mmtheorems77 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7601-7700 - Page 77 of 123

Type	Label	Description
Statement
Theorem	elt3OLD 7601	Euler's constant e = 2.71828... is less than 3.
|- e < 3
Theorem	egt2lt3 7602	Euler's constant e = 2.71828... is bounded by 2 and 3.
|- (2 < e /\ e < 3)
The exponential, sine, and cosine functions (cont.)
Theorem	abspef01tlubi 7603	An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the punctured closed unit disc projected onto the real or imaginary axis. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = {<.j, y>. | (j e. NN0 /\ y = (((i x. A)^j) / (!` j)))}   &   |- (P = Re \/ P = Im)   =>   |- ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ ((A^M) x. ((M + 1) / ((!` M) x. M))))
Theorem	efsepi 7604	Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}   &   |- A e. CC   &   |- M e. NN0   &   |- B e. CC   &   |- (exp` A) = (B + sum_k e. (ZZ>=` M)(F` k))   &   |- (F` M) = C   &   |- N = (M + 1)   &   |- D = (B + C)   =>   |- (exp` A) = (D + sum_k e. (ZZ>=` N)(F` k))
Theorem	effsumlei 7605	The partial sums of the series expansion of the exponential function of a nonnegative real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}   &   |- A e. RR   &   |- N e. NN0   =>   |- (0 <_ A -> (( + seq0 F)` N) <_ (exp` A))
Theorem	eft0vali 7606	The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}   &   |- A e. CC   =>   |- (F` 0) = 1
Theorem	ef4pi 7607	Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}   &   |- A e. CC   =>   |- (exp` A) = ((((1 + A) + ((A^2) / 2)) + ((A^3) / 6)) + sum_k e. (ZZ>=` 4)(F` k))
Theorem	ef4p 7608	Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}   =>   |- (A e. CC -> (exp` A) = ((((1 + A) + ((A^2) / 2)) + ((A^3) / 6)) + sum_k e. (ZZ>=` 4)(F` k)))
Theorem	efge1i 7609	The exponential function of a nonnegative real number is greater than or equal to 1. (Contributed by Paul Chapman, 21-Aug-2007.)
|- A e. RR   =>   |- (0 <_ A -> 1 <_ (exp` A))
Theorem	efge1pi 7610	The exponential function of a nonnegative real number is greater than or equal to 1 plus that number. (Contributed by Paul Chapman, 18-Oct-2007.)
|- A e. RR   =>   |- (0 <_ A -> (1 + A) <_ (exp` A))
Theorem	efgt1i 7611	The exponential function of a positive real number is greater than 1. (Contributed by Paul Chapman, 21-Aug-2007.)
|- A e. RR   =>   |- (0 < A -> 1 < (exp` A))
Theorem	efgt0i 7612	The exponential function of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.)
|- A e. RR   =>   |- 0 < (exp` A)
Theorem	efgt0 7613	The exponential function of a real number is greater than 0. (Contributed by Paul Chapman, 13-Sep-2007.)
|- (A e. RR -> 0 < (exp` A))
Theorem	eflti 7614	The exponential function on the reals is strictly monotonic. (Contributed by Paul Chapman, 21-Aug-2007.)
|- A e. RR   &   |- B e. RR   =>   |- (A < B -> (exp` A) < (exp` B))
Theorem	efltbi 7615	The exponential function on the reals is strictly monotonic. (Contributed by Steve Rodriguez, 22-Aug-2007.)
|- A e. RR   &   |- B e. RR   =>   |- (A < B <-> (exp` A) < (exp` B))
Theorem	reef11i 7616	The exponential function on real numbers is one-to-one. (Contributed by Steve Rodriguez, 25-Aug-2007.)
|- A e. RR   &   |- B e. RR   =>   |- ((exp` A) = (exp` B) <-> A = B)
Theorem	reef11 7617	The exponential function on real numbers is one-to-one.
|- ((A e. RR /\ B e. RR) -> ((exp` A) = (exp` B) <-> A = B))
Theorem	reeff1 7618	The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.)
|- (exp |` RR):RR-1-1->(0(,) +oo)
Theorem	efm1limi 7619	Series convergence to the exponential function minus 1. (Contributed by Paul Chapman, 11-Sep-2007.)
|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}   &   |- A e. CC   =>   |- (<.1, + >. seq F) ~~> ((exp` A) - 1)
Theorem	absefm1lei 7620	The absolute value of the exponential function minus 1 is less than or equal to the exponential function minus 1 of the absolute value. (Contributed by Paul Chapman, 11-Sep-2007.)
|- A e. CC   =>   |- (abs` ((exp` A) - 1)) <_ ((exp` (abs` A)) - 1)
Theorem	eflegeolem1 7621	Lemma for eflegeoi 7623.
Theorem	eflegeolem2 7622	Lemma for eflegeoi 7623.
