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Mirrors > Home > MPE Home > Th. List > halfpire | Structured version Visualization version GIF version |
Description: π / 2 is real. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
halfpire | ⊢ (π / 2) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 25626 | . 2 ⊢ π ∈ ℝ | |
2 | 1 | rehalfcli 12233 | 1 ⊢ (π / 2) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 (class class class)co 7272 ℝcr 10881 / cdiv 11643 2c2 12039 πcpi 15787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-er 8490 df-map 8609 df-pm 8610 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-ioo 13094 df-ioc 13095 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-fl 13523 df-seq 13733 df-exp 13794 df-fac 13999 df-bc 14028 df-hash 14056 df-shft 14789 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-limsup 15191 df-clim 15208 df-rlim 15209 df-sum 15409 df-ef 15788 df-sin 15790 df-cos 15791 df-pi 15793 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-pt 17166 df-prds 17169 df-xrs 17224 df-qtop 17229 df-imas 17230 df-xps 17232 df-mre 17306 df-mrc 17307 df-acs 17309 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-mulg 18712 df-cntz 18934 df-cmn 19399 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-fbas 20605 df-fg 20606 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cld 22181 df-ntr 22182 df-cls 22183 df-nei 22260 df-lp 22298 df-perf 22299 df-cn 22389 df-cnp 22390 df-haus 22477 df-tx 22724 df-hmeo 22917 df-fil 23008 df-fm 23100 df-flim 23101 df-flf 23102 df-xms 23484 df-ms 23485 df-tms 23486 df-cncf 24052 df-limc 25041 df-dv 25042 |
This theorem is referenced by: neghalfpire 25633 cosneghalfpi 25638 sinhalfpip 25660 sinhalfpim 25661 coshalfpip 25662 coshalfpim 25663 sincosq1lem 25665 sincosq1sgn 25666 sincosq2sgn 25667 sincosq3sgn 25668 sincosq4sgn 25669 coseq00topi 25670 coseq0negpitopi 25671 tanabsge 25674 cosq14gt0 25678 cosq14ge0 25679 sincos6thpi 25683 cosne0 25696 sinord 25701 resinf1o 25703 tanord1 25704 tanord 25705 tanregt0 25706 efif1olem4 25712 argregt0 25776 argrege0 25777 logimul 25780 isosctrlem1 25979 asinsinlem 26052 asinsin 26053 acoscos 26054 asin1 26055 reasinsin 26057 acosbnd 26061 asinrecl 26063 acosrecl 26064 atanlogsublem 26076 atantan 26084 atanbndlem 26086 atanbnd 26087 atan1 26089 basellem4 26244 logi 33709 iexpire 33710 cos2h 35777 tan2h 35778 dvacos 35871 fourierdlem62 43691 |
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