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	cosnegt 7401	The cosines of a number and its negative are the same.
|- (A e. CC -> (cos` -uA) = (cos` A))
Theorem	sin0 7402	Value of the sine function at 0. (Contributed by Steve Rodriguez, 5-Jul-2006.)
|- (sin` 0) = 0
Theorem	sin0ALT 7403	Value of the sine function at 0.
|- (sin` 0) = 0
Theorem	cos0 7404	Value of the cosine function at 0.
|- (cos` 0) = 1
Theorem	efivalt 7405	The exponential function in terms of sine and cosine.
|- (A e. CC -> (exp` (i x. A)) = ((cos` A) + (i x. (sin` A))))
Theorem	efmivalt 7406	The exponential function in terms of sine and cosine.
|- (A e. CC -> (exp` (-ui x. A)) = ((cos` A) - (i x. (sin` A))))
Theorem	efeult 7407	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 7408	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 7409	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 7410	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 7411	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 7412	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 7413	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 7414	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 7415	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 7416	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 7417	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 7418	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 7419	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 7420	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 7421	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 7422	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 7423	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 7424	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 7425	Lemma for sin01bnd 7430 and cos01bnd 7431.
Theorem	sin01bndlem2 7426	Lemma for sin01bnd 7430.
Theorem	sin01bndlem3 7427	Lemma for sin01bnd 7430.
Theorem	cos01bndlem2 7428	Lemma for cos01bnd 7431.
Theorem	cos01bndlem3 7429	Lemma for cos01bnd 7431.
Theorem	sin01bnd 7430	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 7431	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 7432	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ((1 / 3) < (cos` 1) /\ (cos` 1) < (2 / 3))
Theorem	cos2bnd 7433	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 7434	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 7435	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 7436	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 7437	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (0 < (sin` 1) /\ 0 < (cos` 1))
Theorem	sincos2sgn 7438	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (0 < (sin` 2) /\ (cos` 2) < 0)
Theorem	sin4lt0 7439	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (sin` 4) < 0
Theorem	absefit 7440	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 7441	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 7442	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)))))
Theorem	demoivreALT 7443	Shorter proof of demoivre 7442 using the exponential function.
|- ((A e. CC /\ N e. NN0) -> (((cos` A) + (i x. (sin` A)))^N) = ((cos` (N x. A)) + (i x. (sin` (N x. A)))))
Axiom of dependent choice
Theorem	acdc3lem 7444	Lemma for acdc3 7445. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that A.u e. (F` x)-. urv, which is unique when r is a well-ordering on A.
Theorem	acdc3 7445	Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value C. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence.
|- A e. V   =>   |- ((F:A-->(P~A \ {(/)}) /\ C e. A) -> E.g(g:NN-->A /\ (g` 1) = C /\ A.k e. NN (g` (k + 1)) e. (F` (g` k))))
Theorem	acdc2lem1 7446	Lemma for acdc2 7448.
Theorem	acdc2lem2 7447	Lemma for acdc2 7448. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that A.u e. (yFx)-. urv, which is unique when r is a well-ordering on A.
Theorem	acdc2 7448	A more general version of acdc 7453 that allows the function F to vary with k.
|- A e. V   =>   |- ((A =/= (/) /\ F:(NN X. A)-->(P~A \ {(/)})) -> E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))
Theorem	acdc5lem1 7449	Lemma for acdc5 7451.
Theorem	acdc5lem2 7450	Lemma for acdc5 7451. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that A.u e. (yFx)-. urv, which is unique when r is a well-ordering on A.
Theorem	acdc5 7451	A more general version of acdc 7453 that has an initial value and where the function F depends on k.
|- A e. V   =>   |- ((F:(NN X. A)-->(P~A \ {(/)}) /\ C e. A) -> E.g(g:NN-->A /\ (g