Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupresre | Structured version Visualization version GIF version |
Description: The supremum limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupresre.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
limsupresre | ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ) | |
2 | pnfxr 10695 | . . . . . . . . . . 11 ⊢ +∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℝ → +∞ ∈ ℝ*) |
4 | icossre 12818 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑘[,)+∞) ⊆ ℝ) | |
5 | 1, 3, 4 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → (𝑘[,)+∞) ⊆ ℝ) |
6 | resima2 5888 | . . . . . . . . 9 ⊢ ((𝑘[,)+∞) ⊆ ℝ → ((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → ((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) |
8 | 7 | ineq1d 4188 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → (((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
9 | 8 | supeq1d 8910 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
10 | 9 | mpteq2ia 5157 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
12 | 11 | rneqd 5808 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
13 | 12 | infeq1d 8941 | . 2 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
14 | limsupresre.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
15 | 14 | resexd 41423 | . . 3 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ V) |
16 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
17 | 16 | limsupval 14831 | . . 3 ⊢ ((𝐹 ↾ ℝ) ∈ V → (lim sup‘(𝐹 ↾ ℝ)) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
18 | 15, 17 | syl 17 | . 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝐹 ↾ ℝ) “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
19 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
20 | 19 | limsupval 14831 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
21 | 14, 20 | syl 17 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
22 | 13, 18, 21 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 “ cima 5558 ‘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-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-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-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-ov 7159 df-oprab 7160 df-mpo 7161 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-le 10681 df-ico 12745 df-limsup 14828 |
This theorem is referenced by: limsupresuz 42004 |
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