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 5213 | . . 3 ⊢ ∅ ∈ V | |
2 | eqid 2823 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
3 | 2 | limsupval 14833 | . . 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 5948 | . . . . . . . . . 10 ⊢ (∅ “ (𝑥[,)+∞)) = ∅ | |
6 | 5 | ineq1i 4187 | . . . . . . . . 9 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = (∅ ∩ ℝ*) |
7 | 0in 4349 | . . . . . . . . 9 ⊢ (∅ ∩ ℝ*) = ∅ | |
8 | 6, 7 | eqtri 2846 | . . . . . . . 8 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = ∅ |
9 | 8 | supeq1i 8913 | . . . . . . 7 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(∅, ℝ*, < ) |
10 | xrsup0 12719 | . . . . . . 7 ⊢ sup(∅, ℝ*, < ) = -∞ | |
11 | 9, 10 | eqtri 2846 | . . . . . 6 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = -∞ |
12 | 11 | mpteq2i 5160 | . . . . 5 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ -∞) |
13 | ren0 41682 | . . . . . 6 ⊢ ℝ ≠ ∅ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ ≠ ∅) |
15 | 12, 14 | rnmptc 6971 | . . . 4 ⊢ (⊤ → ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞}) |
16 | 15 | mptru 1544 | . . 3 ⊢ ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞} |
17 | 16 | infeq1i 8944 | . 2 ⊢ inf(ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf({-∞}, ℝ*, < ) |
18 | xrltso 12537 | . . 3 ⊢ < Or ℝ* | |
19 | mnfxr 10700 | . . 3 ⊢ -∞ ∈ ℝ* | |
20 | infsn 8971 | . . 3 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → inf({-∞}, ℝ*, < ) = -∞) | |
21 | 18, 19, 20 | mp2an 690 | . 2 ⊢ inf({-∞}, ℝ*, < ) = -∞ |
22 | 4, 17, 21 | 3eqtri 2850 | 1 ⊢ (lim sup‘∅) = -∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∩ cin 3937 ∅c0 4293 {csn 4569 ↦ cmpt 5148 Or wor 5475 ran crn 5558 “ cima 5560 ‘cfv 6357 (class class class)co 7158 supcsup 8906 infcinf 8907 ℝcr 10538 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 [,)cico 12743 lim supclsp 14829 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-limsup 14830 |
This theorem is referenced by: climlimsupcex 42057 liminf0 42081 liminflelimsupcex 42085 |
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