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Mirrors > Home > MPE Home > Th. List > itgz | Structured version Visualization version GIF version |
Description: The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
itgz | ⊢ ∫𝐴0 d𝑥 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . 3 ⊢ (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘))) | |
2 | 1 | dfitg 23581 | . 2 ⊢ ∫𝐴0 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)))) |
3 | ax-icn 10033 | . . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ | |
4 | elfznn0 12471 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
5 | expcl 12918 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 696 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ) |
7 | elfzelz 12380 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
8 | ine0 10503 | . . . . . . . . . . . . . . . 16 ⊢ i ≠ 0 | |
9 | expne0i 12932 | . . . . . . . . . . . . . . . 16 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) | |
10 | 3, 8, 9 | mp3an12 1454 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0) |
11 | 7, 10 | syl 17 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ≠ 0) |
12 | 6, 11 | div0d 10838 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → (0 / (i↑𝑘)) = 0) |
13 | 12 | fveq2d 6233 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘0)) |
14 | re0 13936 | . . . . . . . . . . . 12 ⊢ (ℜ‘0) = 0 | |
15 | 13, 14 | syl6eq 2701 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = 0) |
16 | 15 | ifeq1d 4137 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...3) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), 0, 0)) |
17 | ifid 4158 | . . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), 0, 0) = 0 | |
18 | 16, 17 | syl6eq 2701 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...3) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0) |
19 | 18 | mpteq2dv 4778 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ 0)) |
20 | fconstmpt 5197 | . . . . . . . 8 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
21 | 19, 20 | syl6eqr 2703 | . . . . . . 7 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (ℝ × {0})) |
22 | 21 | fveq2d 6233 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) = (∫2‘(ℝ × {0}))) |
23 | itg20 23549 | . . . . . 6 ⊢ (∫2‘(ℝ × {0})) = 0 | |
24 | 22, 23 | syl6eq 2701 | . . . . 5 ⊢ (𝑘 ∈ (0...3) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) = 0) |
25 | 24 | oveq2d 6706 | . . . 4 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)))) = ((i↑𝑘) · 0)) |
26 | 6 | mul01d 10273 | . . . 4 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · 0) = 0) |
27 | 25, 26 | eqtrd 2685 | . . 3 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)))) = 0) |
28 | 27 | sumeq2i 14473 | . 2 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)))) = Σ𝑘 ∈ (0...3)0 |
29 | fzfi 12811 | . . . 4 ⊢ (0...3) ∈ Fin | |
30 | 29 | olci 405 | . . 3 ⊢ ((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) |
31 | sumz 14497 | . . 3 ⊢ (((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) → Σ𝑘 ∈ (0...3)0 = 0) | |
32 | 30, 31 | ax-mp 5 | . 2 ⊢ Σ𝑘 ∈ (0...3)0 = 0 |
33 | 2, 28, 32 | 3eqtri 2677 | 1 ⊢ ∫𝐴0 d𝑥 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ⊆ wss 3607 ifcif 4119 {csn 4210 class class class wbr 4685 ↦ cmpt 4762 × cxp 5141 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 ℂcc 9972 ℝcr 9973 0cc0 9974 ici 9976 · cmul 9979 ≤ cle 10113 / cdiv 10722 3c3 11109 ℕ0cn0 11330 ℤcz 11415 ℤ≥cuz 11725 ...cfz 12364 ↑cexp 12900 ℜcre 13881 Σcsu 14460 ∫2citg2 23430 ∫citg 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-ofr 6940 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-xadd 11985 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461 df-xmet 19787 df-met 19788 df-ovol 23279 df-vol 23280 df-mbf 23433 df-itg1 23434 df-itg2 23435 df-itg 23437 df-0p 23482 |
This theorem is referenced by: itgge0 23622 itgfsum 23638 |
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