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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 2823 | . . 3 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))) | |
2 | 1 | dfitg 24372 | . 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 13344 | . . 3 ⊢ (𝜑 → (0...3) ∈ Fin) | |
4 | ax-icn 10598 | . . . . 5 ⊢ i ∈ ℂ | |
5 | elfznn0 13003 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0) | |
6 | 5 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑘 ∈ ℕ0) |
7 | expcl 13450 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (i↑𝑘) ∈ ℂ) | |
8 | 4, 6, 7 | sylancr 589 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (i↑𝑘) ∈ ℂ) |
9 | elfzelz 12911 | . . . . . 6 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ) | |
10 | eqidd 2824 | . . . . . . 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 2824 | . . . . . . 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 24371 | . . . . . 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 594 | . . . . 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 10671 | . . . 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 10663 | . . 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 15092 | . 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 2919 | 1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 ici 10541 · cmul 10544 ≤ cle 10678 / cdiv 11299 3c3 11696 ℕ0cn0 11900 ℤcz 11984 ...cfz 12895 ↑cexp 13432 ℜcre 14458 Σcsu 15044 ∫2citg2 24219 𝐿1cibl 24220 ∫citg 24221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-ibl 24225 df-itg 24226 |
This theorem is referenced by: itgneg 24406 itgaddlem2 24426 itgadd 24427 itgsub 24428 itgfsum 24429 itgmulc2lem2 24435 itgmulc2 24436 itgabs 24437 itgsplitioo 24440 ditgcl 24458 ditgswap 24459 ftc1lem1 24634 ftc1lem2 24635 ftc1a 24636 ftc1lem4 24638 ftc2 24643 itgparts 24646 itgsubstlem 24647 itgulm 24998 itgaddnclem2 34953 itgaddnc 34954 itgsubnc 34956 itgmulc2nclem2 34961 itgmulc2nc 34962 itgabsnc 34963 ftc1cnnclem 34967 ftc1anc 34977 ftc2nc 34978 itgpowd 39828 itgsinexplem1 42246 itgsinexp 42247 itgspltprt 42271 fourierdlem30 42429 fourierdlem47 42445 fourierdlem73 42471 fourierdlem83 42481 fourierdlem87 42485 fourierdlem95 42493 fourierdlem103 42501 fourierdlem104 42502 fourierdlem107 42505 fourierdlem112 42510 sqwvfoura 42520 etransclem23 42549 |
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