Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > negpilt0 | Structured version Visualization version GIF version |
Description: Negative π is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
negpilt0 | ⊢ -π < 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pipos 25374 | . 2 ⊢ 0 < π | |
2 | pire 25372 | . . 3 ⊢ π ∈ ℝ | |
3 | lt0neg2 11363 | . . 3 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (0 < π ↔ -π < 0) |
5 | 1, 4 | mpbi 233 | 1 ⊢ -π < 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 class class class wbr 5067 ℝcr 10752 0cc0 10753 < clt 10891 -cneg 11087 πcpi 15652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 ax-addf 10832 ax-mulf 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-of 7487 df-om 7663 df-1st 7779 df-2nd 7780 df-supp 7924 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-2o 8223 df-er 8411 df-map 8530 df-pm 8531 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fsupp 9010 df-fi 9051 df-sup 9082 df-inf 9083 df-oi 9150 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-7 11922 df-8 11923 df-9 11924 df-n0 12115 df-z 12201 df-dec 12318 df-uz 12463 df-q 12569 df-rp 12611 df-xneg 12728 df-xadd 12729 df-xmul 12730 df-ioo 12963 df-ioc 12964 df-ico 12965 df-icc 12966 df-fz 13120 df-fzo 13263 df-fl 13391 df-seq 13599 df-exp 13660 df-fac 13864 df-bc 13893 df-hash 13921 df-shft 14654 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-limsup 15056 df-clim 15073 df-rlim 15074 df-sum 15274 df-ef 15653 df-sin 15655 df-cos 15656 df-pi 15658 df-struct 16724 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-mulr 16840 df-starv 16841 df-sca 16842 df-vsca 16843 df-ip 16844 df-tset 16845 df-ple 16846 df-ds 16848 df-unif 16849 df-hom 16850 df-cco 16851 df-rest 16951 df-topn 16952 df-0g 16970 df-gsum 16971 df-topgen 16972 df-pt 16973 df-prds 16976 df-xrs 17031 df-qtop 17036 df-imas 17037 df-xps 17039 df-mre 17113 df-mrc 17114 df-acs 17116 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-submnd 18243 df-mulg 18513 df-cntz 18735 df-cmn 19196 df-psmet 20379 df-xmet 20380 df-met 20381 df-bl 20382 df-mopn 20383 df-fbas 20384 df-fg 20385 df-cnfld 20388 df-top 21815 df-topon 21832 df-topsp 21854 df-bases 21867 df-cld 21940 df-ntr 21941 df-cls 21942 df-nei 22019 df-lp 22057 df-perf 22058 df-cn 22148 df-cnp 22149 df-haus 22236 df-tx 22483 df-hmeo 22676 df-fil 22767 df-fm 22859 df-flim 22860 df-flf 22861 df-xms 23242 df-ms 23243 df-tms 23244 df-cncf 23799 df-limc 24787 df-dv 24788 |
This theorem is referenced by: fourierdlem62 43412 fourierdlem83 43433 fourierdlem94 43444 fourierdlem102 43452 fourierdlem103 43453 fourierdlem104 43454 fourierdlem111 43461 fourierdlem112 43462 fourierdlem113 43463 fourierdlem114 43464 sqwvfoura 43472 sqwvfourb 43473 fouriersw 43475 |
Copyright terms: Public domain | W3C validator |