Theorem limsuppnf 46994

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 2926	. . 3 ⊢ Ⅎ𝑙𝐹
2		limsuppnf.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		limsuppnf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
4	1, 2, 3	limsuppnflem 46993	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
5		breq1 5111	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙))
6	5	anbi1d 643	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
7	6	rexbidv 3190	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))))
8		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
9		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
10		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
11		limsuppnf.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
12		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
13	11, 12	nffv 6926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
14	9, 10, 13	nfbr 5158	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑦 ≤ (𝐹‘𝑙)
15	8, 14	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))
16		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))
17		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
18		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
19	18	breq2d 5120	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑦 ≤ (𝐹‘𝑗)))
20	17, 19	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
21	15, 16, 20	cbvrexw 3309	. . . . . . . . 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 282	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗))))
24	23	cbvralvw 3244	. . . . . 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 5111	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑗) ↔ 𝑥 ≤ (𝐹‘𝑗)))
27	26	anbi2d 642	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
28	27	rexbidv 3190	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
29	28	ralbidv 3189	. . . . 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 282	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
31	30	cbvralvw 3244	. . 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 282	1 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnfc 2911  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108  ⟶wf 6562  ‘cfv 6566  ℝcr 11238  +∞cpnf 11379  ℝ^*cxr 11381   ≤ cle 11383  lim supclsp 15681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  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-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 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-po 5566  df-so 5567  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-ico 13522  df-limsup 15682

This theorem is used by:  limsupre2lem  47007