mmtheorems87 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10658

Statement List for Metamath Proof Explorer - 8601-8700 - Page 87 of 107

Type	Label	Description
Statement
Theorem	spweu 8601	A supremum is unique.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ E.x e. X ph) -> E!x e. X ph)
Theorem	spwpr2 8602	Property of supremum defining condition for an unordered pair.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- (((R e. T /\ A = {B, C}) /\ (B e. U /\ C e. W)) -> (ph <-> ((BRx /\ CRx) /\ A.y e. X ((BRy /\ CRy) -> xRy))))
Theorem	spwval 8603	Value of a supremum.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W /\ E.x e. X ph) -> (R supw A) = U.{x e. X | ph})
Theorem	spwcl 8604	Closure of a supremum.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W /\ E.x e. X ph) -> (R supw A) e. X)
Theorem	spwnex 8605	Non-closure when the supremum doesn't exist.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)
Real and complex numbers (cont.)
The exponential, sine, and cosine functions (cont.)
Theorem	sincolem 8606	Lemma for sinco 8608 and cosco 8609.
Theorem	sincnlem 8607	Lemma for sincn 8610 and coscn 8611.
Theorem	sinco 8608	Sine expressed as a function composition. (Contributed by Paul Chapman, 28-Nov-2007.)
|- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}   &   |- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}   &   |- J = {<.x, y>. | (x e. CC /\ y = (x / (2 x. i)))}   &   |- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w) - ((exp o. G)` w)))}   =>   |- sin = (J o. H)
Theorem	cosco 8609	Cosine expressed as a function composition. (Contributed by Paul Chapman, 28-Nov-2007.)
|- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}   &   |- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}   &   |- J = {<.x, y>. | (x e. CC /\ y = (x / 2))}   &   |- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w) + ((exp o. G)` w)))}   =>   |- cos = (J o. H)
Theorem	sincn 8610	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- sin e. (CC-cn->CC)
Theorem	coscn 8611	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- cos e. (CC-cn->CC)
Properties of pi = 3.14159...
Theorem	pilem1 8612	Lemma for pire 8618, pigt2lt4 8616 and sinpi 8617.
Theorem	pilem2 8613	Lemma for pire 8618, pigt2lt4 8616 and sinpi 8617.
Theorem	pilem3 8614	Lemma for pire 8618, pigt2lt4 8616 and sinpi 8617.
Theorem	pilem4 8615	Lemma for pire 8618, pigt2lt4 8616 and sinpi 8617.
Theorem	pigt2lt4 8616	pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (2 < pi /\ pi < 4)
Theorem	sinpi 8617	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` pi) = 0
Theorem	pire 8618	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	pipos 8619	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.)
|- 0 < pi
Theorem	sinhalfpilem 8620	Lemma for sinhalfpi 8621 and coshalfpi 8622.
Theorem	sinhalfpi 8621	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (pi / 2)) = 1
Theorem	coshalfpi 8622	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (pi / 2)) = 0
Theorem	cospi 8623	The cosine of pi is -u1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` pi) = -u1
Theorem	eulerid 8624	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((exp` (i x. pi)) + 1) = 0
Theorem	sin2pi 8625	The sine of 2pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (2 x. pi)) = 0
Theorem	cos2pi 8626	The cosine of 2pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (2 x. pi)) = 1
Theorem	sinperlem1 8627	Lemma for sin2kpi 8629 and cos2kpi 8630.
Theorem	sinperlem2 8628	Lemma for sin2kpi 8629 and cos2kpi 8630.
