Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cos02pilt1 | Structured version Visualization version GIF version |
Description: Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 23-Mar-2024.) |
Ref | Expression |
---|---|
cos02pilt1 | ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 12965 | . . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℝ) | |
2 | 1 | recoscld 15705 | . 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ∈ ℝ) |
3 | 1red 10834 | . 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ∈ ℝ) | |
4 | cosbnd 15742 | . . . 4 ⊢ (𝐴 ∈ ℝ → (-1 ≤ (cos‘𝐴) ∧ (cos‘𝐴) ≤ 1)) | |
5 | 4 | simprd 499 | . . 3 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ≤ 1) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ≤ 1) |
7 | 0zd 12188 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℤ) | |
8 | 2re 11904 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
9 | pire 25348 | . . . . . . . . 9 ⊢ π ∈ ℝ | |
10 | 8, 9 | remulcli 10849 | . . . . . . . 8 ⊢ (2 · π) ∈ ℝ |
11 | 10 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ) |
12 | 0xr 10880 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
13 | 10 | rexri 10891 | . . . . . . . . 9 ⊢ (2 · π) ∈ ℝ* |
14 | elioo2 12976 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ*) → (𝐴 ∈ (0(,)(2 · π)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (2 · π)))) | |
15 | 12, 13, 14 | mp2an 692 | . . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (2 · π))) |
16 | 15 | simp2bi 1148 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < 𝐴) |
17 | 2rp 12591 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
18 | pirp 25351 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
19 | rpmulcl 12609 | . . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ π ∈ ℝ+) → (2 · π) ∈ ℝ+) | |
20 | 17, 18, 19 | mp2an 692 | . . . . . . . 8 ⊢ (2 · π) ∈ ℝ+ |
21 | rpgt0 12598 | . . . . . . . 8 ⊢ ((2 · π) ∈ ℝ+ → 0 < (2 · π)) | |
22 | 20, 21 | mp1i 13 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (2 · π)) |
23 | 1, 11, 16, 22 | divgt0d 11767 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (𝐴 / (2 · π))) |
24 | 20 | a1i 11 | . . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ+) |
25 | 15 | simp3bi 1149 | . . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 < (2 · π)) |
26 | 1, 11, 24, 25 | ltdiv1dd 12685 | . . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < ((2 · π) / (2 · π))) |
27 | 11 | recnd 10861 | . . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℂ) |
28 | 22 | gt0ne0d 11396 | . . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ≠ 0) |
29 | 27, 28 | dividd 11606 | . . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((2 · π) / (2 · π)) = 1) |
30 | 26, 29 | breqtrd 5079 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < 1) |
31 | 0p1e1 11952 | . . . . . . 7 ⊢ (0 + 1) = 1 | |
32 | 30, 31 | breqtrrdi 5095 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < (0 + 1)) |
33 | btwnnz 12253 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 0 < (𝐴 / (2 · π)) ∧ (𝐴 / (2 · π)) < (0 + 1)) → ¬ (𝐴 / (2 · π)) ∈ ℤ) | |
34 | 7, 23, 32, 33 | syl3anc 1373 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (𝐴 / (2 · π)) ∈ ℤ) |
35 | 1 | recnd 10861 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℂ) |
36 | coseq1 25414 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ)) | |
37 | 35, 36 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ)) |
38 | 34, 37 | mtbird 328 | . . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (cos‘𝐴) = 1) |
39 | 38 | neqned 2947 | . . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ≠ 1) |
40 | 39 | necomd 2996 | . 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ≠ (cos‘𝐴)) |
41 | 2, 3, 6, 40 | leneltd 10986 | 1 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 -cneg 11063 / cdiv 11489 2c2 11885 ℤcz 12176 ℝ+crp 12586 (,)cioo 12935 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: cosq34lt1 25416 cos0pilt1 25421 |
Copyright terms: Public domain | W3C validator |