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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sinaover2ne0 | Structured version Visualization version GIF version |
Description: If 𝐴 in (0, 2π) then sin(𝐴 / 2) is not 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
sinaover2ne0 | ⊢ (𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 12500 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 10392 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℂ) |
3 | 2cnd 11436 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 2 ∈ ℂ) | |
4 | picn 24618 | . . . . . 6 ⊢ π ∈ ℂ | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → π ∈ ℂ) |
6 | 2ne0 11469 | . . . . . 6 ⊢ 2 ≠ 0 | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 2 ≠ 0) |
8 | pire 24617 | . . . . . . 7 ⊢ π ∈ ℝ | |
9 | pipos 24619 | . . . . . . 7 ⊢ 0 < π | |
10 | 8, 9 | gt0ne0ii 10895 | . . . . . 6 ⊢ π ≠ 0 |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → π ≠ 0) |
12 | 2, 3, 5, 7, 11 | divdiv1d 11165 | . . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((𝐴 / 2) / π) = (𝐴 / (2 · π))) |
13 | 0zd 11723 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℤ) | |
14 | 2re 11432 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
15 | 14, 8 | remulcli 10380 | . . . . . . 7 ⊢ (2 · π) ∈ ℝ |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ) |
17 | 0xr 10410 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℝ*) |
19 | 16 | rexrd 10413 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ*) |
20 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ (0(,)(2 · π))) | |
21 | ioogtlb 40514 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ* ∧ 𝐴 ∈ (0(,)(2 · π))) → 0 < 𝐴) | |
22 | 18, 19, 20, 21 | syl3anc 1494 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < 𝐴) |
23 | 2pos 11468 | . . . . . . . 8 ⊢ 0 < 2 | |
24 | 14, 8, 23, 9 | mulgt0ii 10496 | . . . . . . 7 ⊢ 0 < (2 · π) |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (2 · π)) |
26 | 1, 16, 22, 25 | divgt0d 11296 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (𝐴 / (2 · π))) |
27 | 1rp 12123 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ∈ ℝ+) |
29 | 16, 25 | elrpd 12160 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ+) |
30 | 2 | div1d 11126 | . . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / 1) = 𝐴) |
31 | iooltub 40530 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ* ∧ 𝐴 ∈ (0(,)(2 · π))) → 𝐴 < (2 · π)) | |
32 | 18, 19, 20, 31 | syl3anc 1494 | . . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 < (2 · π)) |
33 | 30, 32 | eqbrtrd 4897 | . . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / 1) < (2 · π)) |
34 | 1, 28, 29, 33 | ltdiv23d 12230 | . . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < 1) |
35 | 1e0p1 11871 | . . . . . 6 ⊢ 1 = (0 + 1) | |
36 | 34, 35 | syl6breq 4916 | . . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < (0 + 1)) |
37 | btwnnz 11788 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 0 < (𝐴 / (2 · π)) ∧ (𝐴 / (2 · π)) < (0 + 1)) → ¬ (𝐴 / (2 · π)) ∈ ℤ) | |
38 | 13, 26, 36, 37 | syl3anc 1494 | . . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (𝐴 / (2 · π)) ∈ ℤ) |
39 | 12, 38 | eqneltrd 2925 | . . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ ((𝐴 / 2) / π) ∈ ℤ) |
40 | 2 | halfcld 11610 | . . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / 2) ∈ ℂ) |
41 | sineq0 24680 | . . . 4 ⊢ ((𝐴 / 2) ∈ ℂ → ((sin‘(𝐴 / 2)) = 0 ↔ ((𝐴 / 2) / π) ∈ ℤ)) | |
42 | 40, 41 | syl 17 | . . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((sin‘(𝐴 / 2)) = 0 ↔ ((𝐴 / 2) / π) ∈ ℤ)) |
43 | 39, 42 | mtbird 317 | . 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (sin‘(𝐴 / 2)) = 0) |
44 | 43 | neqned 3006 | 1 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 ℝcr 10258 0cc0 10259 1c1 10260 + caddc 10262 · cmul 10264 ℝ*cxr 10397 < clt 10398 / cdiv 11016 2c2 11413 ℤcz 11711 ℝ+crp 12119 (,)cioo 12470 sincsin 15173 πcpi 15176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-fi 8592 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ioo 12474 df-ioc 12475 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-fl 12895 df-mod 12971 df-seq 13103 df-exp 13162 df-fac 13361 df-bc 13390 df-hash 13418 df-shft 14191 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-limsup 14586 df-clim 14603 df-rlim 14604 df-sum 14801 df-ef 15177 df-sin 15179 df-cos 15180 df-pi 15182 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-rest 16443 df-topn 16444 df-0g 16462 df-gsum 16463 df-topgen 16464 df-pt 16465 df-prds 16468 df-xrs 16522 df-qtop 16527 df-imas 16528 df-xps 16530 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-mulg 17902 df-cntz 18107 df-cmn 18555 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-fbas 20110 df-fg 20111 df-cnfld 20114 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-cld 21201 df-ntr 21202 df-cls 21203 df-nei 21280 df-lp 21318 df-perf 21319 df-cn 21409 df-cnp 21410 df-haus 21497 df-tx 21743 df-hmeo 21936 df-fil 22027 df-fm 22119 df-flim 22120 df-flf 22121 df-xms 22502 df-ms 22503 df-tms 22504 df-cncf 23058 df-limc 24036 df-dv 24037 |
This theorem is referenced by: fourierdlem43 41159 fourierdlem44 41160 |
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