cosne0 - Metamath Proof Explorer

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Theorem cosne0 26454

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 26389	. . . . . 6 ⊢ (π / 2) ∈ ℝ
2	1	recni 11148	. . . . 5 ⊢ (π / 2) ∈ ℂ
3		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ)
4		nncan 11411	. . . . 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 6830	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (cos‘𝐴))
7		subcl 11380	. . . . 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 26420	. . . 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 2766	. 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 26383	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ∈ ℂ)
15		pire 26382	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
16		pipos 26384	. . . . . . . . . . . . . 14 ⊢ 0 < π
17	15, 16	gt0ne0ii 11674	. . . . . . . . . . . . 13 ⊢ π ≠ 0
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ≠ 0)
19	8, 14, 18	divcan1d 11919	. . . . . . . . . . 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 12493	. . . . . . . . . . . 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 11113	. . . . . . . . . . 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 2829	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − 𝐴) ∈ ℝ)
26		resubcl 11446	. . . . . . . . 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 2829	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ℝ)
29	28	rered 15149	. . . . . 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 2829	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ (-(π / 2)(,)(π / 2)))
32		0zd 12501	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 ∈ ℤ)
33		elioore 13296	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
34		resubcl 11446	. . . . . . . 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 13326	. . . . . . . . 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 11631	. . . . . . . . 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 12078	. . . . . 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 11450	. . . . . . . . . . . 12 ⊢ -(π / 2) ∈ ℂ
46	13, 2	negsubi 11460	. . . . . . . . . . . . 13 ⊢ (π + -(π / 2)) = (π − (π / 2))
47		pidiv2halves 26392	. . . . . . . . . . . . . 14 ⊢ ((π / 2) + (π / 2)) = π
48	13, 2, 2, 47	subaddrii 11471	. . . . . . . . . . . . 13 ⊢ (π − (π / 2)) = (π / 2)
49	46, 48	eqtri 2752	. . . . . . . . . . . 12 ⊢ (π + -(π / 2)) = (π / 2)
50	2, 13, 45, 49	subaddrii 11471	. . . . . . . . . . 11 ⊢ ((π / 2) − π) = -(π / 2)
51	37	simpld 494	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
52	50, 51	eqbrtrid 5130	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
53	44, 36, 33, 52	ltsub23d 11743	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
54	13	mulridi 11138	. . . . . . . . 9 ⊢ (π · 1) = π
55	53, 54	breqtrrdi 5137	. . . . . . . 8 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < (π · 1))
56		1red 11135	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 1 ∈ ℝ)
57		ltdivmul 12018	. . . . . . . . 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 1376	. . . . . . . 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 12651	. . . . . . 7 ⊢ 1 = (0 + 1)
61	59, 60	breqtrdi 5136	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < (0 + 1))
62		btwnnz 12570	. . . . . 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 1373	. . . . 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 26449	. . . . 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 2961	. . 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 2992	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 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 − cmin 11365 -cneg 11366 / cdiv 11795 2c2 12201 ℤcz 12489 (,)cioo 13266 ℜcre 15022 sincsin 15988 cosccos 15989 πcpi 15991

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784

This theorem is referenced by: tanord 26463 tanregt0 26464 atantan 26849 tan2h 37591 fourierdlem62 46150

W3C validator