cosne0 - Metamath Proof Explorer

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Theorem cosne0 26890

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 26826	. . . . . 6 ⊢ (π / 2) ∈ ℝ
2	1	recni 11353	. . . . 5 ⊢ (π / 2) ∈ ℂ
3		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ)
4		nncan 11617	. . . . 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 6915	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → (cos‘((π / 2) − ((π / 2) − 𝐴))) = (cos‘𝐴))
7		subcl 11586	. . . . 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 26857	. . . 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 2800	. 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 26818	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → π ∈ ℂ)
15		pire 26816	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
16		pipos 26820	. . . . . . . . . . . . . 14 ⊢ 0 < π
17	15, 16	gt0ne0ii 11882	. . . . . . . . . . . . 13 ⊢ π ≠ 0
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → π ≠ 0)
19	8, 14, 18	divcan1d 12124	. . . . . . . . . . 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 12727	. . . . . . . . . . . 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 11315	. . . . . . . . . . 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 2865	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → ((π / 2) − 𝐴) ∈ ℝ)
26		resubcl 11652	. . . . . . . . 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 2865	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ℝ)
29	28	rered 15423	. . . . . 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 2865	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) ∧ (((π / 2) − 𝐴) / π) ∈ ℤ) → 𝐴 ∈ ( -(π / 2)(,)(π / 2)))
32		0zd 12735	. . . . . 6 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 0 ∈ ℤ)
33		elioore 13536	. . . . . . . 8 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
34		resubcl 11652	. . . . . . . 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 13566	. . . . . . . . 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 11839	. . . . . . . . 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 12282	. . . . . 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 11656	. . . . . . . . . . . 12 ⊢ -(π / 2) ∈ ℂ
46	13, 2	negsubi 11666	. . . . . . . . . . . . 13 ⊢ (π + -(π / 2)) = (π − (π / 2))
47		pidiv2halves 26829	. . . . . . . . . . . . . 14 ⊢ ((π / 2) + (π / 2)) = π
48	13, 2, 2, 47	subaddrii 11677	. . . . . . . . . . . . 13 ⊢ (π − (π / 2)) = (π / 2)
49	46, 48	eqtri 2786	. . . . . . . . . . . 12 ⊢ (π + -(π / 2)) = (π / 2)
50	2, 13, 45, 49	subaddrii 11677	. . . . . . . . . . 11 ⊢ ((π / 2) − π) = -(π / 2)
51	37	simpld 500	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
52	50, 51	eqbrtrid 5144	. . . . . . . . . 10 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
53	44, 36, 33, 52	ltsub23d 11951	. . . . . . . . 9 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
54	13	mulridi 11343	. . . . . . . . 9 ⊢ (π · 1) = π
55	53, 54	breqtrrdi 5151	. . . . . . . 8 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < (π · 1))
56		1red 11339	. . . . . . . . 9 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 1 ∈ ℝ)
57		ltdivmul 12222	. . . . . . . . 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 1402	. . . . . . . 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 12891	. . . . . . 7 ⊢ 1 = (0 + 1)
61	59, 60	breqtrdi 5150	. . . . . 6 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) / π) < (0 + 1))
62		btwnnz 12805	. . . . . 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 1399	. . . . 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 26885	. . . . 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 2995	. . 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 3026	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 2147 ≠ wne 2959 class class class wbr 5106 ‘cfv 6561 (class class class)co 7446 ℂcc 11228 ℝcr 11229 0cc0 11230 1c1 11231 + caddc 11233 · cmul 11235 < clt 11373 − cmin 11571 -cneg 11572 / cdiv 12003 2c2 12427 ℤcz 12723 (,)cioo 13506 ℜcre 15296 sincsin 16261 cosccos 16262 πcpi 16264

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308 ax-addf 11309

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-map 8880 df-pm 8881 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-q 13106 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-ioo 13510 df-ioc 13511 df-ico 13512 df-icc 13513 df-fz 13670 df-fzo 13819 df-fl 13962 df-mod 14040 df-seq 14175 df-exp 14235 df-fac 14448 df-bc 14477 df-hash 14505 df-shft 15252 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-limsup 15670 df-clim 15687 df-rlim 15688 df-sum 15886 df-ef 16265 df-sin 16267 df-cos 16268 df-pi 16270 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-starv 17475 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-unif 17483 df-hom 17484 df-cco 17485 df-rest 17625 df-topn 17626 df-0g 17644 df-gsum 17645 df-topgen 17646 df-pt 17647 df-prds 17650 df-xrs 17706 df-qtop 17711 df-imas 17712 df-xps 17714 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-mulg 19310 df-cntz 19564 df-cmn 20029 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-mopn 21707 df-fbas 21708 df-fg 21709 df-cnfld 21712 df-top 23245 df-topon 23262 df-topsp 23284 df-bases 23297 df-cld 23370 df-ntr 23371 df-cls 23372 df-nei 23449 df-lp 23487 df-perf 23488 df-cn 23578 df-cnp 23579 df-haus 23666 df-tx 23914 df-hmeo 24107 df-fil 24198 df-fm 24290 df-flim 24291 df-flf 24292 df-xms 24672 df-ms 24673 df-tms 24674 df-cncf 25232 df-limc 26219 df-dv 26220

This theorem is used by: tanord 26899 tanregt0 26900 atantan 27284 tan2h 38740 fourierdlem62 47393

W3C validator