Theorem limsuppnf 45814

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 2894	. . 3 ⊢ Ⅎ𝑙𝐹
2		limsuppnf.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsuppnf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsuppnflem 45813	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
5		breq1 5096	. . . . . . . . . 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 3156	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
8		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
9		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
10		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
11		limsuppnf.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
12		nfcv 2894	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
13	11, 12	nffv 6838	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
14	9, 10, 13	nfbr 5140	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑦 ≤ (𝐹‘𝑙)
15	8, 14	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))
16		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))
17		breq2 5097	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
18		fveq2 6828	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
19	18	breq2d 5105	. . . . . . . . . . 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 3275	. . . . . . . . 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 3210	. . . . . 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 5096	. . . . . . . 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 3156	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
29	28	ralbidv 3155	. . . . 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 3210	. . 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 1541  Ⅎwnfc 2879  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897   class class class wbr 5093  ⟶wf 6483  ‘cfv 6487  ℝcr 11011  +∞cpnf 11149  ℝ^*cxr 11151   ≤ cle 11153  lim supclsp 15383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089  ax-pre-sup 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-sup 9332  df-inf 9333  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-ico 13257  df-limsup 15384

This theorem is referenced by:  limsupre2lem  45827