Theorem limsuppnf 45659

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 2897	. . 3 ⊢ Ⅎ𝑙𝐹
2		limsuppnf.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsuppnf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsuppnflem 45658	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
5		breq1 5126	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙))
6	5	anbi1d 631	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
7	6	rexbidv 3166	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
8		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
9		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
10		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
11		limsuppnf.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
12		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
13	11, 12	nffv 6895	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
14	9, 10, 13	nfbr 5170	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑦 ≤ (𝐹‘𝑙)
15	8, 14	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))
16		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))
17		breq2 5127	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
18		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
19	18	breq2d 5135	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑦 ≤ (𝐹‘𝑗)))
20	17, 19	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
21	15, 16, 20	cbvrexw 3290	. . . . . . . . 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 279	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
24	23	cbvralvw 3223	. . . . . 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 5126	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑗) ↔ 𝑥 ≤ (𝐹‘𝑗)))
27	26	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
28	27	rexbidv 3166	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
29	28	ralbidv 3165	. . . . 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 279	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
31	30	cbvralvw 3223	. . 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 279	1 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  Ⅎwnfc 2882  ∀wral 3050  ∃wrex 3059   ⊆ wss 3931   class class class wbr 5123  ⟶wf 6536  ‘cfv 6540  ℝcr 11135  +∞cpnf 11273  ℝ^*cxr 11275   ≤ cle 11277  lim supclsp 15487

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 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-po 5572  df-so 5573  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-sup 9463  df-inf 9464  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11475  df-neg 11476  df-ico 13374  df-limsup 15488

This theorem is referenced by:  limsupre2lem  45672