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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsup0 | Structured version Visualization version GIF version |
Description: The superior limit of the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsup0 | ⊢ (lim sup‘∅) = -∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
2 | eqid 2733 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
3 | 2 | limsupval 15415 | . . 3 ⊢ (∅ ∈ V → (lim sup‘∅) = inf(ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (lim sup‘∅) = inf(ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) |
5 | 0ima 6075 | . . . . . . . . . 10 ⊢ (∅ “ (𝑥[,)+∞)) = ∅ | |
6 | 5 | ineq1i 4208 | . . . . . . . . 9 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = (∅ ∩ ℝ*) |
7 | 0in 4393 | . . . . . . . . 9 ⊢ (∅ ∩ ℝ*) = ∅ | |
8 | 6, 7 | eqtri 2761 | . . . . . . . 8 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = ∅ |
9 | 8 | supeq1i 9439 | . . . . . . 7 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(∅, ℝ*, < ) |
10 | xrsup0 13299 | . . . . . . 7 ⊢ sup(∅, ℝ*, < ) = -∞ | |
11 | 9, 10 | eqtri 2761 | . . . . . 6 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = -∞ |
12 | 11 | mpteq2i 5253 | . . . . 5 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ -∞) |
13 | ren0 44099 | . . . . . 6 ⊢ ℝ ≠ ∅ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ ≠ ∅) |
15 | 12, 14 | rnmptc 7205 | . . . 4 ⊢ (⊤ → ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞}) |
16 | 15 | mptru 1549 | . . 3 ⊢ ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞} |
17 | 16 | infeq1i 9470 | . 2 ⊢ inf(ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf({-∞}, ℝ*, < ) |
18 | xrltso 13117 | . . 3 ⊢ < Or ℝ* | |
19 | mnfxr 11268 | . . 3 ⊢ -∞ ∈ ℝ* | |
20 | infsn 9497 | . . 3 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → inf({-∞}, ℝ*, < ) = -∞) | |
21 | 18, 19, 20 | mp2an 691 | . 2 ⊢ inf({-∞}, ℝ*, < ) = -∞ |
22 | 4, 17, 21 | 3eqtri 2765 | 1 ⊢ (lim sup‘∅) = -∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∩ cin 3947 ∅c0 4322 {csn 4628 ↦ cmpt 5231 Or wor 5587 ran crn 5677 “ cima 5679 ‘cfv 6541 (class class class)co 7406 supcsup 9432 infcinf 9433 ℝcr 11106 +∞cpnf 11242 -∞cmnf 11243 ℝ*cxr 11244 < clt 11245 [,)cico 13323 lim supclsp 15411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-limsup 15412 |
This theorem is referenced by: climlimsupcex 44472 liminf0 44496 liminflelimsupcex 44500 |
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