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Mirrors > Home > ILE Home > Th. List > cosq23lt0 | Unicode version |
Description: The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.) |
Ref | Expression |
---|---|
cosq23lt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 9856 | . . . 4 | |
2 | 1 | recnd 7935 | . . 3 |
3 | sinhalfpip 13494 | . . 3 | |
4 | 2, 3 | syl 14 | . 2 |
5 | halfpire 13466 | . . . . . 6 | |
6 | 5 | a1i 9 | . . . . 5 |
7 | 6, 1 | readdcld 7936 | . . . 4 |
8 | pidiv2halves 13469 | . . . . 5 | |
9 | 5 | rexri 7964 | . . . . . . . 8 |
10 | 3re 8939 | . . . . . . . . . 10 | |
11 | 10, 5 | remulcli 7921 | . . . . . . . . 9 |
12 | 11 | rexri 7964 | . . . . . . . 8 |
13 | elioo2 9865 | . . . . . . . 8 | |
14 | 9, 12, 13 | mp2an 424 | . . . . . . 7 |
15 | 14 | simp2bi 1008 | . . . . . 6 |
16 | 6, 1, 6, 15 | ltadd2dd 8328 | . . . . 5 |
17 | 8, 16 | eqbrtrrid 4023 | . . . 4 |
18 | 11 | a1i 9 | . . . . . 6 |
19 | 14 | simp3bi 1009 | . . . . . 6 |
20 | 1, 18, 6, 19 | ltadd2dd 8328 | . . . . 5 |
21 | ax-1cn 7854 | . . . . . . . 8 | |
22 | 3cn 8940 | . . . . . . . 8 | |
23 | 5 | recni 7919 | . . . . . . . 8 |
24 | 21, 22, 23 | adddiri 7918 | . . . . . . 7 |
25 | 3p1e4 9000 | . . . . . . . . 9 | |
26 | 22, 21, 25 | addcomli 8051 | . . . . . . . 8 |
27 | 26 | oveq1i 5860 | . . . . . . 7 |
28 | 23 | mulid2i 7910 | . . . . . . . 8 |
29 | 28 | oveq1i 5860 | . . . . . . 7 |
30 | 24, 27, 29 | 3eqtr3ri 2200 | . . . . . 6 |
31 | 4cn 8943 | . . . . . . 7 | |
32 | 2cn 8936 | . . . . . . . 8 | |
33 | 2ap0 8958 | . . . . . . . 8 # | |
34 | 32, 33 | pm3.2i 270 | . . . . . . 7 # |
35 | picn 13461 | . . . . . . 7 | |
36 | div32ap 8596 | . . . . . . 7 # | |
37 | 31, 34, 35, 36 | mp3an 1332 | . . . . . 6 |
38 | 4d2e2 9025 | . . . . . . 7 | |
39 | 38 | oveq1i 5860 | . . . . . 6 |
40 | 30, 37, 39 | 3eqtr2i 2197 | . . . . 5 |
41 | 20, 40 | breqtrdi 4028 | . . . 4 |
42 | pire 13460 | . . . . . 6 | |
43 | 42 | rexri 7964 | . . . . 5 |
44 | 2re 8935 | . . . . . . 7 | |
45 | 44, 42 | remulcli 7921 | . . . . . 6 |
46 | 45 | rexri 7964 | . . . . 5 |
47 | elioo2 9865 | . . . . 5 | |
48 | 43, 46, 47 | mp2an 424 | . . . 4 |
49 | 7, 17, 41, 48 | syl3anbrc 1176 | . . 3 |
50 | sinq34lt0t 13505 | . . 3 | |
51 | 49, 50 | syl 14 | . 2 |
52 | 4, 51 | eqbrtrrd 4011 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 cfv 5196 (class class class)co 5850 cc 7759 cr 7760 cc0 7761 c1 7762 caddc 7764 cmul 7766 cxr 7940 clt 7941 # cap 8487 cdiv 8576 c2 8916 c3 8917 c4 8918 cioo 9832 csin 11594 ccos 11595 cpi 11597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 ax-pre-suploc 7882 ax-addf 7883 ax-mulf 7884 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-of 6058 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-oadd 6396 df-er 6509 df-map 6624 df-pm 6625 df-en 6715 df-dom 6716 df-fin 6717 df-sup 6957 df-inf 6958 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-5 8927 df-6 8928 df-7 8929 df-8 8930 df-9 8931 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-xneg 9716 df-xadd 9717 df-ioo 9836 df-ioc 9837 df-ico 9838 df-icc 9839 df-fz 9953 df-fzo 10086 df-seqfrec 10389 df-exp 10463 df-fac 10647 df-bc 10669 df-ihash 10697 df-shft 10766 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-sumdc 11304 df-ef 11598 df-sin 11600 df-cos 11601 df-pi 11603 df-rest 12568 df-topgen 12587 df-psmet 12740 df-xmet 12741 df-met 12742 df-bl 12743 df-mopn 12744 df-top 12749 df-topon 12762 df-bases 12794 df-ntr 12849 df-cn 12941 df-cnp 12942 df-tx 13006 df-cncf 13311 df-limced 13378 df-dvap 13379 |
This theorem is referenced by: coseq0q4123 13508 cos02pilt1 13525 |
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