| 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 2737 | . . . . 5 ⊢ (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
| 5 | 4 | limsuple 15514 | . . . 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 7438 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑗[,)+∞) = (𝑖[,)+∞)) | |
| 8 | 7 | imaeq2d 6078 | . . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝐹 “ (𝑗[,)+∞)) = (𝐹 “ (𝑖[,)+∞))) |
| 9 | 8 | ineq1d 4219 | . . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*)) |
| 10 | 9 | supeq1d 9486 | . . . . . 6 ⊢ (𝑗 = 𝑖 → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
| 11 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → 𝑖 ∈ ℝ) | |
| 12 | xrltso 13183 | . . . . . . . 8 ⊢ < Or ℝ* | |
| 13 | 12 | supex 9503 | . . . . . . 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 7039 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
| 16 | 15 | breq2d 5155 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (𝐴 ≤ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) ↔ 𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
| 17 | 16 | ralbidva 3176 | . . 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 7438 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝑖[,)+∞) = (𝑘[,)+∞)) | |
| 20 | 19 | imaeq2d 6078 | . . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) |
| 21 | 20 | ineq1d 4219 | . . . . . 6 ⊢ (𝑖 = 𝑘 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
| 22 | 21 | supeq1d 9486 | . . . . 5 ⊢ (𝑖 = 𝑘 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
| 23 | 22 | breq2d 5155 | . . . 4 ⊢ (𝑖 = 𝑘 → (𝐴 ≤ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ↔ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
| 24 | 23 | cbvralvw 3237 | . . 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 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 ℝcr 11154 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 lim supclsp 15506 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-limsup 15507 |
| This theorem is referenced by: (None) |
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