cosne0 - Metamath Proof Explorer

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Theorem cosne0 26572

Description: The cosine function has no zeroes within the vertical strip of the complex plane between real part -π / 2 and π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.)

**Assertion**
Ref	Expression
cosne0	⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) ≠ 0)

**Proof of Theorem cosne0**
Step	Hyp	Ref	Expression
1		halfpire 26507	. . . . . 6 ⊢ (π / 2) ∈ ℝ
2	1	recni 11276	. . . . 5 ⊢ (π / 2) ∈ ℂ
3		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ)
4		nncan 11539	. . . . 5 ⊢ (((π / 2) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((π / 2) − ((π / 2) − 𝐴)) = 𝐴)
5	2, 3, 4	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((π / 2) − ((π / 2) − 𝐴)) = 𝐴)
6	5	fveq2d 6909	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (cos‘𝐴))
7		subcl 11508	. . . . 5 ⊢ (((π / 2) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((π / 2) − 𝐴) ∈ ℂ)
8	2, 3, 7	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((π / 2) − 𝐴) ∈ ℂ)
9		coshalfpim 26538	. . . 4 ⊢ (((π / 2) − 𝐴) ∈ ℂ → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (sin‘((π / 2) − 𝐴)))
10	8, 9	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (sin‘((π / 2) − 𝐴)))
11	6, 10	eqtr3d 2778	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) = (sin‘((π / 2) − 𝐴)))
12	5	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − ((π / 2) − 𝐴)) = 𝐴)
13		picn 26502	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ∈ ℂ)
15		pire 26501	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
16		pipos 26503	. . . . . . . . . . . . . 14 ⊢ 0 < π
17	15, 16	gt0ne0ii 11800	. . . . . . . . . . . . 13 ⊢ π ≠ 0
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ≠ 0)
19	8, 14, 18	divcan1d 12045	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((((π / 2) − 𝐴) / π) · π) = ((π / 2) − 𝐴))
20	19	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((((π / 2) − 𝐴) / π) · π) = ((π / 2) − 𝐴))
21		zre 12619	. . . . . . . . . . . 12 ⊢ ((((π / 2) − 𝐴) / π) ∈ ℤ → (((π / 2) − 𝐴) / π) ∈ ℝ)
22	21	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → (((π / 2) − 𝐴) / π) ∈ ℝ)
23		remulcl 11241	. . . . . . . . . . 11 ⊢ (((((π / 2) − 𝐴) / π) ∈ ℝ ∧ π ∈ ℝ) → ((((π / 2) − 𝐴) / π) · π) ∈ ℝ)
24	22, 15, 23	sylancl 586	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((((π / 2) − 𝐴) / π) · π) ∈ ℝ)
25	20, 24	eqeltrrd 2841	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − 𝐴) ∈ ℝ)
26		resubcl 11574	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ ((π / 2) − 𝐴) ∈ ℝ) → ((π / 2) − ((π / 2) − 𝐴)) ∈ ℝ)
27	1, 25, 26	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − ((π / 2) − 𝐴)) ∈ ℝ)
28	12, 27	eqeltrrd 2841	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ℝ)
29	28	rered 15264	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → (ℜ‘𝐴) = 𝐴)
30		simplr 768	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
31	29, 30	eqeltrrd 2841	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ (-(π / 2)(,)(π / 2)))
32		0zd 12627	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 ∈ ℤ)
33		elioore 13418	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
34		resubcl 11574	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
35	1, 33, 34	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ ℝ)
36	15	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → π ∈ ℝ)
37		eliooord 13447	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
38	37	simprd 495	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
39		posdif 11757	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
40	33, 1, 39	sylancl 586	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
41	38, 40	mpbid 232	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < ((π / 2) − 𝐴))
42	16	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < π)
43	35, 36, 41, 42	divgt0d 12204	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (((π / 2) − 𝐴) / π))
44	1	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (π / 2) ∈ ℝ)
45	2	negcli 11578	. . . . . . . . . . . 12 ⊢ -(π / 2) ∈ ℂ
46	13, 2	negsubi 11588	. . . . . . . . . . . . 13 ⊢ (π + -(π / 2)) = (π − (π / 2))
47		pidiv2halves 26510	. . . . . . . . . . . . . 14 ⊢ ((π / 2) + (π / 2)) = π
48	13, 2, 2, 47	subaddrii 11599	. . . . . . . . . . . . 13 ⊢ (π − (π / 2)) = (π / 2)
49	46, 48	eqtri 2764	. . . . . . . . . . . 12 ⊢ (π + -(π / 2)) = (π / 2)
50	2, 13, 45, 49	subaddrii 11599	. . . . . . . . . . 11 ⊢ ((π / 2) − π) = -(π / 2)
51	37	simpld 494	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
52	50, 51	eqbrtrid 5177	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
53	44, 36, 33, 52	ltsub23d 11869	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
54	13	mulridi 11266	. . . . . . . . 9 ⊢ (π · 1) = π
55	53, 54	breqtrrdi 5184	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < (π · 1))
56		1red 11263	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 1 ∈ ℝ)
57		ltdivmul 12144	. . . . . . . . 9 ⊢ ((((π / 2) − 𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → ((((π / 2) − 𝐴) / π) < 1 ↔ ((π / 2) − 𝐴) < (π · 1)))
58	35, 56, 36, 42, 57	syl112anc 1375	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((((π / 2) − 𝐴) / π) < 1 ↔ ((π / 2) − 𝐴) < (π · 1)))
59	55, 58	mpbird 257	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < 1)
60		1e0p1 12777	. . . . . . 7 ⊢ 1 = (0 + 1)
61	59, 60	breqtrdi 5183	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < (0 + 1))
62		btwnnz 12696	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 0 < (((π / 2) − 𝐴) / π) ∧ (((π / 2) − 𝐴) / π) < (0 + 1)) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
63	32, 43, 61, 62	syl3anc 1372	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
64	31, 63	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
65	64	pm2.01da 798	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
66		sineq0 26567	. . . . 5 ⊢ (((π / 2) − 𝐴) ∈ ℂ → ((sin‘((π / 2) − 𝐴)) = 0 ↔ (((π / 2) − 𝐴) / π) ∈ ℤ))
67	8, 66	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((sin‘((π / 2) − 𝐴)) = 0 ↔ (((π / 2) − 𝐴) / π) ∈ ℤ))
68	67	necon3abid 2976	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((sin‘((π / 2) − 𝐴)) ≠ 0 ↔ ¬ (((π / 2) − 𝐴) / π) ∈ ℤ))
69	65, 68	mpbird 257	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (sin‘((π / 2) − 𝐴)) ≠ 0)
70	11, 69	eqnetrd 3007	1 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 < clt 11296 − cmin 11493 -cneg 11494 / cdiv 11921 2c2 12322 ℤcz 12615 (,)cioo 13388 ℜcre 15137 sincsin 16100 cosccos 16101 πcpi 16103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903

This theorem is referenced by: tanord 26581 tanregt0 26582 atantan 26967 tan2h 37620 fourierdlem62 46188

W3C validator