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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 26486 | . 2 ⊢ π ∈ ℝ | |
2 | 1 | recni 11278 | 1 ⊢ π ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ℂcc 11156 πcpi 16068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 |
This theorem is referenced by: negpicn 26490 pidiv2halves 26495 efhalfpi 26499 cospi 26500 efipi 26501 sin2pi 26503 cos2pi 26504 ef2pi 26505 ef2kpi 26506 efper 26507 sinperlem 26508 sin2kpi 26511 cos2kpi 26512 sin2pim 26513 cos2pim 26514 sinmpi 26515 cosmpi 26516 sinppi 26517 cosppi 26518 efimpi 26519 ptolemy 26524 sinq12gt0 26535 sinq34lt0t 26537 cosq14gt0 26538 cosq14ge0 26539 sincosq1eq 26540 sincos6thpi 26543 sincos3rdpi 26544 abssinper 26548 sinkpi 26549 coskpi 26550 sineq0 26551 coseq1 26552 efeq1 26555 cosne0 26556 resinf1o 26563 eff1o 26576 logi 26614 logneg 26615 logm1 26616 eflogeq 26629 argimgt0 26639 logneg2 26642 logf1o2 26677 cxpsqrt 26730 abscxpbnd 26781 root1eq1 26783 cxpeq 26785 ang180lem1 26837 ang180lem2 26838 ang180lem3 26839 ang180lem4 26840 acosf 26902 acosneg 26915 acoscos 26921 acos1 26923 sinacos 26933 atanlogsublem 26943 atanlogsub 26944 atantan 26951 atanbndlem 26953 basellem1 27109 efmul2picn 34442 itgexpif 34452 vtscl 34484 vtsprod 34485 circlemeth 34486 cos2h 37312 tan2h 37313 areacirc 37414 ef11d 42135 cxp112d 42137 cxp111d 42138 cxpi11d 42139 acos1half 42328 proot1ex 42861 coseq0 45485 coskpi2 45487 cosnegpi 45488 sinaover2ne0 45489 cosknegpi 45490 itgsinexplem1 45575 wallispilem4 45689 wallispi 45691 stirlinglem15 45709 dirker2re 45713 dirkerdenne0 45714 dirkerper 45717 dirkertrigeqlem1 45719 dirkertrigeqlem2 45720 dirkertrigeqlem3 45721 dirkertrigeq 45722 dirkeritg 45723 dirkercncflem1 45724 dirkercncflem2 45725 fourierdlem62 45789 fourierdlem66 45793 fourierdlem94 45821 fourierdlem95 45822 fourierdlem101 45828 fourierdlem102 45829 fourierdlem103 45830 fourierdlem111 45838 fourierdlem112 45839 fourierdlem113 45840 fourierdlem114 45841 sqwvfoura 45849 sqwvfourb 45850 fourierswlem 45851 fouriersw 45852 |
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