sinaover2ne0 - Mathbox for Glauco Siliprandi

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Theorem sinaover2ne0 45897

Description: If 𝐴 in (0, 2π) then sin(𝐴 / 2) is not 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
sinaover2ne0	⊢ (𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0)

**Proof of Theorem sinaover2ne0**
Step	Hyp	Ref	Expression
1		elioore 13392	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℝ)
2	1	recnd 11263	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℂ)
3		2cnd 12318	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 2 ∈ ℂ)
4		picn 26419	. . . . . 6 ⊢ π ∈ ℂ
5	4	a1i 11	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → π ∈ ℂ)
6		2ne0 12344	. . . . . 6 ⊢ 2 ≠ 0
7	6	a1i 11	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 2 ≠ 0)
8		pire 26418	. . . . . . 7 ⊢ π ∈ ℝ
9		pipos 26420	. . . . . . 7 ⊢ 0 < π
10	8, 9	gt0ne0ii 11773	. . . . . 6 ⊢ π ≠ 0
11	10	a1i 11	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → π ≠ 0)
12	2, 3, 5, 7, 11	divdiv1d 12048	. . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((𝐴 / 2) / π) = (𝐴 / (2 · π)))
13		0zd 12600	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℤ)
14		2re 12314	. . . . . . . 8 ⊢ 2 ∈ ℝ
15	14, 8	remulcli 11251	. . . . . . 7 ⊢ (2 · π) ∈ ℝ
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ)
17		0xr 11282	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
18	17	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℝ^*)
19	16	rexrd 11285	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ^*)
20		id 22	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ (0(,)(2 · π)))
21		ioogtlb 45524	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ (2 · π) ∈ ℝ^* ∧ 𝐴 ∈ (0(,)(2 · π))) → 0 < 𝐴)
22	18, 19, 20, 21	syl3anc 1373	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < 𝐴)
23		2pos 12343	. . . . . . . 8 ⊢ 0 < 2
24	14, 8, 23, 9	mulgt0ii 11368	. . . . . . 7 ⊢ 0 < (2 · π)
25	24	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (2 · π))
26	1, 16, 22, 25	divgt0d 12177	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (𝐴 / (2 · π)))
27		1rp 13012	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
28	27	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ∈ ℝ⁺)
29	16, 25	elrpd 13048	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ⁺)
30	2	div1d 12009	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / 1) = 𝐴)
31		iooltub 45539	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ (2 · π) ∈ ℝ^* ∧ 𝐴 ∈ (0(,)(2 · π))) → 𝐴 < (2 · π))
32	18, 19, 20, 31	syl3anc 1373	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 < (2 · π))
33	30, 32	eqbrtrd 5141	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / 1) < (2 · π))
34	1, 28, 29, 33	ltdiv23d 13118	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < 1)
35		1e0p1 12750	. . . . . 6 ⊢ 1 = (0 + 1)
36	34, 35	breqtrdi 5160	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < (0 + 1))
37		btwnnz 12669	. . . . 5 ⊢ ((0 ∈ ℤ ∧ 0 < (𝐴 / (2 · π)) ∧ (𝐴 / (2 · π)) < (0 + 1)) → ¬ (𝐴 / (2 · π)) ∈ ℤ)
38	13, 26, 36, 37	syl3anc 1373	. . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (𝐴 / (2 · π)) ∈ ℤ)
39	12, 38	eqneltrd 2854	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ ((𝐴 / 2) / π) ∈ ℤ)
40	2	halfcld 12486	. . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / 2) ∈ ℂ)
41		sineq0 26485	. . . 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 325	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (sin‘(𝐴 / 2)) = 0)
44	43	neqned 2939	1 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 ℝ^*cxr 11268 < clt 11269 / cdiv 11894 2c2 12295 ℤcz 12588 ℝ⁺crp 13008 (,)cioo 13362 sincsin 16079 πcpi 16082

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820

This theorem is referenced by: fourierdlem43 46179 fourierdlem44 46180

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