Theorem	sin2kpi 8629	If K is an integer, the sine of 2Kpi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (K e. ZZ -> (sin` (K x. (2 x. pi))) = 0)
Theorem	cos2kpi 8630	If K is an integer, the cosine of 2Kpi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (K e. ZZ -> (cos` (K x. (2 x. pi))) = 1)
Theorem	sinper 8631	The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((A e. CC /\ K e. ZZ) -> (sin` (A + (K x. (2 x. pi)))) = (sin` A))
Theorem	cosper 8632	The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((A e. CC /\ K e. ZZ) -> (cos` (A + (K x. (2 x. pi)))) = (cos` A))
Theorem	sin2pim 8633	Sine of a number subtracted from 2 x. pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (sin` ((2 x. pi) - A)) = -u(sin` A))
Theorem	cos2pim 8634	Cosine of a number subtracted from 2 x. pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (cos` ((2 x. pi) - A)) = (cos` A))
Theorem	sinmpi 8635	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (sin` (A - pi)) = -u(sin` A))
Theorem	cosmpi 8636	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (cos` (A - pi)) = -u(cos` A))
Theorem	efimpi 8637	The exponential function of i times a real number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (exp` (i x. (A - pi))) = -u(exp` (i x. A)))
Theorem	sinhalfpip 8638	The sine of pi / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. CC -> (sin` ((pi / 2) + A)) = (cos` A))
Theorem	sinhalfpim 8639	The sine of pi / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. CC -> (sin` ((pi / 2) - A)) = (cos` A))
Theorem	coshalfpip 8640	The cosine of pi / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. CC -> (cos` ((pi / 2) + A)) = -u(sin` A))
Theorem	coshalfpim 8641	The cosine of pi / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. CC -> (cos` ((pi / 2) - A)) = (sin` A))
Theorem	sincosq1lem 8642	Lemma for sincosq1sgn 8643.
Theorem	sincosq1sgn 8643	The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. (0(,)(pi / 2)) -> (0 < (sin` A) /\ 0 < (cos` A)))
Theorem	sincosq2sgn 8644	The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. ((pi / 2)(,)pi) -> (0 < (sin` A) /\ (cos` A) < 0))
Theorem	sincosq3sgn 8645	The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. (pi(,)(3 x. (pi / 2))) -> ((sin` A) < 0 /\ (cos` A) < 0))
Theorem	sincosq4sgn 8646	The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. ((3 x. (pi / 2))(,)(2 x. pi)) -> ((sin` A) < 0 /\ 0 < (cos` A)))
Theorem	sinq12gt0t 8647	The sine of a number strictly between 0 and pi is positive. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. (0(,)pi) -> 0 < (sin` A))
Theorem	sincosq1eq 8648	Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)
|- ((A e. CC /\ B e. CC /\ (A + B) = 1) -> (sin` (A x. (pi / 2))) = (cos` (B x. (pi / 2))))
Theorem	sincos4thpi 8649	The sine and cosine of pi / 4. (Contributed by Paul Chapman, 25-Jan-2008.)
|- ((sin` (pi / 4)) = (1 / (sqr` 2)) /\ (cos` (pi / 4)) = (1 / (sqr` 2)))
Theorem	sincos6thpi 8650	The sine and cosine of pi / 6. (Contributed by Paul Chapman, 25-Jan-2008.)
|- ((sin` (pi / 6)) = (1 / 2) /\ (cos` (pi / 6)) = ((sqr` 3) / 2))
Theorem	cosh111lem1 8651	Lemma for cosh111t 8654.
Theorem	cosh111lem2 8652	Lemma for cosh111t 8654.
Theorem	cosh111lem3 8653	Lemma for cosh111t 8654.
Theorem	cosh111t 8654	Cosine is one-to-one over the closed-below, open-above interval from 0 to pi. (Contributed by Paul Chapman, 16-Mar-2008.)
|- ((A e. (0[,)pi) /\ B e. (0[,)pi)) -> (A = B <-> (cos` A) = (cos` B)))
Mapping of the exponential function
Theorem	efgh 8655	The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.)
|- F = {<.x, y>. | (x e. CC /\ y = (exp` (A x. x)))}   =>   |- ((A e. CC /\ B e. CC /\ C e. CC) -> (F` (B + C)) = ((F` B) x. (F` C)))
Theorem	efghgrpilem 8656	Lemma for efghgrpi 8657,
Theorem	efghgrpi 8657	The image of a subgroup of the group +, under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.)
|- S = {y | E.x e. X y = (exp` (A x. x))}   &   |- G = ( x. |` (S X. S))   &   |- A e. CC   &   |- X (_