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Mirrors > Home > ILE Home > Th. List > ef2pi | GIF version |
Description: The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.) |
Ref | Expression |
---|---|
ef2pi | ⊢ (exp‘(i · (2 · π))) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 8904 | . . . 4 ⊢ 2 ∈ ℂ | |
2 | picn 13119 | . . . 4 ⊢ π ∈ ℂ | |
3 | 1, 2 | mulcli 7883 | . . 3 ⊢ (2 · π) ∈ ℂ |
4 | efival 11629 | . . 3 ⊢ ((2 · π) ∈ ℂ → (exp‘(i · (2 · π))) = ((cos‘(2 · π)) + (i · (sin‘(2 · π))))) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (exp‘(i · (2 · π))) = ((cos‘(2 · π)) + (i · (sin‘(2 · π)))) |
6 | cos2pi 13136 | . . . 4 ⊢ (cos‘(2 · π)) = 1 | |
7 | sin2pi 13135 | . . . . . 6 ⊢ (sin‘(2 · π)) = 0 | |
8 | 7 | oveq2i 5835 | . . . . 5 ⊢ (i · (sin‘(2 · π))) = (i · 0) |
9 | it0e0 9054 | . . . . 5 ⊢ (i · 0) = 0 | |
10 | 8, 9 | eqtri 2178 | . . . 4 ⊢ (i · (sin‘(2 · π))) = 0 |
11 | 6, 10 | oveq12i 5836 | . . 3 ⊢ ((cos‘(2 · π)) + (i · (sin‘(2 · π)))) = (1 + 0) |
12 | 1p0e1 8949 | . . 3 ⊢ (1 + 0) = 1 | |
13 | 11, 12 | eqtri 2178 | . 2 ⊢ ((cos‘(2 · π)) + (i · (sin‘(2 · π)))) = 1 |
14 | 5, 13 | eqtri 2178 | 1 ⊢ (exp‘(i · (2 · π))) = 1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 ‘cfv 5170 (class class class)co 5824 ℂcc 7730 0cc0 7732 1c1 7733 ici 7734 + caddc 7735 · cmul 7737 2c2 8884 expce 11539 sincsin 11541 cosccos 11542 πcpi 11544 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 ax-pre-suploc 7853 ax-addf 7854 ax-mulf 7855 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-disj 3943 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-of 6032 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-oadd 6367 df-er 6480 df-map 6595 df-pm 6596 df-en 6686 df-dom 6687 df-fin 6688 df-sup 6928 df-inf 6929 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-5 8895 df-6 8896 df-7 8897 df-8 8898 df-9 8899 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-xneg 9679 df-xadd 9680 df-ioo 9796 df-ioc 9797 df-ico 9798 df-icc 9799 df-fz 9913 df-fzo 10042 df-seqfrec 10345 df-exp 10419 df-fac 10600 df-bc 10622 df-ihash 10650 df-shft 10715 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-sumdc 11251 df-ef 11545 df-sin 11547 df-cos 11548 df-pi 11550 df-rest 12364 df-topgen 12383 df-psmet 12398 df-xmet 12399 df-met 12400 df-bl 12401 df-mopn 12402 df-top 12407 df-topon 12420 df-bases 12452 df-ntr 12507 df-cn 12599 df-cnp 12600 df-tx 12664 df-cncf 12969 df-limced 13036 df-dvap 13037 |
This theorem is referenced by: ef2kpi 13138 |
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