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Mirrors > Home > ILE Home > Th. List > cosper | Unicode version |
Description: The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.) |
Ref | Expression |
---|---|
cosper |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosval 11753 |
. 2
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2 | cosval 11753 |
. 2
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3 | 1, 2 | sinperlem 14732 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 ax-caucvg 7966 ax-pre-suploc 7967 ax-addf 7968 ax-mulf 7969 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-disj 3999 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-isom 5247 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-of 6110 df-1st 6169 df-2nd 6170 df-recs 6334 df-irdg 6399 df-frec 6420 df-1o 6445 df-oadd 6449 df-er 6563 df-map 6680 df-pm 6681 df-en 6771 df-dom 6772 df-fin 6773 df-sup 7017 df-inf 7018 df-pnf 8030 df-mnf 8031 df-xr 8032 df-ltxr 8033 df-le 8034 df-sub 8166 df-neg 8167 df-reap 8568 df-ap 8575 df-div 8666 df-inn 8956 df-2 9014 df-3 9015 df-4 9016 df-5 9017 df-6 9018 df-7 9019 df-8 9020 df-9 9021 df-n0 9213 df-z 9290 df-uz 9565 df-q 9657 df-rp 9691 df-xneg 9809 df-xadd 9810 df-ioo 9929 df-ioc 9930 df-ico 9931 df-icc 9932 df-fz 10046 df-fzo 10180 df-seqfrec 10486 df-exp 10561 df-fac 10748 df-bc 10770 df-ihash 10798 df-shft 10866 df-cj 10893 df-re 10894 df-im 10895 df-rsqrt 11049 df-abs 11050 df-clim 11329 df-sumdc 11404 df-ef 11698 df-sin 11700 df-cos 11701 df-pi 11703 df-rest 12758 df-topgen 12777 df-psmet 13882 df-xmet 13883 df-met 13884 df-bl 13885 df-mopn 13886 df-top 13984 df-topon 13997 df-bases 14029 df-ntr 14082 df-cn 14174 df-cnp 14175 df-tx 14239 df-cncf 14544 df-limced 14628 df-dvap 14629 |
This theorem is referenced by: cos2kpi 14736 cos2pim 14738 cosq34lt1 14774 |
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