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Mirrors > Home > ILE Home > Th. List > sincos4thpi | Unicode version |
Description: The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.) |
Ref | Expression |
---|---|
sincos4thpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfcn 8934 | . . . . . . . . . 10 | |
2 | ax-1cn 7713 | . . . . . . . . . . 11 | |
3 | 2halves 8949 | . . . . . . . . . . 11 | |
4 | 2, 3 | ax-mp 5 | . . . . . . . . . 10 |
5 | sincosq1eq 12920 | . . . . . . . . . 10 | |
6 | 1, 1, 4, 5 | mp3an 1315 | . . . . . . . . 9 |
7 | 6 | oveq2i 5785 | . . . . . . . 8 |
8 | 7 | oveq2i 5785 | . . . . . . 7 |
9 | 2cn 8791 | . . . . . . . . . . . 12 | |
10 | pire 12867 | . . . . . . . . . . . . 13 | |
11 | 10 | recni 7778 | . . . . . . . . . . . 12 |
12 | 2ap0 8813 | . . . . . . . . . . . 12 # | |
13 | 2, 9, 11, 9, 12, 12 | divmuldivapi 8532 | . . . . . . . . . . 11 |
14 | 11 | mulid2i 7769 | . . . . . . . . . . . 12 |
15 | 2t2e4 8874 | . . . . . . . . . . . 12 | |
16 | 14, 15 | oveq12i 5786 | . . . . . . . . . . 11 |
17 | 13, 16 | eqtri 2160 | . . . . . . . . . 10 |
18 | 17 | fveq2i 5424 | . . . . . . . . 9 |
19 | 18, 18 | oveq12i 5786 | . . . . . . . 8 |
20 | 19 | oveq2i 5785 | . . . . . . 7 |
21 | 9, 12 | recidapi 8503 | . . . . . . . . . . 11 |
22 | 21 | oveq1i 5784 | . . . . . . . . . 10 |
23 | 2re 8790 | . . . . . . . . . . . . 13 | |
24 | 10, 23, 12 | redivclapi 8539 | . . . . . . . . . . . 12 |
25 | 24 | recni 7778 | . . . . . . . . . . 11 |
26 | 9, 1, 25 | mulassi 7775 | . . . . . . . . . 10 |
27 | 25 | mulid2i 7769 | . . . . . . . . . 10 |
28 | 22, 26, 27 | 3eqtr3i 2168 | . . . . . . . . 9 |
29 | 28 | fveq2i 5424 | . . . . . . . 8 |
30 | 1, 25 | mulcli 7771 | . . . . . . . . 9 |
31 | sin2t 11456 | . . . . . . . . 9 | |
32 | 30, 31 | ax-mp 5 | . . . . . . . 8 |
33 | sinhalfpi 12877 | . . . . . . . 8 | |
34 | 29, 32, 33 | 3eqtr3i 2168 | . . . . . . 7 |
35 | 8, 20, 34 | 3eqtr3i 2168 | . . . . . 6 |
36 | 35 | fveq2i 5424 | . . . . 5 |
37 | 4re 8797 | . . . . . . . . 9 | |
38 | 4ap0 8819 | . . . . . . . . 9 # | |
39 | 10, 37, 38 | redivclapi 8539 | . . . . . . . 8 |
40 | resincl 11427 | . . . . . . . 8 | |
41 | 39, 40 | ax-mp 5 | . . . . . . 7 |
42 | 41, 41 | remulcli 7780 | . . . . . 6 |
43 | 0le2 8810 | . . . . . 6 | |
44 | 41 | msqge0i 8379 | . . . . . 6 |
45 | 23, 42, 43, 44 | sqrtmulii 10906 | . . . . 5 |
46 | sqrt1 10818 | . . . . 5 | |
47 | 36, 45, 46 | 3eqtr3ri 2169 | . . . 4 |
48 | 42 | sqrtcli 10892 | . . . . . . 7 |
49 | 44, 48 | ax-mp 5 | . . . . . 6 |
50 | 49 | recni 7778 | . . . . 5 |
51 | sqrt2re 11841 | . . . . . . 7 | |
52 | 51 | recni 7778 | . . . . . 6 |
53 | 2pos 8811 | . . . . . . . 8 | |
54 | 23, 53 | sqrtgt0ii 10903 | . . . . . . 7 |
55 | 51, 54 | gt0ap0ii 8390 | . . . . . 6 # |
56 | 52, 55 | pm3.2i 270 | . . . . 5 # |
57 | divmulap2 8436 | . . . . 5 # | |
58 | 2, 50, 56, 57 | mp3an 1315 | . . . 4 |
59 | 47, 58 | mpbir 145 | . . 3 |
60 | 0re 7766 | . . . . 5 | |
61 | pipos 12869 | . . . . . . . 8 | |
62 | 4pos 8817 | . . . . . . . 8 | |
63 | 10, 37, 61, 62 | divgt0ii 8677 | . . . . . . 7 |
64 | 1re 7765 | . . . . . . . 8 | |
65 | pigt2lt4 12865 | . . . . . . . . . . 11 | |
66 | 65 | simpri 112 | . . . . . . . . . 10 |
67 | 10, 37, 37, 62 | ltdiv1ii 8687 | . . . . . . . . . 10 |
68 | 66, 67 | mpbi 144 | . . . . . . . . 9 |
69 | 37 | recni 7778 | . . . . . . . . . 10 |
70 | 69, 38 | dividapi 8505 | . . . . . . . . 9 |
71 | 68, 70 | breqtri 3953 | . . . . . . . 8 |
72 | 39, 64, 71 | ltleii 7866 | . . . . . . 7 |
73 | 0xr 7812 | . . . . . . . 8 | |
74 | elioc2 9719 | . . . . . . . 8 | |
75 | 73, 64, 74 | mp2an 422 | . . . . . . 7 |
76 | 39, 63, 72, 75 | mpbir3an 1163 | . . . . . 6 |
77 | sin01gt0 11468 | . . . . . 6 | |
78 | 76, 77 | ax-mp 5 | . . . . 5 |
79 | 60, 41, 78 | ltleii 7866 | . . . 4 |
80 | 41 | sqrtmsqi 10894 | . . . 4 |
81 | 79, 80 | ax-mp 5 | . . 3 |
82 | 59, 81 | eqtr2i 2161 | . 2 |
83 | 59, 81 | eqtri 2160 | . . 3 |
84 | 17 | fveq2i 5424 | . . . 4 |
85 | 6, 18, 84 | 3eqtr3i 2168 | . . 3 |
86 | 83, 85 | eqtr2i 2161 | . 2 |
87 | 82, 86 | pm3.2i 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 cxr 7799 clt 7800 cle 7801 # cap 8343 cdiv 8432 c2 8771 c4 8773 cioc 9672 csqrt 10768 csin 11350 ccos 11351 cpi 11353 |
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 ax-pre-suploc 7741 ax-addf 7742 ax-mulf 7743 |
This theorem depends on definitions: df-bi 116 df-stab 816 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-disj 3907 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-of 5982 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-map 6544 df-pm 6545 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-inf 6872 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-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-ioo 9675 df-ioc 9676 df-ico 9677 df-icc 9678 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-bc 10494 df-ihash 10522 df-shft 10587 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 df-pi 11359 df-rest 12122 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-met 12158 df-bl 12159 df-mopn 12160 df-top 12165 df-topon 12178 df-bases 12210 df-ntr 12265 df-cn 12357 df-cnp 12358 df-tx 12422 df-cncf 12727 df-limced 12794 df-dvap 12795 |
This theorem is referenced by: tan4thpi 12922 |
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