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Mirrors > Home > MPE Home > Th. List > itg0 | Structured version Visualization version GIF version |
Description: The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
itg0 | ⊢ ∫∅𝐴 d𝑥 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (ℜ‘(𝐴 / (i↑𝑘))) = (ℜ‘(𝐴 / (i↑𝑘))) | |
2 | 1 | dfitg 24621 | . 2 ⊢ ∫∅𝐴 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)))) |
3 | ifan 4478 | . . . . . . . . . . 11 ⊢ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0) = if(𝑥 ∈ ∅, if(0 ≤ (ℜ‘(𝐴 / (i↑𝑘))), (ℜ‘(𝐴 / (i↑𝑘))), 0), 0) | |
4 | noel 4231 | . . . . . . . . . . . 12 ⊢ ¬ 𝑥 ∈ ∅ | |
5 | 4 | iffalsei 4435 | . . . . . . . . . . 11 ⊢ if(𝑥 ∈ ∅, if(0 ≤ (ℜ‘(𝐴 / (i↑𝑘))), (ℜ‘(𝐴 / (i↑𝑘))), 0), 0) = 0 |
6 | 3, 5 | eqtri 2759 | . . . . . . . . . 10 ⊢ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0) = 0 |
7 | 6 | mpteq2i 5132 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ 0) |
8 | fconstmpt 5596 | . . . . . . . . 9 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
9 | 7, 8 | eqtr4i 2762 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)) = (ℝ × {0}) |
10 | 9 | fveq2i 6698 | . . . . . . 7 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0))) = (∫2‘(ℝ × {0})) |
11 | itg20 24589 | . . . . . . 7 ⊢ (∫2‘(ℝ × {0})) = 0 | |
12 | 10, 11 | eqtri 2759 | . . . . . 6 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0))) = 0 |
13 | 12 | oveq2i 7202 | . . . . 5 ⊢ ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)))) = ((i↑𝑘) · 0) |
14 | ax-icn 10753 | . . . . . . 7 ⊢ i ∈ ℂ | |
15 | elfznn0 13170 | . . . . . . 7 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
16 | expcl 13618 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
17 | 14, 15, 16 | sylancr 590 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ) |
18 | 17 | mul01d 10996 | . . . . 5 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · 0) = 0) |
19 | 13, 18 | syl5eq 2783 | . . . 4 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)))) = 0) |
20 | 19 | sumeq2i 15228 | . . 3 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)))) = Σ𝑘 ∈ (0...3)0 |
21 | fzfi 13510 | . . . . 5 ⊢ (0...3) ∈ Fin | |
22 | 21 | olci 866 | . . . 4 ⊢ ((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) |
23 | sumz 15251 | . . . 4 ⊢ (((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) → Σ𝑘 ∈ (0...3)0 = 0) | |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ Σ𝑘 ∈ (0...3)0 = 0 |
25 | 20, 24 | eqtri 2759 | . 2 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(𝐴 / (i↑𝑘)))), (ℜ‘(𝐴 / (i↑𝑘))), 0)))) = 0 |
26 | 2, 25 | eqtri 2759 | 1 ⊢ ∫∅𝐴 d𝑥 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∅c0 4223 ifcif 4425 {csn 4527 class class class wbr 5039 ↦ cmpt 5120 × cxp 5534 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 ℂcc 10692 ℝcr 10693 0cc0 10694 ici 10696 · cmul 10699 ≤ cle 10833 / cdiv 11454 3c3 11851 ℕ0cn0 12055 ℤ≥cuz 12403 ...cfz 13060 ↑cexp 13600 ℜcre 14625 Σcsu 15214 ∫2citg2 24467 ∫citg 24469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xadd 12670 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 df-xmet 20310 df-met 20311 df-ovol 24315 df-vol 24316 df-mbf 24470 df-itg1 24471 df-itg2 24472 df-itg 24474 df-0p 24521 |
This theorem is referenced by: itgsplitioo 24689 ditg0 24704 ditgneg 24708 ftc2 24895 ftc2nc 35545 areacirc 35556 itgvol0 43127 |
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