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Mirrors > Home > ILE Home > Th. List > efival | Unicode version |
Description: The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
efival |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7715 | . . . . . 6 | |
2 | mulcl 7747 | . . . . . 6 | |
3 | 1, 2 | mpan 420 | . . . . 5 |
4 | efcl 11370 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | negicn 7963 | . . . . . 6 | |
7 | mulcl 7747 | . . . . . 6 | |
8 | 6, 7 | mpan 420 | . . . . 5 |
9 | efcl 11370 | . . . . 5 | |
10 | 8, 9 | syl 14 | . . . 4 |
11 | 5, 10 | addcld 7785 | . . 3 |
12 | 5, 10 | subcld 8073 | . . 3 |
13 | 2cn 8791 | . . . . 5 | |
14 | 2ap0 8813 | . . . . 5 # | |
15 | 13, 14 | pm3.2i 270 | . . . 4 # |
16 | divdirap 8457 | . . . 4 # | |
17 | 15, 16 | mp3an3 1304 | . . 3 |
18 | 11, 12, 17 | syl2anc 408 | . 2 |
19 | 10, 5 | pncan3d 8076 | . . . . . 6 |
20 | 19 | oveq2d 5790 | . . . . 5 |
21 | 5, 10, 12 | addassd 7788 | . . . . 5 |
22 | 5 | 2timesd 8962 | . . . . 5 |
23 | 20, 21, 22 | 3eqtr4d 2182 | . . . 4 |
24 | 23 | oveq1d 5789 | . . 3 |
25 | divcanap3 8458 | . . . . 5 # | |
26 | 13, 14, 25 | mp3an23 1307 | . . . 4 |
27 | 5, 26 | syl 14 | . . 3 |
28 | 24, 27 | eqtr2d 2173 | . 2 |
29 | cosval 11410 | . . 3 | |
30 | 2mulicn 8942 | . . . . . . 7 | |
31 | 2muliap0 8944 | . . . . . . 7 # | |
32 | 30, 31 | pm3.2i 270 | . . . . . 6 # |
33 | div12ap 8454 | . . . . . 6 # | |
34 | 1, 32, 33 | mp3an13 1306 | . . . . 5 |
35 | 12, 34 | syl 14 | . . . 4 |
36 | sinval 11409 | . . . . 5 | |
37 | 36 | oveq2d 5790 | . . . 4 |
38 | divrecap 8448 | . . . . . . 7 # | |
39 | 13, 14, 38 | mp3an23 1307 | . . . . . 6 |
40 | 12, 39 | syl 14 | . . . . 5 |
41 | 1 | mulid2i 7769 | . . . . . . . 8 |
42 | 41 | oveq1i 5784 | . . . . . . 7 |
43 | iap0 8943 | . . . . . . . . . . 11 # | |
44 | 1, 43 | dividapi 8505 | . . . . . . . . . 10 |
45 | 44 | oveq2i 5785 | . . . . . . . . 9 |
46 | ax-1cn 7713 | . . . . . . . . . 10 | |
47 | 46, 13, 1, 1, 14, 43 | divmuldivapi 8532 | . . . . . . . . 9 |
48 | 45, 47 | eqtr3i 2162 | . . . . . . . 8 |
49 | halfcn 8934 | . . . . . . . . 9 | |
50 | 49 | mulid1i 7768 | . . . . . . . 8 |
51 | 48, 50 | eqtr3i 2162 | . . . . . . 7 |
52 | 42, 51 | eqtr3i 2162 | . . . . . 6 |
53 | 52 | oveq2i 5785 | . . . . 5 |
54 | 40, 53 | syl6eqr 2190 | . . . 4 |
55 | 35, 37, 54 | 3eqtr4d 2182 | . . 3 |
56 | 29, 55 | oveq12d 5792 | . 2 |
57 | 18, 28, 56 | 3eqtr4d 2182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 ci 7622 caddc 7623 cmul 7625 cmin 7933 cneg 7934 # cap 8343 cdiv 8432 c2 8771 ce 11348 csin 11350 ccos 11351 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 df-sin 11356 df-cos 11357 |
This theorem is referenced by: efmival 11440 efeul 11441 efieq 11442 sinadd 11443 cosadd 11444 absefi 11475 demoivre 11479 efhalfpi 12880 efipi 12882 ef2pi 12886 efimpi 12900 |
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