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Mirrors > Home > MPE Home > Th. List > cos0pilt1 | Structured version Visualization version GIF version |
Description: Cosine is between minus one and one on the open interval between zero and π. (Contributed by Jim Kingdon, 7-May-2024.) |
Ref | Expression |
---|---|
cos0pilt1 | ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ (-1(,)1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 12965 | . . 3 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 ∈ ℝ) | |
2 | 1 | recoscld 15705 | . 2 ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ ℝ) |
3 | cospi 25362 | . . 3 ⊢ (cos‘π) = -1 | |
4 | ioossicc 13021 | . . . . 5 ⊢ (0(,)π) ⊆ (0[,]π) | |
5 | 4 | sseli 3896 | . . . 4 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 ∈ (0[,]π)) |
6 | 0xr 10880 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
7 | pire 25348 | . . . . . . 7 ⊢ π ∈ ℝ | |
8 | 7 | rexri 10891 | . . . . . 6 ⊢ π ∈ ℝ* |
9 | 0re 10835 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
10 | pipos 25350 | . . . . . . 7 ⊢ 0 < π | |
11 | 9, 7, 10 | ltleii 10955 | . . . . . 6 ⊢ 0 ≤ π |
12 | ubicc2 13053 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ π ∈ ℝ* ∧ 0 ≤ π) → π ∈ (0[,]π)) | |
13 | 6, 8, 11, 12 | mp3an 1463 | . . . . 5 ⊢ π ∈ (0[,]π) |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ (0(,)π) → π ∈ (0[,]π)) |
15 | eliooord 12994 | . . . . 5 ⊢ (𝐴 ∈ (0(,)π) → (0 < 𝐴 ∧ 𝐴 < π)) | |
16 | 15 | simprd 499 | . . . 4 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 < π) |
17 | 5, 14, 16 | cosordlem 25419 | . . 3 ⊢ (𝐴 ∈ (0(,)π) → (cos‘π) < (cos‘𝐴)) |
18 | 3, 17 | eqbrtrrid 5089 | . 2 ⊢ (𝐴 ∈ (0(,)π) → -1 < (cos‘𝐴)) |
19 | 2re 11904 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
20 | 19, 7 | remulcli 10849 | . . . . . 6 ⊢ (2 · π) ∈ ℝ |
21 | 20 | rexri 10891 | . . . . 5 ⊢ (2 · π) ∈ ℝ* |
22 | 1le2 12039 | . . . . . 6 ⊢ 1 ≤ 2 | |
23 | lemulge12 11695 | . . . . . 6 ⊢ (((π ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ π ∧ 1 ≤ 2)) → π ≤ (2 · π)) | |
24 | 7, 19, 11, 22, 23 | mp4an 693 | . . . . 5 ⊢ π ≤ (2 · π) |
25 | iooss2 12971 | . . . . 5 ⊢ (((2 · π) ∈ ℝ* ∧ π ≤ (2 · π)) → (0(,)π) ⊆ (0(,)(2 · π))) | |
26 | 21, 24, 25 | mp2an 692 | . . . 4 ⊢ (0(,)π) ⊆ (0(,)(2 · π)) |
27 | 26 | sseli 3896 | . . 3 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 ∈ (0(,)(2 · π))) |
28 | cos02pilt1 25415 | . . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1) | |
29 | 27, 28 | syl 17 | . 2 ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) < 1) |
30 | neg1rr 11945 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 10891 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 1re 10833 | . . . 4 ⊢ 1 ∈ ℝ | |
33 | 32 | rexri 10891 | . . 3 ⊢ 1 ∈ ℝ* |
34 | elioo2 12976 | . . 3 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((cos‘𝐴) ∈ (-1(,)1) ↔ ((cos‘𝐴) ∈ ℝ ∧ -1 < (cos‘𝐴) ∧ (cos‘𝐴) < 1))) | |
35 | 31, 33, 34 | mp2an 692 | . 2 ⊢ ((cos‘𝐴) ∈ (-1(,)1) ↔ ((cos‘𝐴) ∈ ℝ ∧ -1 < (cos‘𝐴) ∧ (cos‘𝐴) < 1)) |
36 | 2, 18, 29, 35 | syl3anbrc 1345 | 1 ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 ∈ wcel 2110 ⊆ wss 3866 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 · cmul 10734 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 -cneg 11063 2c2 11885 (,)cioo 12935 [,]cicc 12938 cosccos 15626 πcpi 15628 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 |
This theorem is referenced by: (None) |
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