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 41428 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | limsupval 14831 | . . 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 41547 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*) |
11 | df-ss 3952 | . . . . . . . . . 10 ⊢ ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) | |
12 | 10, 11 | sylib 220 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) |
13 | 12 | eqcomd 2827 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
14 | 13 | supeq1d 8910 | . . . . . . 7 ⊢ (𝜑 → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
15 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 9, 15 | mpteq2da 5160 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
17 | 8, 16 | eqtr2d 2857 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺) |
18 | 17 | rneqd 5808 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺) |
19 | 18 | infeq1d 8941 | . 2 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran 𝐺, ℝ*, < )) |
20 | 6, 19 | eqtrd 2856 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 ↦ cmpt 5146 ran crn 5556 “ cima 5558 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supcsup 8904 infcinf 8905 ℝcr 10536 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 [,)cico 12741 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-pre-lttri 10611 ax-pre-lttrn 10612 |
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-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-limsup 14828 |
This theorem is referenced by: limsupmnflem 42050 limsup10ex 42103 |
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