cosne0 - Metamath Proof Explorer

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Theorem cosne0 26839

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 26775	. . . . . 6 ⊢ (π / 2) ∈ ℝ
2	1	recni 11304	. . . . 5 ⊢ (π / 2) ∈ ℂ
3		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ)
4		nncan 11568	. . . . 5 ⊢ (((π / 2) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((π / 2) − ((π / 2) − 𝐴)) = 𝐴)
5	2, 3, 4	sylancr 599	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((π / 2) − ((π / 2) − 𝐴)) = 𝐴)
6	5	fveq2d 6881	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (cos‘𝐴))
7		subcl 11537	. . . . 5 ⊢ (((π / 2) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((π / 2) − 𝐴) ∈ ℂ)
8	2, 3, 7	sylancr 599	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((π / 2) − 𝐴) ∈ ℂ)
9		coshalfpim 26806	. . . 4 ⊢ (((π / 2) − 𝐴) ∈ ℂ → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (sin‘((π / 2) − 𝐴)))
10	8, 9	syl 18	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (sin‘((π / 2) − 𝐴)))
11	6, 10	eqtr3d 2798	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) = (sin‘((π / 2) − 𝐴)))
12	5	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − ((π / 2) − 𝐴)) = 𝐴)
13		picn 26767	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ∈ ℂ)
15		pire 26765	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
16		pipos 26769	. . . . . . . . . . . . . 14 ⊢ 0 < π
17	15, 16	gt0ne0ii 11833	. . . . . . . . . . . . 13 ⊢ π ≠ 0
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ≠ 0)
19	8, 14, 18	divcan1d 12075	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((((π / 2) − 𝐴) / π) · π) = ((π / 2) − 𝐴))
20	19	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((((π / 2) − 𝐴) / π) · π) = ((π / 2) − 𝐴))
21		zre 12678	. . . . . . . . . . . 12 ⊢ ((((π / 2) − 𝐴) / π) ∈ ℤ → (((π / 2) − 𝐴) / π) ∈ ℝ)
22	21	adantl 487	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → (((π / 2) − 𝐴) / π) ∈ ℝ)
23		remulcl 11266	. . . . . . . . . . 11 ⊢ (((((π / 2) − 𝐴) / π) ∈ ℝ ∧ π ∈ ℝ) → ((((π / 2) − 𝐴) / π) · π) ∈ ℝ)
24	22, 15, 23	sylancl 598	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((((π / 2) − 𝐴) / π) · π) ∈ ℝ)
25	20, 24	eqeltrrd 2862	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − 𝐴) ∈ ℝ)
26		resubcl 11603	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ ((π / 2) − 𝐴) ∈ ℝ) → ((π / 2) − ((π / 2) − 𝐴)) ∈ ℝ)
27	1, 25, 26	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − ((π / 2) − 𝐴)) ∈ ℝ)
28	12, 27	eqeltrrd 2862	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ℝ)
29	28	rered 15371	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → (ℜ‘𝐴) = 𝐴)
30		simplr 781	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
31	29, 30	eqeltrrd 2862	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ (-(π / 2)(,)(π / 2)))
32		0zd 12686	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 ∈ ℤ)
33		elioore 13487	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
34		resubcl 11603	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
35	1, 33, 34	sylancr 599	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ ℝ)
36	15	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → π ∈ ℝ)
37		eliooord 13517	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
38	37	simprd 501	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
39		posdif 11790	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
40	33, 1, 39	sylancl 598	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
41	38, 40	mpbid 235	. . . . . . 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 12233	. . . . . 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 11607	. . . . . . . . . . . 12 ⊢ -(π / 2) ∈ ℂ
46	13, 2	negsubi 11617	. . . . . . . . . . . . 13 ⊢ (π + -(π / 2)) = (π − (π / 2))
47		pidiv2halves 26778	. . . . . . . . . . . . . 14 ⊢ ((π / 2) + (π / 2)) = π
48	13, 2, 2, 47	subaddrii 11628	. . . . . . . . . . . . 13 ⊢ (π − (π / 2)) = (π / 2)
49	46, 48	eqtri 2784	. . . . . . . . . . . 12 ⊢ (π + -(π / 2)) = (π / 2)
50	2, 13, 45, 49	subaddrii 11628	. . . . . . . . . . 11 ⊢ ((π / 2) − π) = -(π / 2)
51	37	simpld 500	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
52	50, 51	eqbrtrid 5140	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
53	44, 36, 33, 52	ltsub23d 11902	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
54	13	mulridi 11294	. . . . . . . . 9 ⊢ (π · 1) = π
55	53, 54	breqtrrdi 5147	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < (π · 1))
56		1red 11290	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 1 ∈ ℝ)
57		ltdivmul 12173	. . . . . . . . 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 1401	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((((π / 2) − 𝐴) / π) < 1 ↔ ((π / 2) − 𝐴) < (π · 1)))
59	55, 58	mpbird 260	. . . . . . 7 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < 1)
60		1e0p1 12842	. . . . . . 7 ⊢ 1 = (0 + 1)
61	59, 60	breqtrdi 5146	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < (0 + 1))
62		btwnnz 12756	. . . . . 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 1398	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
64	31, 63	syl 18	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
65	64	pm2.01da 811	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ¬ (((π / 2) − 𝐴) / π) ∈ ℤ)
66		sineq0 26834	. . . . 5 ⊢ (((π / 2) − 𝐴) ∈ ℂ → ((sin‘((π / 2) − 𝐴)) = 0 ↔ (((π / 2) − 𝐴) / π) ∈ ℤ))
67	8, 66	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((sin‘((π / 2) − 𝐴)) = 0 ↔ (((π / 2) − 𝐴) / π) ∈ ℤ))
68	67	necon3abid 2992	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((sin‘((π / 2) − 𝐴)) ≠ 0 ↔ ¬ (((π / 2) − 𝐴) / π) ∈ ℤ))
69	65, 68	mpbird 260	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (sin‘((π / 2) − 𝐴)) ≠ 0)
70	11, 69	eqnetrd 3023	1 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) ≠ 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 < clt 11324 − cmin 11522 -cneg 11523 / cdiv 11954 2c2 12378 ℤcz 12674 (,)cioo 13457 ℜcre 15244 sincsin 16209 cosccos 16210 πcpi 16212

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-pi 16218 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167

This theorem is used by: tanord 26848 tanregt0 26849 atantan 27233 tan2h 38503 fourierdlem62 47122

W3C validator