| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 25804 | . 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 11214 | . . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ | |
| 4 | elfznn0 13660 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
| 5 | expcl 14120 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
| 6 | 3, 4, 5 | sylancr 587 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ) |
| 7 | ine0 11698 | . . . . . . . . . . . . . . 15 ⊢ i ≠ 0 | |
| 8 | elfzelz 13564 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
| 9 | expne0i 14135 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) | |
| 10 | 3, 7, 8, 9 | mp3an12i 1467 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ≠ 0) |
| 11 | 6, 10 | div0d 12042 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → (0 / (i↑𝑘)) = 0) |
| 12 | 11 | fveq2d 6910 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘0)) |
| 13 | re0 15191 | . . . . . . . . . . . 12 ⊢ (ℜ‘0) = 0 | |
| 14 | 12, 13 | eqtrdi 2793 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = 0) |
| 15 | 14 | ifeq1d 4545 | . . . . . . . . . 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 4566 | . . . . . . . . . 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 5244 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ 0)) |
| 19 | fconstmpt 5747 | . . . . . . . 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 6910 | . . . . . 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 25772 | . . . . . 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 7447 | . . . 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 11460 | . . . 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 15734 | . 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 14013 | . . . 4 ⊢ (0...3) ∈ Fin | |
| 29 | 28 | olci 867 | . . 3 ⊢ ((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) |
| 30 | sumz 15758 | . . 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 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ifcif 4525 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 ici 11157 · cmul 11160 ≤ cle 11296 / cdiv 11920 3c3 12322 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 ↑cexp 14102 ℜcre 15136 Σcsu 15722 ∫2citg2 25651 ∫citg 25653 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-itg 25658 df-0p 25705 |
| This theorem is referenced by: itgge0 25846 itgfsum 25862 |
| Copyright terms: Public domain | W3C validator |