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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 2737 | . . 3 ⊢ (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘))) | |
2 | 1 | dfitg 25085 | . 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 11068 | . . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ | |
4 | elfznn0 13488 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
5 | expcl 13939 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 587 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ) |
7 | ine0 11548 | . . . . . . . . . . . . . . 15 ⊢ i ≠ 0 | |
8 | elfzelz 13395 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
9 | expne0i 13954 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) | |
10 | 3, 7, 8, 9 | mp3an12i 1465 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ≠ 0) |
11 | 6, 10 | div0d 11888 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → (0 / (i↑𝑘)) = 0) |
12 | 11 | fveq2d 6843 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘0)) |
13 | re0 14996 | . . . . . . . . . . . 12 ⊢ (ℜ‘0) = 0 | |
14 | 12, 13 | eqtrdi 2793 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = 0) |
15 | 14 | ifeq1d 4503 | . . . . . . . . . 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)) |
16 | ifid 4524 | . . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), 0, 0) = 0 | |
17 | 15, 16 | eqtrdi 2793 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...3) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0) |
18 | 17 | mpteq2dv 5205 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ 0)) |
19 | fconstmpt 5692 | . . . . . . . 8 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
20 | 18, 19 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (ℝ × {0})) |
21 | 20 | fveq2d 6843 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) = (∫2‘(ℝ × {0}))) |
22 | itg20 25053 | . . . . . 6 ⊢ (∫2‘(ℝ × {0})) = 0 | |
23 | 21, 22 | eqtrdi 2793 | . . . . 5 ⊢ (𝑘 ∈ (0...3) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) = 0) |
24 | 23 | oveq2d 7367 | . . . 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)) |
25 | 6 | mul01d 11312 | . . . 4 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · 0) = 0) |
26 | 24, 25 | eqtrd 2777 | . . 3 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)))) = 0) |
27 | 26 | sumeq2i 15543 | . 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 |
28 | fzfi 13831 | . . . 4 ⊢ (0...3) ∈ Fin | |
29 | 28 | olci 864 | . . 3 ⊢ ((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) |
30 | sumz 15566 | . . 3 ⊢ (((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) → Σ𝑘 ∈ (0...3)0 = 0) | |
31 | 29, 30 | ax-mp 5 | . 2 ⊢ Σ𝑘 ∈ (0...3)0 = 0 |
32 | 2, 27, 31 | 3eqtri 2769 | 1 ⊢ ∫𝐴0 d𝑥 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ⊆ wss 3908 ifcif 4484 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 × cxp 5629 ‘cfv 6493 (class class class)co 7351 Fincfn 8841 ℂcc 11007 ℝcr 11008 0cc0 11009 ici 11011 · cmul 11014 ≤ cle 11148 / cdiv 11770 3c3 12167 ℕ0cn0 12371 ℤcz 12457 ℤ≥cuz 12721 ...cfz 13378 ↑cexp 13921 ℜcre 14941 Σcsu 15529 ∫2citg2 24931 ∫citg 24933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-disj 5069 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-oi 9404 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-xadd 12988 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-seq 13861 df-exp 13922 df-hash 14184 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-clim 15329 df-sum 15530 df-xmet 20741 df-met 20742 df-ovol 24779 df-vol 24780 df-mbf 24934 df-itg1 24935 df-itg2 24936 df-itg 24938 df-0p 24985 |
This theorem is referenced by: itgge0 25126 itgfsum 25142 |
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