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Mirrors > Home > MPE Home > Th. List > ibl0 | Structured version Visualization version GIF version |
Description: The zero function is integrable on any measurable set. (Unlike iblconst 24412, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
ibl0 | ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10627 | . . 3 ⊢ 0 ∈ ℂ | |
2 | mbfconst 24228 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 0 ∈ ℂ) → (𝐴 × {0}) ∈ MblFn) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ MblFn) |
4 | ax-icn 10590 | . . . . . . . 8 ⊢ i ∈ ℂ | |
5 | ine0 11069 | . . . . . . . 8 ⊢ i ≠ 0 | |
6 | elfzelz 12902 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
7 | 6 | ad2antlr 725 | . . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ) |
8 | expclz 13448 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ) | |
9 | expne0i 13455 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) | |
10 | 8, 9 | div0d 11409 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (0 / (i↑𝑘)) = 0) |
11 | 4, 5, 7, 10 | mp3an12i 1461 | . . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (0 / (i↑𝑘)) = 0) |
12 | 11 | fveq2d 6668 | . . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘0)) |
13 | re0 14505 | . . . . . 6 ⊢ (ℜ‘0) = 0 | |
14 | 12, 13 | syl6eq 2872 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(0 / (i↑𝑘))) = 0) |
15 | 14 | itgvallem3 24380 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) = 0) |
16 | 0re 10637 | . . . 4 ⊢ 0 ∈ ℝ | |
17 | 15, 16 | eqeltrdi 2921 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) ∈ ℝ) |
18 | 17 | ralrimiva 3182 | . 2 ⊢ (𝐴 ∈ dom vol → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) ∈ ℝ) |
19 | eqidd 2822 | . . 3 ⊢ (𝐴 ∈ dom vol → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) | |
20 | eqidd 2822 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘)))) | |
21 | c0ex 10629 | . . . . 5 ⊢ 0 ∈ V | |
22 | 21 | fconst 6559 | . . . 4 ⊢ (𝐴 × {0}):𝐴⟶{0} |
23 | fdm 6516 | . . . 4 ⊢ ((𝐴 × {0}):𝐴⟶{0} → dom (𝐴 × {0}) = 𝐴) | |
24 | 22, 23 | mp1i 13 | . . 3 ⊢ (𝐴 ∈ dom vol → dom (𝐴 × {0}) = 𝐴) |
25 | 21 | fvconst2 6960 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
26 | 25 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
27 | 19, 20, 24, 26 | isibl 24360 | . 2 ⊢ (𝐴 ∈ dom vol → ((𝐴 × {0}) ∈ 𝐿1 ↔ ((𝐴 × {0}) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) ∈ ℝ))) |
28 | 3, 18, 27 | mpbir2and 711 | 1 ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ifcif 4466 {csn 4560 class class class wbr 5058 ↦ cmpt 5138 × cxp 5547 dom cdm 5549 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 ici 10533 ≤ cle 10670 / cdiv 11291 3c3 11687 ℤcz 11975 ...cfz 12886 ↑cexp 13423 ℜcre 14450 volcvol 24058 MblFncmbf 24209 ∫2citg2 24211 𝐿1cibl 24212 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 df-ibl 24217 df-0p 24265 |
This theorem is referenced by: itgge0 24405 itgfsum 24421 bddiblnc 34956 |
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