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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsuplesup | Structured version Visualization version GIF version |
Description: An upper bound for the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsuplesup.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
limsuplesup.2 | ⊢ (𝜑 → 𝐾 ∈ ℝ) |
Ref | Expression |
---|---|
limsuplesup | ⊢ (𝜑 → (lim sup‘𝐹) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsuplesup.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | eqid 2732 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
3 | 2 | limsupval 15414 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
5 | nfv 1917 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
6 | inss2 4228 | . . . . 5 ⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ* | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ*) |
8 | 7 | supxrcld 43781 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
9 | limsuplesup.2 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ) | |
10 | inss2 4228 | . . . . 5 ⊢ ((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*) ⊆ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*) ⊆ ℝ*) |
12 | 11 | supxrcld 43781 | . . 3 ⊢ (𝜑 → sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
13 | oveq1 7412 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘[,)+∞) = (𝐾[,)+∞)) | |
14 | 13 | imaeq2d 6057 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝐾[,)+∞))) |
15 | 14 | ineq1d 4210 | . . . 4 ⊢ (𝑘 = 𝐾 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*)) |
16 | 15 | supeq1d 9437 | . . 3 ⊢ (𝑘 = 𝐾 → sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
17 | 5, 8, 9, 12, 16 | infxrlbrnmpt2 44106 | . 2 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
18 | 4, 17 | eqbrtrd 5169 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 “ cima 5678 ‘cfv 6540 (class class class)co 7405 supcsup 9431 infcinf 9432 ℝcr 11105 +∞cpnf 11241 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 [,)cico 13322 lim supclsp 15410 |
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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-limsup 15411 |
This theorem is referenced by: (None) |
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