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Mirrors > Home > ILE Home > Th. List > sin2pi | Unicode version |
Description: The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.) |
Ref | Expression |
---|---|
sin2pi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | picn 13463 | . . 3 | |
2 | sin2t 11701 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | sinpi 13461 | . . . . . 6 | |
5 | cospi 13476 | . . . . . 6 | |
6 | 4, 5 | oveq12i 5863 | . . . . 5 |
7 | neg1cn 8972 | . . . . . 6 | |
8 | 7 | mul02i 8298 | . . . . 5 |
9 | 6, 8 | eqtri 2191 | . . . 4 |
10 | 9 | oveq2i 5862 | . . 3 |
11 | 2t0e0 9026 | . . 3 | |
12 | 10, 11 | eqtri 2191 | . 2 |
13 | 3, 12 | eqtri 2191 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wcel 2141 cfv 5196 (class class class)co 5851 cc 7761 cc0 7763 c1 7764 cmul 7768 cneg 8080 c2 8918 csin 11596 ccos 11597 cpi 11599 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 ax-arch 7882 ax-caucvg 7883 ax-pre-suploc 7884 ax-addf 7885 ax-mulf 7886 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-of 6059 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6510 df-map 6625 df-pm 6626 df-en 6716 df-dom 6717 df-fin 6718 df-sup 6958 df-inf 6959 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-5 8929 df-6 8930 df-7 8931 df-8 8932 df-9 8933 df-n0 9125 df-z 9202 df-uz 9477 df-q 9568 df-rp 9600 df-xneg 9718 df-xadd 9719 df-ioo 9838 df-ioc 9839 df-ico 9840 df-icc 9841 df-fz 9955 df-fzo 10088 df-seqfrec 10391 df-exp 10465 df-fac 10649 df-bc 10671 df-ihash 10699 df-shft 10768 df-cj 10795 df-re 10796 df-im 10797 df-rsqrt 10951 df-abs 10952 df-clim 11231 df-sumdc 11306 df-ef 11600 df-sin 11602 df-cos 11603 df-pi 11605 df-rest 12570 df-topgen 12589 df-psmet 12742 df-xmet 12743 df-met 12744 df-bl 12745 df-mopn 12746 df-top 12751 df-topon 12764 df-bases 12796 df-ntr 12851 df-cn 12943 df-cnp 12944 df-tx 13008 df-cncf 13313 df-limced 13380 df-dvap 13381 |
This theorem is referenced by: ef2pi 13481 |
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