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Mirrors > Home > ILE Home > Th. List > halfpire | Unicode version |
Description: is real. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
halfpire |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 12877 | . 2 | |
2 | 1 | rehalfcli 8975 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 1480 (class class class)co 5774 cr 7626 cdiv 8439 c2 8778 cpi 11360 |
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 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 ax-arch 7746 ax-caucvg 7747 ax-pre-suploc 7748 ax-addf 7749 ax-mulf 7750 |
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 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 df-inn 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 df-9 8793 df-n0 8985 df-z 9062 df-uz 9334 df-q 9419 df-rp 9449 df-xneg 9566 df-xadd 9567 df-ioo 9682 df-ioc 9683 df-ico 9684 df-icc 9685 df-fz 9798 df-fzo 9927 df-seqfrec 10226 df-exp 10300 df-fac 10479 df-bc 10501 df-ihash 10529 df-shft 10594 df-cj 10621 df-re 10622 df-im 10623 df-rsqrt 10777 df-abs 10778 df-clim 11055 df-sumdc 11130 df-ef 11361 df-sin 11363 df-cos 11364 df-pi 11366 df-rest 12132 df-topgen 12151 df-psmet 12166 df-xmet 12167 df-met 12168 df-bl 12169 df-mopn 12170 df-top 12175 df-topon 12188 df-bases 12220 df-ntr 12275 df-cn 12367 df-cnp 12368 df-tx 12432 df-cncf 12737 df-limced 12804 df-dvap 12805 |
This theorem is referenced by: neghalfpire 12884 cosneghalfpi 12889 sinhalfpip 12911 sinhalfpim 12912 coshalfpip 12913 coshalfpim 12914 sincosq1lem 12916 sincosq1sgn 12917 sincosq2sgn 12918 sincosq3sgn 12919 sincosq4sgn 12920 cosq14gt0 12923 cosq23lt0 12924 coseq0q4123 12925 coseq00topi 12926 coseq0negpitopi 12927 sincosq1eq 12930 sincos6thpi 12933 cos02pilt1 12942 |
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