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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 13350 | . . 3 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 ∈ ℝ) | |
2 | 1 | recoscld 16083 | . 2 ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ ℝ) |
3 | cospi 25964 | . . 3 ⊢ (cos‘π) = -1 | |
4 | ioossicc 13406 | . . . . 5 ⊢ (0(,)π) ⊆ (0[,]π) | |
5 | 4 | sseli 3977 | . . . 4 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 ∈ (0[,]π)) |
6 | 0xr 11257 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
7 | pire 25950 | . . . . . . 7 ⊢ π ∈ ℝ | |
8 | 7 | rexri 11268 | . . . . . 6 ⊢ π ∈ ℝ* |
9 | 0re 11212 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
10 | pipos 25952 | . . . . . . 7 ⊢ 0 < π | |
11 | 9, 7, 10 | ltleii 11333 | . . . . . 6 ⊢ 0 ≤ π |
12 | ubicc2 13438 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ π ∈ ℝ* ∧ 0 ≤ π) → π ∈ (0[,]π)) | |
13 | 6, 8, 11, 12 | mp3an 1462 | . . . . 5 ⊢ π ∈ (0[,]π) |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ (0(,)π) → π ∈ (0[,]π)) |
15 | eliooord 13379 | . . . . 5 ⊢ (𝐴 ∈ (0(,)π) → (0 < 𝐴 ∧ 𝐴 < π)) | |
16 | 15 | simprd 497 | . . . 4 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 < π) |
17 | 5, 14, 16 | cosordlem 26021 | . . 3 ⊢ (𝐴 ∈ (0(,)π) → (cos‘π) < (cos‘𝐴)) |
18 | 3, 17 | eqbrtrrid 5183 | . 2 ⊢ (𝐴 ∈ (0(,)π) → -1 < (cos‘𝐴)) |
19 | 2re 12282 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
20 | 19, 7 | remulcli 11226 | . . . . . 6 ⊢ (2 · π) ∈ ℝ |
21 | 20 | rexri 11268 | . . . . 5 ⊢ (2 · π) ∈ ℝ* |
22 | 1le2 12417 | . . . . . 6 ⊢ 1 ≤ 2 | |
23 | lemulge12 12073 | . . . . . 6 ⊢ (((π ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ π ∧ 1 ≤ 2)) → π ≤ (2 · π)) | |
24 | 7, 19, 11, 22, 23 | mp4an 692 | . . . . 5 ⊢ π ≤ (2 · π) |
25 | iooss2 13356 | . . . . 5 ⊢ (((2 · π) ∈ ℝ* ∧ π ≤ (2 · π)) → (0(,)π) ⊆ (0(,)(2 · π))) | |
26 | 21, 24, 25 | mp2an 691 | . . . 4 ⊢ (0(,)π) ⊆ (0(,)(2 · π)) |
27 | 26 | sseli 3977 | . . 3 ⊢ (𝐴 ∈ (0(,)π) → 𝐴 ∈ (0(,)(2 · π))) |
28 | cos02pilt1 26017 | . . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1) | |
29 | 27, 28 | syl 17 | . 2 ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) < 1) |
30 | neg1rr 12323 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 11268 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 1re 11210 | . . . 4 ⊢ 1 ∈ ℝ | |
33 | 32 | rexri 11268 | . . 3 ⊢ 1 ∈ ℝ* |
34 | elioo2 13361 | . . 3 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((cos‘𝐴) ∈ (-1(,)1) ↔ ((cos‘𝐴) ∈ ℝ ∧ -1 < (cos‘𝐴) ∧ (cos‘𝐴) < 1))) | |
35 | 31, 33, 34 | mp2an 691 | . 2 ⊢ ((cos‘𝐴) ∈ (-1(,)1) ↔ ((cos‘𝐴) ∈ ℝ ∧ -1 < (cos‘𝐴) ∧ (cos‘𝐴) < 1)) |
36 | 2, 18, 29, 35 | syl3anbrc 1344 | 1 ⊢ (𝐴 ∈ (0(,)π) → (cos‘𝐴) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 -cneg 11441 2c2 12263 (,)cioo 13320 [,]cicc 13323 cosccos 16004 πcpi 16006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-fbas 20926 df-fg 20927 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-nei 22584 df-lp 22622 df-perf 22623 df-cn 22713 df-cnp 22714 df-haus 22801 df-tx 23048 df-hmeo 23241 df-fil 23332 df-fm 23424 df-flim 23425 df-flf 23426 df-xms 23808 df-ms 23809 df-tms 23810 df-cncf 24376 df-limc 25365 df-dv 25366 |
This theorem is referenced by: (None) |
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