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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 1874 | . . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ | |
3 | 1, 2 | nfan 1863 | . . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ) |
4 | nfv 1874 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ | |
5 | 3, 4 | nfan 1863 | . . . . 5 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) |
6 | limsupub2.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
7 | 6 | ffvelrnda 6674 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*) |
8 | 7 | ad5ant14 746 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ∈ ℝ*) |
9 | rexr 10484 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
10 | 9 | ad4antlr 721 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*) |
11 | pnfxr 10492 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → +∞ ∈ ℝ*) |
13 | simpr 477 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥) | |
14 | ltpnf 12330 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
15 | 14 | ad4antlr 721 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < +∞) |
16 | 8, 10, 12, 13, 15 | xrlelttrd 12368 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) < +∞) |
17 | 16 | ex 405 | . . . . . 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 3160 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
20 | 19 | reximdva 3212 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
21 | 20 | imp 398 | . 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 41450 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
26 | 21, 25 | r19.29a 3227 | 1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 Ⅎwnf 1747 ∈ wcel 2051 Ⅎwnfc 2909 ≠ wne 2960 ∀wral 3081 ∃wrex 3082 ⊆ wss 3822 class class class wbr 4925 ⟶wf 6181 ‘cfv 6185 ℝcr 10332 +∞cpnf 10469 ℝ*cxr 10471 < clt 10472 ≤ cle 10473 lim supclsp 14686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-ico 12558 df-limsup 14687 |
This theorem is referenced by: limsupubuz2 41559 |
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