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Mirrors > Home > MPE Home > Th. List > itgcl | Structured version Visualization version GIF version |
Description: The integral of an integrable function is a complex number. This is Metamath 100 proof #86. (Contributed by Mario Carneiro, 29-Jun-2014.) |
Ref | Expression |
---|---|
itgmpt.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
itgcl.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
Ref | Expression |
---|---|
itgcl | ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))) | |
2 | 1 | dfitg 25790 | . 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) |
3 | fzfid 13993 | . . 3 ⊢ (𝜑 → (0...3) ∈ Fin) | |
4 | ax-icn 11217 | . . . . 5 ⊢ i ∈ ℂ | |
5 | elfznn0 13648 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
6 | 5 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑘 ∈ ℕ0) |
7 | expcl 14099 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
8 | 4, 6, 7 | sylancr 585 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (i↑𝑘) ∈ ℂ) |
9 | elfzelz 13555 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
10 | eqidd 2727 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) | |
11 | eqidd 2727 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘)))) | |
12 | itgcl.2 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
13 | itgmpt.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
14 | 10, 11, 12, 13 | iblitg 25789 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
15 | 9, 14 | sylan2 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
16 | 15 | recnd 11292 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℂ) |
17 | 8, 16 | mulcld 11284 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) ∈ ℂ) |
18 | 3, 17 | fsumcl 15737 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) ∈ ℂ) |
19 | 2, 18 | eqeltrid 2830 | 1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ifcif 4533 class class class wbr 5153 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 ici 11160 · cmul 11163 ≤ cle 11299 / cdiv 11921 3c3 12320 ℕ0cn0 12524 ℤcz 12610 ...cfz 13538 ↑cexp 14081 ℜcre 15102 Σcsu 15690 ∫2citg2 25636 𝐿1cibl 25637 ∫citg 25638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 df-ibl 25642 df-itg 25643 |
This theorem is referenced by: itgneg 25824 itgaddlem2 25844 itgadd 25845 itgsub 25846 itgfsum 25847 itgmulc2lem2 25853 itgmulc2 25854 itgabs 25855 itgsplitioo 25858 ditgcl 25878 ditgswap 25879 ftc1lem1 26061 ftc1lem2 26062 ftc1a 26063 ftc1lem4 26065 ftc2 26070 itgparts 26073 itgsubstlem 26074 itgpowd 26076 itgulm 26437 itgaddnclem2 37380 itgaddnc 37381 itgsubnc 37383 itgmulc2nclem2 37388 itgmulc2nc 37389 itgabsnc 37390 ftc1cnnclem 37392 ftc1anc 37402 ftc2nc 37403 lcmineqlem10 41737 itgsinexplem1 45575 itgsinexp 45576 itgspltprt 45600 fourierdlem30 45758 fourierdlem47 45774 fourierdlem73 45800 fourierdlem83 45810 fourierdlem87 45814 fourierdlem95 45822 fourierdlem103 45830 fourierdlem104 45831 fourierdlem107 45834 fourierdlem112 45839 sqwvfoura 45849 etransclem23 45878 |
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