Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupge | Structured version Visualization version GIF version |
Description: The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupge.b | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
limsupge.f | ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) |
limsupge.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
limsupge | ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupge.b | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
2 | limsupge.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) | |
3 | limsupge.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | eqid 2736 | . . . . 5 ⊢ (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | limsuple 15004 | . . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑖 ∈ ℝ 𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖))) |
6 | 1, 2, 3, 5 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑖 ∈ ℝ 𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖))) |
7 | oveq1 7198 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑗[,)+∞) = (𝑖[,)+∞)) | |
8 | 7 | imaeq2d 5914 | . . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝐹 “ (𝑗[,)+∞)) = (𝐹 “ (𝑖[,)+∞))) |
9 | 8 | ineq1d 4112 | . . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*)) |
10 | 9 | supeq1d 9040 | . . . . . 6 ⊢ (𝑗 = 𝑖 → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
11 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → 𝑖 ∈ ℝ) | |
12 | xrltso 12696 | . . . . . . . 8 ⊢ < Or ℝ* | |
13 | 12 | supex 9057 | . . . . . . 7 ⊢ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V) |
15 | 4, 10, 11, 14 | fvmptd3 6819 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 15 | breq2d 5051 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) ↔ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
17 | 16 | ralbidva 3107 | . . 3 ⊢ (𝜑 → (∀𝑖 ∈ ℝ 𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) ↔ ∀𝑖 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
18 | 6, 17 | bitrd 282 | . 2 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑖 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
19 | oveq1 7198 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝑖[,)+∞) = (𝑘[,)+∞)) | |
20 | 19 | imaeq2d 5914 | . . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) |
21 | 20 | ineq1d 4112 | . . . . . 6 ⊢ (𝑖 = 𝑘 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
22 | 21 | supeq1d 9040 | . . . . 5 ⊢ (𝑖 = 𝑘 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
23 | 22 | breq2d 5051 | . . . 4 ⊢ (𝑖 = 𝑘 → (𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ↔ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
24 | 23 | cbvralvw 3348 | . . 3 ⊢ (∀𝑖 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
25 | 24 | a1i 11 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
26 | 18, 25 | bitrd 282 | 1 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 class class class wbr 5039 ↦ cmpt 5120 “ cima 5539 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 supcsup 9034 ℝcr 10693 +∞cpnf 10829 ℝ*cxr 10831 < clt 10832 ≤ cle 10833 [,)cico 12902 lim supclsp 14996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-limsup 14997 |
This theorem is referenced by: (None) |
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