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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupval3 | Structured version Visualization version GIF version |
Description: The superior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupval3.1 | ⊢ Ⅎ𝑘𝜑 |
limsupval3.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
limsupval3.3 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
limsupval3.4 | ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) |
Ref | Expression |
---|---|
limsupval3 | ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupval3.3 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
2 | limsupval3.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 1, 2 | fexd 6967 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | eqid 2798 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | limsupval 14823 | . . 3 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
7 | limsupval3.4 | . . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))) |
9 | limsupval3.1 | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
10 | 1 | fimassd 41864 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*) |
11 | df-ss 3898 | . . . . . . . . . 10 ⊢ ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) | |
12 | 10, 11 | sylib 221 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) |
13 | 12 | eqcomd 2804 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
14 | 13 | supeq1d 8894 | . . . . . . 7 ⊢ (𝜑 → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
15 | 14 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 9, 15 | mpteq2da 5124 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
17 | 8, 16 | eqtr2d 2834 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺) |
18 | 17 | rneqd 5772 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺) |
19 | 18 | infeq1d 8925 | . 2 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran 𝐺, ℝ*, < )) |
20 | 6, 19 | eqtrd 2833 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ↦ cmpt 5110 ran crn 5520 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 infcinf 8889 ℝcr 10525 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 [,)cico 12728 lim supclsp 14819 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-limsup 14820 |
This theorem is referenced by: limsupmnflem 42362 limsup10ex 42415 |
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