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Mirrors > Home > ILE Home > Th. List > sincos2sgn | Unicode version |
Description: The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
sincos2sgn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8555 |
. . . 4
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2 | 2pos 8576 |
. . . 4
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3 | 1 | leidi 8026 |
. . . 4
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4 | 0xr 7597 |
. . . . 5
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5 | elioc2 9417 |
. . . . 5
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6 | 4, 1, 5 | mp2an 418 |
. . . 4
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7 | 1, 2, 3, 6 | mpbir3an 1126 |
. . 3
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8 | sin02gt0 11117 |
. . 3
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9 | 7, 8 | ax-mp 7 |
. 2
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10 | cos2bnd 11114 |
. . . 4
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11 | 10 | simpri 112 |
. . 3
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12 | 9re 8572 |
. . . . 5
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13 | 9pos 8589 |
. . . . 5
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14 | 12, 13 | recgt0ii 8431 |
. . . 4
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15 | 12, 13 | gt0ap0ii 8167 |
. . . . . 6
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16 | 12, 15 | rerecclapi 8307 |
. . . . 5
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17 | lt0neg2 8010 |
. . . . 5
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18 | 16, 17 | ax-mp 7 |
. . . 4
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19 | 14, 18 | mpbi 144 |
. . 3
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20 | recoscl 11075 |
. . . . 5
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21 | 1, 20 | ax-mp 7 |
. . . 4
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22 | 16 | renegcli 7807 |
. . . 4
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23 | 0re 7551 |
. . . 4
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24 | 21, 22, 23 | lttri 7652 |
. . 3
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25 | 11, 19, 24 | mp2an 418 |
. 2
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26 | 9, 25 | pm3.2i 267 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 ax-arch 7527 ax-caucvg 7528 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-disj 3831 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-isom 5039 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-irdg 6151 df-frec 6172 df-1o 6197 df-oadd 6201 df-er 6308 df-en 6514 df-dom 6515 df-fin 6516 df-sup 6735 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-5 8547 df-6 8548 df-7 8549 df-8 8550 df-9 8551 df-n0 8737 df-z 8814 df-uz 9083 df-q 9168 df-rp 9198 df-ioc 9374 df-ico 9375 df-fz 9488 df-fzo 9617 df-iseq 9916 df-seq3 9917 df-exp 10018 df-fac 10197 df-bc 10219 df-ihash 10247 df-shft 10312 df-cj 10339 df-re 10340 df-im 10341 df-rsqrt 10494 df-abs 10495 df-clim 10730 df-isum 10806 df-ef 11001 df-sin 11003 df-cos 11004 |
This theorem is referenced by: sin4lt0 11120 |
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