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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 2735 | . . 3 ⊢ (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘))) | |
2 | 1 | dfitg 25819 | . 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 11212 | . . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ | |
4 | elfznn0 13657 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
5 | expcl 14117 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 587 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ) |
7 | ine0 11696 | . . . . . . . . . . . . . . 15 ⊢ i ≠ 0 | |
8 | elfzelz 13561 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
9 | expne0i 14132 | . . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) | |
10 | 3, 7, 8, 9 | mp3an12i 1464 | . . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ≠ 0) |
11 | 6, 10 | div0d 12040 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → (0 / (i↑𝑘)) = 0) |
12 | 11 | fveq2d 6911 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘0)) |
13 | re0 15188 | . . . . . . . . . . . 12 ⊢ (ℜ‘0) = 0 | |
14 | 12, 13 | eqtrdi 2791 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...3) → (ℜ‘(0 / (i↑𝑘))) = 0) |
15 | 14 | ifeq1d 4550 | . . . . . . . . . 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 4571 | . . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), 0, 0) = 0 | |
17 | 15, 16 | eqtrdi 2791 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...3) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0) |
18 | 17 | mpteq2dv 5250 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ 0)) |
19 | fconstmpt 5751 | . . . . . . . 8 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
20 | 18, 19 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑘 ∈ (0...3) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) = (ℝ × {0})) |
21 | 20 | fveq2d 6911 | . . . . . 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 25787 | . . . . . 6 ⊢ (∫2‘(ℝ × {0})) = 0 | |
23 | 21, 22 | eqtrdi 2791 | . . . . 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 11458 | . . . 4 ⊢ (𝑘 ∈ (0...3) → ((i↑𝑘) · 0) = 0) |
26 | 24, 25 | eqtrd 2775 | . . 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 15731 | . 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 14010 | . . . 4 ⊢ (0...3) ∈ Fin | |
29 | 28 | olci 866 | . . 3 ⊢ ((0...3) ⊆ (ℤ≥‘0) ∨ (0...3) ∈ Fin) |
30 | sumz 15755 | . . 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 2767 | 1 ⊢ ∫𝐴0 d𝑥 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ifcif 4531 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 ici 11155 · cmul 11158 ≤ cle 11294 / cdiv 11918 3c3 12320 ℕ0cn0 12524 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 ↑cexp 14099 ℜcre 15133 Σcsu 15719 ∫2citg2 25665 ∫citg 25667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-itg 25672 df-0p 25719 |
This theorem is referenced by: itgge0 25861 itgfsum 25877 |
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