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Mirrors > Home > MPE Home > Th. List > picn | Structured version Visualization version GIF version |
Description: π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.) |
Ref | Expression |
---|---|
picn | ⊢ π ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 24255 | . 2 ⊢ π ∈ ℝ | |
2 | 1 | recni 10090 | 1 ⊢ π ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 ℂcc 9972 πcpi 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 |
This theorem is referenced by: negpicn 24259 pidiv2halves 24264 efhalfpi 24268 cospi 24269 efipi 24270 sin2pi 24272 cos2pi 24273 ef2pi 24274 ef2kpi 24275 efper 24276 sinperlem 24277 sin2kpi 24280 cos2kpi 24281 sin2pim 24282 cos2pim 24283 sinmpi 24284 cosmpi 24285 sinppi 24286 cosppi 24287 efimpi 24288 sinhalfpip 24289 sinhalfpim 24290 coshalfpip 24291 coshalfpim 24292 ptolemy 24293 sinq12gt0 24304 sinq34lt0t 24306 cosq14gt0 24307 cosq14ge0 24308 sincosq1eq 24309 sincos6thpi 24312 sincos3rdpi 24313 abssinper 24315 sinkpi 24316 coskpi 24317 sineq0 24318 coseq1 24319 efeq1 24320 cosne0 24321 resinf1o 24327 eff1o 24340 logneg 24379 logm1 24380 eflogeq 24393 argimgt0 24403 logneg2 24406 logf1o2 24441 cxpsqrt 24494 abscxpbnd 24539 root1eq1 24541 cxpeq 24543 ang180lem1 24584 ang180lem2 24585 ang180lem3 24586 ang180lem4 24587 acosf 24646 acosneg 24659 acoscos 24665 acos1 24667 sinacos 24677 atanlogsublem 24687 atanlogsub 24688 atantan 24695 atanbndlem 24697 basellem1 24852 efmul2picn 30802 itgexpif 30812 vtscl 30844 vtsprod 30845 circlemeth 30846 logi 31746 cos2h 33530 tan2h 33531 areacirc 33635 proot1ex 38096 coseq0 40393 coskpi2 40395 cosnegpi 40396 sinaover2ne0 40397 cosknegpi 40398 itgsinexplem1 40487 wallispilem4 40603 wallispi 40605 stirlinglem15 40623 dirker2re 40627 dirkerdenne0 40628 dirkerper 40631 dirkertrigeqlem1 40633 dirkertrigeqlem2 40634 dirkertrigeqlem3 40635 dirkertrigeq 40636 dirkeritg 40637 dirkercncflem1 40638 dirkercncflem2 40639 fourierdlem62 40703 fourierdlem66 40707 fourierdlem94 40735 fourierdlem95 40736 fourierdlem101 40742 fourierdlem102 40743 fourierdlem103 40744 fourierdlem111 40752 fourierdlem112 40753 fourierdlem113 40754 fourierdlem114 40755 sqwvfoura 40763 sqwvfourb 40764 fourierswlem 40765 fouriersw 40766 |
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