Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 25302 | . 2 ⊢ π ∈ ℝ | |
2 | 1 | recni 10812 | 1 ⊢ π ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 ℂcc 10692 πcpi 15591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cncf 23729 df-limc 24717 df-dv 24718 |
This theorem is referenced by: negpicn 25306 pidiv2halves 25311 efhalfpi 25315 cospi 25316 efipi 25317 sin2pi 25319 cos2pi 25320 ef2pi 25321 ef2kpi 25322 efper 25323 sinperlem 25324 sin2kpi 25327 cos2kpi 25328 sin2pim 25329 cos2pim 25330 sinmpi 25331 cosmpi 25332 sinppi 25333 cosppi 25334 efimpi 25335 ptolemy 25340 sinq12gt0 25351 sinq34lt0t 25353 cosq14gt0 25354 cosq14ge0 25355 sincosq1eq 25356 sincos6thpi 25359 sincos3rdpi 25360 abssinper 25364 sinkpi 25365 coskpi 25366 sineq0 25367 coseq1 25368 efeq1 25371 cosne0 25372 resinf1o 25379 eff1o 25392 logneg 25430 logm1 25431 eflogeq 25444 argimgt0 25454 logneg2 25457 logf1o2 25492 cxpsqrt 25545 abscxpbnd 25593 root1eq1 25595 cxpeq 25597 ang180lem1 25646 ang180lem2 25647 ang180lem3 25648 ang180lem4 25649 acosf 25711 acosneg 25724 acoscos 25730 acos1 25732 sinacos 25742 atanlogsublem 25752 atanlogsub 25753 atantan 25760 atanbndlem 25762 basellem1 25917 efmul2picn 32242 itgexpif 32252 vtscl 32284 vtsprod 32285 circlemeth 32286 logi 33369 cos2h 35454 tan2h 35455 areacirc 35556 acos1half 39833 proot1ex 40670 coseq0 43023 coskpi2 43025 cosnegpi 43026 sinaover2ne0 43027 cosknegpi 43028 itgsinexplem1 43113 wallispilem4 43227 wallispi 43229 stirlinglem15 43247 dirker2re 43251 dirkerdenne0 43252 dirkerper 43255 dirkertrigeqlem1 43257 dirkertrigeqlem2 43258 dirkertrigeqlem3 43259 dirkertrigeq 43260 dirkeritg 43261 dirkercncflem1 43262 dirkercncflem2 43263 fourierdlem62 43327 fourierdlem66 43331 fourierdlem94 43359 fourierdlem95 43360 fourierdlem101 43366 fourierdlem102 43367 fourierdlem103 43368 fourierdlem111 43376 fourierdlem112 43377 fourierdlem113 43378 fourierdlem114 43379 sqwvfoura 43387 sqwvfourb 43388 fourierswlem 43389 fouriersw 43390 |
Copyright terms: Public domain | W3C validator |