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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 26500 | . 2 ⊢ π ∈ ℝ | |
| 2 | 1 | recni 11275 | 1 ⊢ π ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ℂcc 11153 πcpi 16102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 |
| This theorem is referenced by: negpicn 26504 pidiv2halves 26509 efhalfpi 26513 cospi 26514 efipi 26515 sin2pi 26517 cos2pi 26518 ef2pi 26519 ef2kpi 26520 efper 26521 sinperlem 26522 sin2kpi 26525 cos2kpi 26526 sin2pim 26527 cos2pim 26528 sinmpi 26529 cosmpi 26530 sinppi 26531 cosppi 26532 efimpi 26533 ptolemy 26538 sinq12gt0 26549 sinq34lt0t 26551 cosq14gt0 26552 cosq14ge0 26553 sincosq1eq 26554 tan4thpi 26556 sincos6thpi 26558 sincos3rdpi 26559 abssinper 26563 sinkpi 26564 coskpi 26565 sineq0 26566 coseq1 26567 efeq1 26570 cosne0 26571 resinf1o 26578 eff1o 26591 logi 26629 logneg 26630 logm1 26631 eflogeq 26644 argimgt0 26654 logneg2 26657 logf1o2 26692 cxpsqrt 26745 abscxpbnd 26796 root1eq1 26798 cxpeq 26800 ang180lem1 26852 ang180lem2 26853 ang180lem3 26854 ang180lem4 26855 acosf 26917 acosneg 26930 acoscos 26936 acos1 26938 sinacos 26948 atanlogsublem 26958 atanlogsub 26959 atantan 26966 atanbndlem 26968 basellem1 27124 efmul2picn 34611 itgexpif 34621 vtscl 34653 vtsprod 34654 circlemeth 34655 cos2h 37618 tan2h 37619 areacirc 37720 ef11d 42375 cxp112d 42377 cxp111d 42378 cxpi11d 42379 tanhalfpim 42385 tan3rdpi 42386 acos1half 42388 proot1ex 43208 coseq0 45879 coskpi2 45881 cosnegpi 45882 sinaover2ne0 45883 cosknegpi 45884 itgsinexplem1 45969 wallispilem4 46083 wallispi 46085 stirlinglem15 46103 dirker2re 46107 dirkerdenne0 46108 dirkerper 46111 dirkertrigeqlem1 46113 dirkertrigeqlem2 46114 dirkertrigeqlem3 46115 dirkertrigeq 46116 dirkeritg 46117 dirkercncflem1 46118 dirkercncflem2 46119 fourierdlem62 46183 fourierdlem66 46187 fourierdlem94 46215 fourierdlem95 46216 fourierdlem101 46222 fourierdlem102 46223 fourierdlem103 46224 fourierdlem111 46232 fourierdlem112 46233 fourierdlem113 46234 fourierdlem114 46235 sqwvfoura 46243 sqwvfourb 46244 fourierswlem 46245 fouriersw 46246 |
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