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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 25044 | . 2 ⊢ π ∈ ℝ | |
2 | 1 | recni 10655 | 1 ⊢ π ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ℂcc 10535 πcpi 15420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 |
This theorem is referenced by: negpicn 25048 pidiv2halves 25053 efhalfpi 25057 cospi 25058 efipi 25059 sin2pi 25061 cos2pi 25062 ef2pi 25063 ef2kpi 25064 efper 25065 sinperlem 25066 sin2kpi 25069 cos2kpi 25070 sin2pim 25071 cos2pim 25072 sinmpi 25073 cosmpi 25074 sinppi 25075 cosppi 25076 efimpi 25077 ptolemy 25082 sinq12gt0 25093 sinq34lt0t 25095 cosq14gt0 25096 cosq14ge0 25097 sincosq1eq 25098 sincos6thpi 25101 sincos3rdpi 25102 abssinper 25106 sinkpi 25107 coskpi 25108 sineq0 25109 coseq1 25110 efeq1 25113 cosne0 25114 resinf1o 25120 eff1o 25133 logneg 25171 logm1 25172 eflogeq 25185 argimgt0 25195 logneg2 25198 logf1o2 25233 cxpsqrt 25286 abscxpbnd 25334 root1eq1 25336 cxpeq 25338 ang180lem1 25387 ang180lem2 25388 ang180lem3 25389 ang180lem4 25390 acosf 25452 acosneg 25465 acoscos 25471 acos1 25473 sinacos 25483 atanlogsublem 25493 atanlogsub 25494 atantan 25501 atanbndlem 25503 basellem1 25658 efmul2picn 31867 itgexpif 31877 vtscl 31909 vtsprod 31910 circlemeth 31911 logi 32966 cos2h 34898 tan2h 34899 areacirc 35002 proot1ex 39850 coseq0 42194 coskpi2 42196 cosnegpi 42197 sinaover2ne0 42198 cosknegpi 42199 itgsinexplem1 42288 wallispilem4 42402 wallispi 42404 stirlinglem15 42422 dirker2re 42426 dirkerdenne0 42427 dirkerper 42430 dirkertrigeqlem1 42432 dirkertrigeqlem2 42433 dirkertrigeqlem3 42434 dirkertrigeq 42435 dirkeritg 42436 dirkercncflem1 42437 dirkercncflem2 42438 fourierdlem62 42502 fourierdlem66 42506 fourierdlem94 42534 fourierdlem95 42535 fourierdlem101 42541 fourierdlem102 42542 fourierdlem103 42543 fourierdlem111 42551 fourierdlem112 42552 fourierdlem113 42553 fourierdlem114 42554 sqwvfoura 42562 sqwvfourb 42563 fourierswlem 42564 fouriersw 42565 |
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