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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 2734 | . . . . 5 ⊢ (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | limsuple 15510 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑖 ∈ ℝ 𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖))) |
7 | oveq1 7437 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑗[,)+∞) = (𝑖[,)+∞)) | |
8 | 7 | imaeq2d 6079 | . . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝐹 “ (𝑗[,)+∞)) = (𝐹 “ (𝑖[,)+∞))) |
9 | 8 | ineq1d 4226 | . . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*)) |
10 | 9 | supeq1d 9483 | . . . . . 6 ⊢ (𝑗 = 𝑖 → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
11 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → 𝑖 ∈ ℝ) | |
12 | xrltso 13179 | . . . . . . . 8 ⊢ < Or ℝ* | |
13 | 12 | supex 9500 | . . . . . . 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 7038 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 15 | breq2d 5159 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) ↔ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
17 | 16 | ralbidva 3173 | . . 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 279 | . 2 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑖 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
19 | oveq1 7437 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝑖[,)+∞) = (𝑘[,)+∞)) | |
20 | 19 | imaeq2d 6079 | . . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) |
21 | 20 | ineq1d 4226 | . . . . . 6 ⊢ (𝑖 = 𝑘 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
22 | 21 | supeq1d 9483 | . . . . 5 ⊢ (𝑖 = 𝑘 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
23 | 22 | breq2d 5159 | . . . 4 ⊢ (𝑖 = 𝑘 → (𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ↔ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
24 | 23 | cbvralvw 3234 | . . 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 279 | 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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 class class class wbr 5147 ↦ cmpt 5230 “ cima 5691 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℝcr 11151 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 [,)cico 13385 lim supclsp 15502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-limsup 15503 |
This theorem is referenced by: (None) |
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