mmtheorems75 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7401-7500 - Page 75 of 107

Type	Label	Description
Statement
Theorem	efi4pt 7401	Separate out the first four terms of the infinite series expansion of the exponential function of a pure imaginary number. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = {<.j, y>. | (j e. NN0 /\ y = (((i x. A)^j) / (!` j)))}   =>   |- (A e. RR -> (exp` (i x. A)) = (((1 - ((A^2) / 2)) + (i x. (A - ((A^3) / 6)))) + sum_k e. (ZZ>` 4)(F` k)))
Theorem	resin4pt 7402	Separate out the first four terms of the infinite series expansion of the sine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = {<.j, y>. | (j e. NN0 /\ y = (((i x. A)^j) / (!` j)))}   =>   |- (A e. RR -> (sin` A) = ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>` 4)(F` k))))
Theorem	recos4pt 7403	Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = {<.j, y>. | (j e. NN0 /\ y = (((i x. A)^j) / (!` j)))}   =>   |- (A e. RR -> (cos` A) = ((1 - ((A^2) / 2)) + (Re` sum_k e. (ZZ>` 4)(F` k))))
Theorem	resinclt 7404	The sine of a real number is real.
|- (A e. RR -> (sin` A) e. RR)
Theorem	recosclt 7405	The cosine of a real number is real.
|- (A e. RR -> (cos` A) e. RR)
Theorem	sinf 7406	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- sin:CC-->CC
Theorem	cosf 7407	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- cos:CC-->CC
Theorem	sinnegt 7408	The sine of a negative is the negative of the sine.
|- (A e. CC -> (sin` -uA) = -u(sin` A))
Theorem	cosnegt 7409	The cosines of a number and its negative are the same.
|- (A e. CC -> (cos` -uA) = (cos` A))
Theorem	sin0 7410	Value of the sine function at 0. (Contributed by Steve Rodriguez, 5-Jul-2006.)
|- (sin` 0) = 0
Theorem	sin0ALT 7411	Value of the sine function at 0.
|- (sin` 0) = 0
Theorem	cos0 7412	Value of the cosine function at 0.
|- (cos` 0) = 1
Theorem	efivalt 7413	The exponential function in terms of sine and cosine.
|- (A e. CC -> (exp` (i x. A)) = ((cos` A) + (i x. (sin` A))))
Theorem	efmivalt 7414	The exponential function in terms of sine and cosine.
|- (A e. CC -> (exp` (-ui x. A)) = ((cos` A) - (i x. (sin` A))))
Theorem	efeult 7415	Eulerian representation of the complex exponential. (Suggested by Jeffrey Hankins, 3-Jul-2006.)
|- (A e. CC -> (exp` A) = ((exp` (Re` A)) x. ((cos` (Im` A)) + (i x. (sin` (Im` A))))))
Theorem	efieq 7416	The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ((A e. RR /\ B e. RR) -> ((exp` (i x. A)) = (exp` (i x. B)) <-> ((cos` A) = (cos` B) /\ (sin` A) = (sin` B))))
Theorem	sinadd 7417	Sine addition formula for complex arguments. Equation 14 of [Gleason] p. 310.
|- A e. CC   &   |- B e. CC   =>   |- (sin` (A + B)) = (((sin` A) x. (cos` B)) + ((cos` A) x. (sin` B)))
Theorem	cosadd 7418	Addition formula for cosine. Equation 15 of [Gleason] p. 310.
|- A e. CC   &   |- B e. CC   =>   |- (cos` (A + B)) = (((cos` A) x. (cos` B)) - ((sin` A) x. (sin` B)))
Theorem	sinaddt 7419	Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.)
|- ((A e. CC /\ B e. CC) -> (sin` (A + B)) = (((sin` A) x. (cos` B)) + ((cos` A) x. (sin` B))))
Theorem	cosaddt 7420	Addition formula for cosine. Equation 15 of [Gleason] p. 310.
