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 1915 | . . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ | |
3 | 1, 2 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ) |
4 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ | |
5 | 3, 4 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) |
6 | limsupub2.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
7 | 6 | ffvelrnda 6851 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*) |
8 | 7 | ad5ant14 756 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ∈ ℝ*) |
9 | rexr 10687 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
10 | 9 | ad4antlr 731 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*) |
11 | pnfxr 10695 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → +∞ ∈ ℝ*) |
13 | simpr 487 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥) | |
14 | ltpnf 12516 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
15 | 14 | ad4antlr 731 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 < +∞) |
16 | 8, 10, 12, 13, 15 | xrlelttrd 12554 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → (𝐹‘𝑗) < +∞) |
17 | 16 | ex 415 | . . . . . 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 3217 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
20 | 19 | reximdva 3274 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
21 | 20 | imp 409 | . 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 42034 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
26 | 21, 25 | r19.29a 3289 | 1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 lim supclsp 14827 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-ico 12745 df-limsup 14828 |
This theorem is referenced by: limsupubuz2 42143 |
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