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Mirrors > Home > ILE Home > Th. List > ef2kpi | GIF version |
Description: If 𝐾 is an integer, then the exponential of 2𝐾πi is 1. (Contributed by Mario Carneiro, 9-May-2014.) |
Ref | Expression |
---|---|
ef2kpi | ⊢ (𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7739 | . . . . 5 ⊢ i ∈ ℂ | |
2 | 2cn 8815 | . . . . . 6 ⊢ 2 ∈ ℂ | |
3 | picn 12916 | . . . . . 6 ⊢ π ∈ ℂ | |
4 | 2, 3 | mulcli 7795 | . . . . 5 ⊢ (2 · π) ∈ ℂ |
5 | 1, 4 | mulcli 7795 | . . . 4 ⊢ (i · (2 · π)) ∈ ℂ |
6 | zcn 9083 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
7 | mulcom 7773 | . . . 4 ⊢ (((i · (2 · π)) ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((i · (2 · π)) · 𝐾) = (𝐾 · (i · (2 · π)))) | |
8 | 5, 6, 7 | sylancr 411 | . . 3 ⊢ (𝐾 ∈ ℤ → ((i · (2 · π)) · 𝐾) = (𝐾 · (i · (2 · π)))) |
9 | 8 | fveq2d 5433 | . 2 ⊢ (𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = (exp‘(𝐾 · (i · (2 · π))))) |
10 | efexp 11425 | . . 3 ⊢ (((i · (2 · π)) ∈ ℂ ∧ 𝐾 ∈ ℤ) → (exp‘(𝐾 · (i · (2 · π)))) = ((exp‘(i · (2 · π)))↑𝐾)) | |
11 | 5, 10 | mpan 421 | . 2 ⊢ (𝐾 ∈ ℤ → (exp‘(𝐾 · (i · (2 · π)))) = ((exp‘(i · (2 · π)))↑𝐾)) |
12 | ef2pi 12934 | . . . 4 ⊢ (exp‘(i · (2 · π))) = 1 | |
13 | 12 | oveq1i 5792 | . . 3 ⊢ ((exp‘(i · (2 · π)))↑𝐾) = (1↑𝐾) |
14 | 1exp 10353 | . . 3 ⊢ (𝐾 ∈ ℤ → (1↑𝐾) = 1) | |
15 | 13, 14 | syl5eq 2185 | . 2 ⊢ (𝐾 ∈ ℤ → ((exp‘(i · (2 · π)))↑𝐾) = 1) |
16 | 9, 11, 15 | 3eqtrd 2177 | 1 ⊢ (𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 1c1 7645 ici 7646 · cmul 7649 2c2 8795 ℤcz 9078 ↑cexp 10323 expce 11385 πcpi 11390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834 |
This theorem is referenced by: efper 12936 |
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