cosne0 - Metamath Proof Explorer

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Theorem cosne0 26863

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 26799	. . . . . 6 ⊢ (π / 2) ∈ ℝ
2	1	recni 11327	. . . . 5 ⊢ (π / 2) ∈ ℂ
3		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ)
4		nncan 11591	. . . . 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 6890	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (cos‘𝐴))
7		subcl 11560	. . . . 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 26830	. . . 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 2797	. 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 26791	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → π ∈ ℂ)
15		pire 26789	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
16		pipos 26793	. . . . . . . . . . . . . 14 ⊢ 0 < π
17	15, 16	gt0ne0ii 11856	. . . . . . . . . . . . 13 ⊢ π ≠ 0
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → π ≠ 0)
19	8, 14, 18	divcan1d 12098	. . . . . . . . . . 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 12701	. . . . . . . . . . . 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 11289	. . . . . . . . . . 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 2861	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − 𝐴) ∈ ℝ)
26		resubcl 11626	. . . . . . . . 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 2861	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ℝ)
29	28	rered 15397	. . . . . 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 2861	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ( -(π / 2)(,)(π / 2)))
32		0zd 12709	. . . . . 6 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 0 ∈ ℤ)
33		elioore 13510	. . . . . . . 8 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
34		resubcl 11626	. . . . . . . 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 13540	. . . . . . . . 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 11813	. . . . . . . . 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 12256	. . . . . 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 11630	. . . . . . . . . . . 12 ⊢ -(π / 2) ∈ ℂ
46	13, 2	negsubi 11640	. . . . . . . . . . . . 13 ⊢ (π + -(π / 2)) = (π − (π / 2))
47		pidiv2halves 26802	. . . . . . . . . . . . . 14 ⊢ ((π / 2) + (π / 2)) = π
48	13, 2, 2, 47	subaddrii 11651	. . . . . . . . . . . . 13 ⊢ (π − (π / 2)) = (π / 2)
49	46, 48	eqtri 2783	. . . . . . . . . . . 12 ⊢ (π + -(π / 2)) = (π / 2)
50	2, 13, 45, 49	subaddrii 11651	. . . . . . . . . . 11 ⊢ ((π / 2) − π) = -(π / 2)
51	37	simpld 500	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
52	50, 51	eqbrtrid 5139	. . . . . . . . . 10 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
53	44, 36, 33, 52	ltsub23d 11925	. . . . . . . . 9 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
54	13	mulridi 11317	. . . . . . . . 9 ⊢ (π · 1) = π
55	53, 54	breqtrrdi 5146	. . . . . . . 8 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < (π · 1))
56		1red 11313	. . . . . . . . 9 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 1 ∈ ℝ)
57		ltdivmul 12196	. . . . . . . . 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 12865	. . . . . . 7 ⊢ 1 = (0 + 1)
61	59, 60	breqtrdi 5145	. . . . . 6 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < (0 + 1))
62		btwnnz 12779	. . . . . 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 26858	. . . . 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 2991	. . 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 3022	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 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 · cmul 11209 < clt 11347 − cmin 11545 -cneg 11546 / cdiv 11977 2c2 12401 ℤcz 12697 (,)cioo 13480 ℜcre 15270 sincsin 16235 cosccos 16236 πcpi 16238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-pi 16244 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193

This theorem is used by: tanord 26872 tanregt0 26873 atantan 27257 tan2h 38629 fourierdlem62 47272

W3C validator