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Mirrors > Home > MPE Home > Th. List > pipos | Structured version Visualization version GIF version |
Description: π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.) |
Ref | Expression |
---|---|
pipos | ⊢ 0 < π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2pos 12366 | . 2 ⊢ 0 < 2 | |
2 | pigt2lt4 26512 | . . 3 ⊢ (2 < π ∧ π < 4) | |
3 | 2 | simpli 483 | . 2 ⊢ 2 < π |
4 | 0re 11260 | . . 3 ⊢ 0 ∈ ℝ | |
5 | 2re 12337 | . . 3 ⊢ 2 ∈ ℝ | |
6 | pire 26514 | . . 3 ⊢ π ∈ ℝ | |
7 | 4, 5, 6 | lttri 11384 | . 2 ⊢ ((0 < 2 ∧ 2 < π) → 0 < π) |
8 | 1, 3, 7 | mp2an 692 | 1 ⊢ 0 < π |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5147 0cc0 11152 < clt 11292 2c2 12318 4c4 12320 πcpi 16098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 |
This theorem is referenced by: pirp 26517 sinhalfpilem 26519 sincos4thpi 26569 sincos6thpi 26572 pigt3 26574 sineq0 26580 coseq1 26581 cosq34lt1 26583 efeq1 26584 cosne0 26585 cos0pilt1 26588 recosf1o 26591 tanord1 26593 efif1olem2 26599 efif1olem4 26601 relogrn 26617 logi 26643 logneg 26644 eflogeq 26658 logneg2 26671 logf1o2 26706 root1eq1 26812 logbrec 26839 ang180lem1 26866 ang180lem2 26867 ang180lem3 26868 asin1 26951 basellem4 27141 itgexpif 34599 bj-pinftyccb 37203 bj-minftyccb 37207 bj-pinftynminfty 37209 tan2h 37598 pine0 42326 asin1half 42365 acos1half 42366 proot1ex 43184 isosctrlem1ALT 44931 sineq0ALT 44934 negpilt0 45230 coseq0 45819 sinaover2ne0 45823 itgsin0pilem1 45905 itgsinexplem1 45909 wallispilem2 46021 wallispi 46025 stirlinglem15 46043 stirlingr 46045 dirker2re 46047 dirkerdenne0 46048 dirkerval2 46049 dirkerre 46050 dirkertrigeqlem1 46053 dirkertrigeqlem2 46054 dirkertrigeqlem3 46055 dirkertrigeq 46056 dirkeritg 46057 dirkercncflem1 46058 dirkercncflem2 46059 dirkercncflem4 46061 fourierdlem16 46078 fourierdlem21 46083 fourierdlem22 46084 fourierdlem24 46086 fourierdlem62 46123 fourierdlem66 46127 fourierdlem83 46144 fourierdlem94 46155 fourierdlem95 46156 fourierdlem102 46163 fourierdlem103 46164 fourierdlem104 46165 fourierdlem111 46172 fourierdlem112 46173 fourierdlem113 46174 fourierdlem114 46175 sqwvfoura 46183 sqwvfourb 46184 fourierswlem 46185 fouriersw 46186 |
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