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 5175 | . . 3 ⊢ ∅ ∈ V | |
2 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
3 | 2 | limsupval 14921 | . . 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 5920 | . . . . . . . . . 10 ⊢ (∅ “ (𝑥[,)+∞)) = ∅ | |
6 | 5 | ineq1i 4099 | . . . . . . . . 9 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = (∅ ∩ ℝ*) |
7 | 0in 4282 | . . . . . . . . 9 ⊢ (∅ ∩ ℝ*) = ∅ | |
8 | 6, 7 | eqtri 2761 | . . . . . . . 8 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = ∅ |
9 | 8 | supeq1i 8984 | . . . . . . 7 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(∅, ℝ*, < ) |
10 | xrsup0 12799 | . . . . . . 7 ⊢ sup(∅, ℝ*, < ) = -∞ | |
11 | 9, 10 | eqtri 2761 | . . . . . 6 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = -∞ |
12 | 11 | mpteq2i 5122 | . . . . 5 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ -∞) |
13 | ren0 42480 | . . . . . 6 ⊢ ℝ ≠ ∅ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ ≠ ∅) |
15 | 12, 14 | rnmptc 6979 | . . . 4 ⊢ (⊤ → ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞}) |
16 | 15 | mptru 1549 | . . 3 ⊢ ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞} |
17 | 16 | infeq1i 9015 | . 2 ⊢ inf(ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf({-∞}, ℝ*, < ) |
18 | xrltso 12617 | . . 3 ⊢ < Or ℝ* | |
19 | mnfxr 10776 | . . 3 ⊢ -∞ ∈ ℝ* | |
20 | infsn 9042 | . . 3 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → inf({-∞}, ℝ*, < ) = -∞) | |
21 | 18, 19, 20 | mp2an 692 | . 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 2114 ≠ wne 2934 Vcvv 3398 ∩ cin 3842 ∅c0 4211 {csn 4516 ↦ cmpt 5110 Or wor 5441 ran crn 5526 “ cima 5528 ‘cfv 6339 (class class class)co 7170 supcsup 8977 infcinf 8978 ℝcr 10614 +∞cpnf 10750 -∞cmnf 10751 ℝ*cxr 10752 < clt 10753 [,)cico 12823 lim supclsp 14917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-limsup 14918 |
This theorem is referenced by: climlimsupcex 42852 liminf0 42876 liminflelimsupcex 42880 |
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