|- ((A e. CC /\ B e. CC) -> (cos` (A + B)) = (((cos` A) x. (cos` B)) - ((sin` A) x. (sin` B))))
Theorem	sinsubt 7421	Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (sin` (A - B)) = (((sin` A) x. (cos` B)) - ((cos` A) x. (sin` B))))
Theorem	cossubt 7422	Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (cos` (A - B)) = (((cos` A) x. (cos` B)) + ((sin` A) x. (sin` B))))
Theorem	addsint 7423	Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> ((sin` A) + (sin` B)) = (2 x. ((sin` ((A + B) / 2)) x. (cos` ((A - B) / 2)))))
Theorem	subsint 7424	Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> ((sin` A) - (sin` B)) = (2 x. ((cos` ((A + B) / 2)) x. (sin` ((A - B) / 2)))))
Theorem	addcost 7425	Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> ((cos` A) + (cos` B)) = (2 x. ((cos` ((A + B) / 2)) x. (cos` ((A - B) / 2)))))
Theorem	subcost 7426	Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> ((cos` A) - (cos` B)) = (-u2 x. ((sin` ((A + B) / 2)) x. (sin` ((A - B) / 2)))))
Theorem	sincossqt 7427	Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded.
|- (A e. CC -> (((sin` A)^2) + ((cos` A)^2)) = 1)
Theorem	sin2tt 7428	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
|- (A e. CC -> (sin` (2 x. A)) = (2 x. ((sin` A) x. (cos` A))))
Theorem	cos2tt 7429	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
|- (A e. CC -> (cos` (2 x. A)) = ((2 x. ((cos` A)^2)) - 1))
Theorem	cos2tOLD 7430	Double-angle formula for cosine. (Contributed by Paul Chapman, 25-Nov-2007.)
|- A e. CC   =>   |- (cos` (2 x. A)) = ((2 x. ((cos` A)^2)) - 1)
Theorem	sinbndt 7431	The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311.
|- (A e. RR -> (-u1 <_ (sin` A) /\ (sin` A) <_ 1))
Theorem	cosbndt 7432	The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311.
|- (A e. RR -> (-u1 <_ (cos` A) /\ (cos` A) <_ 1))
Theorem	sin01bndlem1 7433	Lemma for sin01bnd 7438 and cos01bnd 7439.
Theorem	sin01bndlem2 7434	Lemma for sin01bnd 7438.
Theorem	sin01bndlem3 7435	Lemma for sin01bnd 7438.
Theorem	cos01bndlem2 7436	Lemma for cos01bnd 7439.
Theorem	cos01bndlem3 7437	Lemma for cos01bnd 7439.
Theorem	sin01bnd 7438	Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (A e. (0(,]1) -> ((A - ((A^3) / 3)) < (sin` A) /\ (sin` A) < A))
Theorem	cos01bnd 7439	Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (A e. (0(,]1) -> ((1 - (2 x. ((A^2) / 3))) < (cos` A) /\ (cos` A) < (1 - ((A^2) / 3))))
Theorem	cos1bnd 7440	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ((1 / 3) < (cos` 1) /\ (cos` 1) < (2 / 3))
Theorem	cos2bnd 7441	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (-u(7 / 9) < (cos` 2) /\ (cos` 2) < -u(1 / 9))
Theorem	sin01gt0 7442	The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (A e. (0(,]1) -> 0 < (sin` A))
Theorem	cos01gt0 7443	The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (A e. (0(,]1) -> 0 < (cos` A))
Theorem	sin02gt0 7444	The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (A e. (0(,]2) -> 0 < (sin` A))
Theorem	sincos1sgn 7445	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (0 < (sin` 1) /\ 0 < (cos` 1))
Theorem	sincos2sgn 7446	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (0 < (sin` 2) /\ (cos` 2) < 0)
Theorem	sin4lt0 7447	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (sin` 4) < 0
Theorem	absefit 7448	The absolute value of the exponential function of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.)
|- (A e. RR -> (abs` (exp` (i x. A))) = 1)
Theorem	abseft 7449	The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
|- (A e. CC -> (abs` (exp` A)) = (exp` (Re` A)))
Theorem	demoivre 7450	De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. (Contributed by Steve Rodriguez, 10-Nov-2006.) Warning: The HTML proof page is 0.6 megabyte in size.
|- ((A e. CC /\ N e. NN0) -> (((cos` A) + (i x. (sin` A)))^N) = ((cos` (N x. A)) + (i x. (sin` (N x. A)))