))
25	24	a1i 11	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
26		breq1 5075	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑗) ↔ 𝑥 ≤ (𝐹‘𝑗)))
27	26	anbi2d 636	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
28	27	rexbidv 3163	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
29	28	ralbidv 3162	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
30	25, 29	bitrd 280	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
31	30	cbvralvw 3217	. . 3 ⊢ (∀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	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
33	4, 32	bitrd 280	1 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  Ⅎwnfc 2886  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883   class class class wbr 5072  ⟶wf 6481  ‘cfv 6485  ℝcr 11028  +∞cpnf 11167  ℝ^*cxr 11169   ≤ cle 11171  lim supclsp 15423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-ico 13295  df-limsup 15424

This theorem is referenced by:  limsupre2lem  46167