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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 25838, 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 11256 | . . 3 ⊢ 0 ∈ ℂ | |
2 | mbfconst 25653 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 0 ∈ ℂ) → (𝐴 × {0}) ∈ MblFn) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ MblFn) |
4 | ax-icn 11217 | . . . . . . . 8 ⊢ i ∈ ℂ | |
5 | ine0 11699 | . . . . . . . 8 ⊢ i ≠ 0 | |
6 | elfzelz 13555 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
7 | 6 | ad2antlr 725 | . . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ) |
8 | expclz 14104 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ) | |
9 | expne0i 14114 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) | |
10 | 8, 9 | div0d 12040 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (0 / (i↑𝑘)) = 0) |
11 | 4, 5, 7, 10 | mp3an12i 1462 | . . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (0 / (i↑𝑘)) = 0) |
12 | 11 | fveq2d 6905 | . . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘0)) |
13 | re0 15157 | . . . . . 6 ⊢ (ℜ‘0) = 0 | |
14 | 12, 13 | eqtrdi 2782 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(0 / (i↑𝑘))) = 0) |
15 | 14 | itgvallem3 25806 | . . . 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 11266 | . . . 4 ⊢ 0 ∈ ℝ | |
17 | 15, 16 | eqeltrdi 2834 | . . 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 3136 | . 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 2727 | . . 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 2727 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘)))) | |
21 | c0ex 11258 | . . . . 5 ⊢ 0 ∈ V | |
22 | 21 | fconst 6788 | . . . 4 ⊢ (𝐴 × {0}):𝐴⟶{0} |
23 | fdm 6737 | . . . 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 7221 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
26 | 25 | adantl 480 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
27 | 19, 20, 24, 26 | isibl 25786 | . 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ifcif 4533 {csn 4633 class class class wbr 5153 ↦ cmpt 5236 × cxp 5680 dom cdm 5682 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 ici 11160 ≤ cle 11299 / cdiv 11921 3c3 12320 ℤcz 12610 ...cfz 13538 ↑cexp 14081 ℜcre 15102 volcvol 25483 MblFncmbf 25634 ∫2citg2 25636 𝐿1cibl 25637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-disj 5119 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-xadd 13147 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 df-xmet 21336 df-met 21337 df-ovol 25484 df-vol 25485 df-mbf 25639 df-itg1 25640 df-itg2 25641 df-ibl 25642 df-0p 25690 |
This theorem is referenced by: itgge0 25831 itgfsum 25847 bddiblnc 25862 |
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