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Mirrors > Home > MPE Home > Th. List > Mathboxes > iblempty | Structured version Visualization version GIF version |
Description: The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iblempty | ⊢ ∅ ∈ 𝐿1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbf0 24234 | . 2 ⊢ ∅ ∈ MblFn | |
2 | fconstmpt 5613 | . . . . . . 7 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
3 | 2 | eqcomi 2830 | . . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ 0) = (ℝ × {0}) |
4 | 3 | fveq2i 6672 | . . . . 5 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = (∫2‘(ℝ × {0})) |
5 | itg20 24337 | . . . . 5 ⊢ (∫2‘(ℝ × {0})) = 0 | |
6 | 4, 5 | eqtri 2844 | . . . 4 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = 0 |
7 | 0re 10642 | . . . 4 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | eqeltri 2909 | . . 3 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
9 | 8 | rgenw 3150 | . 2 ⊢ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
10 | noel 4295 | . . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅ | |
11 | 10 | intnanr 490 | . . . . . . . 8 ⊢ ¬ (𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))) |
12 | 11 | iffalsei 4476 | . . . . . . 7 ⊢ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0 |
13 | 12 | eqcomi 2830 | . . . . . 6 ⊢ 0 = if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 = if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) |
15 | 14 | mpteq2dva 5160 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ 0) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) |
16 | eqidd 2822 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘)))) | |
17 | dm0 5789 | . . . . 5 ⊢ dom ∅ = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → dom ∅ = ∅) |
19 | 10 | intnan 489 | . . . . 5 ⊢ ¬ (⊤ ∧ 𝑥 ∈ ∅) |
20 | 19 | pm2.21i 119 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (∅‘𝑥) = 0) |
21 | 15, 16, 18, 20 | isibl 24365 | . . 3 ⊢ (⊤ → (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ))) |
22 | 21 | mptru 1540 | . 2 ⊢ (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ)) |
23 | 1, 9, 22 | mpbir2an 709 | 1 ⊢ ∅ ∈ 𝐿1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ∀wral 3138 ∅c0 4290 ifcif 4466 {csn 4566 class class class wbr 5065 ↦ cmpt 5145 × cxp 5552 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 0cc0 10536 ici 10538 ≤ cle 10675 / cdiv 11296 3c3 11692 ...cfz 12891 ↑cexp 13428 ℜcre 14455 MblFncmbf 24214 ∫2citg2 24216 𝐿1cibl 24217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xadd 12507 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-xmet 20537 df-met 20538 df-ovol 24064 df-vol 24065 df-mbf 24219 df-itg1 24220 df-itg2 24221 df-ibl 24222 df-0p 24270 |
This theorem is referenced by: itgvol0 42251 |
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