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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 1917 | . . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ | |
3 | 1, 2 | nfan 1902 | . . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ) |
4 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ ℝ | |
5 | 3, 4 | nfan 1902 | . . . . 5 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) |
6 | limsupub2.4 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
7 | 6 | ffvelcdmda 7072 | . . . . . . . . 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 11244 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
10 | 9 | ad4antlr 731 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝐹‘𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*) |
11 | pnfxr 11252 | . . . . . . . . 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 13084 | . . . . . . . . 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 13123 | . . . . . . 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 3257 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞))) |
20 | 19 | reximdva 3168 | . . 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 44257 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
26 | 21, 25 | r19.29a 3162 | 1 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3945 class class class wbr 5142 ⟶wf 6529 ‘cfv 6533 ℝcr 11093 +∞cpnf 11229 ℝ*cxr 11231 < clt 11232 ≤ cle 11233 lim supclsp 15398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-sup 9421 df-inf 9422 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-ico 13314 df-limsup 15399 |
This theorem is referenced by: limsupubuz2 44366 |
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