Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupub2 | Structured version Visualization version GIF version |
Description: A extended real valued function, with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
limsupub2.1 | ⊢ Ⅎ𝑗𝜑 |
limsupub2.2 | ⊢ Ⅎ𝑗𝐹 |
limsupub2.3 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
limsupub2.4 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
limsupub2.5 | ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) |
Ref | Expression |
---|---|
limsupub2 | ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupub2.1 | . . . . . . 7 ⊢ Ⅎ𝑗𝜑 | |
2 | nfv 1906 | . . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ | |
3 | 1, 2 | nfan 1891 | . . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ) |
4 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ | |
5 | 3, 4 | nfan 1891 | . . . . 5 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) |
6 | limsupub2.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
7 | 6 | ffvelrnda 6843 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*) |
8 | 7 | ad5ant14 754 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ∈ ℝ*) |
9 | rexr 10675 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
10 | 9 | ad4antlr 729 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*) |
11 | pnfxr 10683 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → +∞ ∈ ℝ*) |
13 | simpr 485 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥) | |
14 | ltpnf 12503 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
15 | 14 | ad4antlr 729 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < +∞) |
16 | 8, 10, 12, 13, 15 | xrlelttrd 12541 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) < +∞) |
17 | 16 | ex 413 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) ≤ 𝑥 → (𝐹‘𝑗) < +∞)) |
18 | 17 | imim2d 57 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
19 | 5, 18 | ralimdaa 3214 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
20 | 19 | reximdva 3271 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
21 | 20 | imp 407 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) |
22 | limsupub2.2 | . . 3 ⊢ Ⅎ𝑗𝐹 | |
23 | limsupub2.3 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
24 | limsupub2.5 | . . 3 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) | |
25 | 1, 22, 23, 6, 24 | limsupub 41861 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
26 | 21, 25 | r19.29a 3286 | 1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 ℝcr 10524 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 lim supclsp 14815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-ico 12732 df-limsup 14816 |
This theorem is referenced by: limsupubuz2 41970 |
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