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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 2733 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
3 | 2 | limsupval 15362 | . . 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 1918 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
6 | inss2 4190 | . . . . 5 ⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ* | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ*) |
8 | 7 | supxrcld 43405 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
9 | limsuplesup.2 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ) | |
10 | inss2 4190 | . . . . 5 ⊢ ((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*) ⊆ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*) ⊆ ℝ*) |
12 | 11 | supxrcld 43405 | . . 3 ⊢ (𝜑 → sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
13 | oveq1 7365 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘[,)+∞) = (𝐾[,)+∞)) | |
14 | 13 | imaeq2d 6014 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝐾[,)+∞))) |
15 | 14 | ineq1d 4172 | . . . 4 ⊢ (𝑘 = 𝐾 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*)) |
16 | 15 | supeq1d 9387 | . . 3 ⊢ (𝑘 = 𝐾 → sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
17 | 5, 8, 9, 12, 16 | infxrlbrnmpt2 43731 | . 2 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
18 | 4, 17 | eqbrtrd 5128 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5106 ↦ cmpt 5189 ran crn 5635 “ cima 5637 ‘cfv 6497 (class class class)co 7358 supcsup 9381 infcinf 9382 ℝcr 11055 +∞cpnf 11191 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 [,)cico 13272 lim supclsp 15358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-limsup 15359 |
This theorem is referenced by: (None) |
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