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 5214 | . . 3 ⊢ ∅ ∈ V | |
2 | eqid 2824 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
3 | 2 | limsupval 14834 | . . 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 5949 | . . . . . . . . . 10 ⊢ (∅ “ (𝑥[,)+∞)) = ∅ | |
6 | 5 | ineq1i 4188 | . . . . . . . . 9 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = (∅ ∩ ℝ*) |
7 | 0in 4350 | . . . . . . . . 9 ⊢ (∅ ∩ ℝ*) = ∅ | |
8 | 6, 7 | eqtri 2847 | . . . . . . . 8 ⊢ ((∅ “ (𝑥[,)+∞)) ∩ ℝ*) = ∅ |
9 | 8 | supeq1i 8914 | . . . . . . 7 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(∅, ℝ*, < ) |
10 | xrsup0 12719 | . . . . . . 7 ⊢ sup(∅, ℝ*, < ) = -∞ | |
11 | 9, 10 | eqtri 2847 | . . . . . 6 ⊢ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ) = -∞ |
12 | 11 | mpteq2i 5161 | . . . . 5 ⊢ (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑥 ∈ ℝ ↦ -∞) |
13 | ren0 41681 | . . . . . 6 ⊢ ℝ ≠ ∅ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ ≠ ∅) |
15 | 12, 14 | rnmptc 6972 | . . . 4 ⊢ (⊤ → ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞}) |
16 | 15 | mptru 1543 | . . 3 ⊢ ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )) = {-∞} |
17 | 16 | infeq1i 8945 | . 2 ⊢ inf(ran (𝑥 ∈ ℝ ↦ sup(((∅ “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf({-∞}, ℝ*, < ) |
18 | xrltso 12537 | . . 3 ⊢ < Or ℝ* | |
19 | mnfxr 10701 | . . 3 ⊢ -∞ ∈ ℝ* | |
20 | infsn 8972 | . . 3 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → inf({-∞}, ℝ*, < ) = -∞) | |
21 | 18, 19, 20 | mp2an 690 | . 2 ⊢ inf({-∞}, ℝ*, < ) = -∞ |
22 | 4, 17, 21 | 3eqtri 2851 | 1 ⊢ (lim sup‘∅) = -∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∩ cin 3938 ∅c0 4294 {csn 4570 ↦ cmpt 5149 Or wor 5476 ran crn 5559 “ cima 5561 ‘cfv 6358 (class class class)co 7159 supcsup 8907 infcinf 8908 ℝcr 10539 +∞cpnf 10675 -∞cmnf 10676 ℝ*cxr 10677 < clt 10678 [,)cico 12743 lim supclsp 14830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-limsup 14831 |
This theorem is referenced by: climlimsupcex 42056 liminf0 42080 liminflelimsupcex 42084 |
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