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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupres | Structured version Visualization version GIF version |
Description: The superior limit of a restriction is less than or equal to the original superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupres.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
limsupres | ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐶)) ≤ (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1912 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | resimass 45184 | . . . . . . . 8 ⊢ ((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ⊆ (𝐹 “ (𝑘[,)+∞)) | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → ((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ⊆ (𝐹 “ (𝑘[,)+∞))) |
4 | 3 | ssrind 4252 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → (((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
5 | 4 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
6 | inss2 4246 | . . . . . 6 ⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ* | |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ*) |
8 | 5, 7 | sstrd 4006 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ*) |
9 | 8 | supxrcld 45047 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
10 | 7 | supxrcld 45047 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
11 | supxrss 13371 | . . . 4 ⊢ (((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ∧ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ*) → sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
12 | 5, 7, 11 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
13 | 1, 9, 10, 12 | infrnmptle 45373 | . 2 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) ≤ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
14 | limsupres.1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
15 | 14 | resexd 6048 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ V) |
16 | eqid 2735 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
17 | 16 | limsupval 15507 | . . . 4 ⊢ ((𝐹 ↾ 𝐶) ∈ V → (lim sup‘(𝐹 ↾ 𝐶)) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
18 | 15, 17 | syl 17 | . . 3 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐶)) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
19 | eqid 2735 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
20 | 19 | limsupval 15507 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
21 | 14, 20 | syl 17 | . . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
22 | 18, 21 | breq12d 5161 | . 2 ⊢ (𝜑 → ((lim sup‘(𝐹 ↾ 𝐶)) ≤ (lim sup‘𝐹) ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ 𝐶) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) ≤ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))) |
23 | 13, 22 | mpbird 257 | 1 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐶)) ≤ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ↾ cres 5691 “ cima 5692 ‘cfv 6563 (class class class)co 7431 supcsup 9478 infcinf 9479 ℝcr 11152 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 [,)cico 13386 lim supclsp 15503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-limsup 15504 |
This theorem is referenced by: (None) |
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