Theorem	eflegeoi 7623	The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
|- A e. RR   &   |- (0 <_ A /\ A < 1)   =>   |- (exp` A) <_ (1 / (1 - A))
Theorem	eflegeo 7624	The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
|- ((A e. RR /\ 0 <_ A /\ A < 1) -> (exp` A) <_ (1 / (1 - A)))
Theorem	efm1legeoi 7625	One less than the exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 15-Sep-2007.)
|- A e. RR   &   |- (0 <_ A /\ A < 1)   =>   |- ((exp` A) - 1) <_ (A / (1 - A))
Theorem	efm1legeo 7626	One less than the exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 15-Sep-2007.)
|- ((A e. RR /\ 0 <_ A /\ A < 1) -> ((exp` A) - 1) <_ (A / (1 - A)))
Theorem	efcnlem1 7627	Lemma for efcn 7631.
Theorem	efcnlem2 7628	Lemma for efcn 7631.
Theorem	efcnlem3 7629	Lemma for efcn 7631.
Theorem	efcnlem4 7630	Lemma for efcn 7631.
Theorem	efcn 7631	The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.)
|- exp e. (CC-cn->CC)
Theorem	reeff1olem1 7632	Lemma for reeff1o 7634.
Theorem	reeff1olem2 7633	Lemma for reeff1o 7634.
Theorem	reeff1o 7634	The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (exp |` RR):RR-1-1-onto->(0(,) +oo)
Theorem	reeff1o2 7635	The real exponential function is one-to-one onto. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (exp |` RR):RR-1-1-onto->RR+
Theorem	reefiso 7636	The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (exp |` RR) Isom < , < (RR, RR+)
Theorem	sinval 7637	Value of the sine function.
|- (A e. CC -> (sin` A) = (((exp` (i x. A)) - (exp` (-ui x. A))) / (2 x. i)))
Theorem	cosval 7638	Value of the cosine function.
|- (A e. CC -> (cos` A) = (((exp` (i x. A)) + (exp` (-ui x. A))) / 2))
Theorem	sincl 7639	Closure of the sine function.
|- (A e. CC -> (sin` A) e. CC)
Theorem	coscl 7640	Closure of the cosine function with a complex argument.
|- (A e. CC -> (cos` A) e. CC)
Theorem	resinval 7641	The sine of a real number in terms of the exponential function.
|- (A e. RR -> (sin` A) = (Im` (exp` (i x. A))))
Theorem	recosval 7642	The cosine of a real number in terms of the exponential function.
|- (A e. RR -> (cos` A) = (Re` (exp` (i x. A))))
Theorem	efi4p 7643	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	resin4p 7644	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	recos4p 7645	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	resincl 7646	The sine of a real number is real.
|- (A e. RR -> (sin` A) e. RR)
Theorem	recoscl 7647	The cosine of a real number is real.
|- (A e. RR -> (cos` A) e. RR)
Theorem	sinf 7648	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- sin:CC-->CC
Theorem	cosf 7649	Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- cos:CC-->CC
Theorem	sinneg 7650	The sine of a negative is the negative of the sine.
|- (A e. CC -> (sin` -uA) = -u(sin` A))
Theorem	cosneg 7651	The cosines of a number and its negative are the same.
|- (A e. CC -> (cos` -uA) = (cos` A))
Theorem	sin0 7652	Value of the sine function at 0. (Contributed by Steve Rodriguez, 5-Jul-2006.)
|- (sin` 0) = 0
Theorem	sin0ALT 7653	Value of the sine function at 0.
|- (sin` 0) = 0
Theorem	cos0 7654	Value of the cosine function at 0.
|- (cos` 0) = 1
Theorem	efival 7655	The exponential function in terms of sine and cosine.
|- (A e. CC -> (exp` (i x. A)) = ((cos` A) + (i x. (sin` A))))
Theorem	efmival 7656	The exponential function in terms of sine and cosine.
|- (A e. CC -> (exp` (-ui x. A)) = ((cos` A) - (i x. (sin` A))))
Theorem	efeul 7657	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 7658	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	sinaddi 7659	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	cosaddi 7660	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	sinadd 7661	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	cosadd 7662	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	sinsub 7663	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	cossub 7664	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	addsin 7665	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	subsin 7666	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	addcos 7667	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	subcos 7668	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	sincossq 7669	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	sin2t 7670	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	cos2t 7671	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	cos2tsin 7672	Double-angle formula for cosine in terms of sine.
|- (A e. CC -> (cos` (2 x. A)) = (1 - (2 x. ((sin` A)^2))))
Theorem	cos2OLD 7673	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	sinbnd 7674	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	cosbnd 7675	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 7676	Lemma for sin01bnd 7681 and cos01bnd 7682.
Theorem	sin01bndlem2 7677	Lemma for sin01bnd 7681.
Theorem	sin01bndlem3 7678	Lemma for sin01bnd 7681.
Theorem	cos01bndlem2 7679	Lemma for cos01bnd 7682.
Theorem	cos01bndlem3 7680	Lemma for cos01bnd 7682.
Theorem	sin01bnd 7681	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 7682	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 7683	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ((1 / 3) < (cos` 1) /\ (cos` 1) < (2 / 3))
Theorem	cos2bnd 7684	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 7685	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 7686	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 7687	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 7688	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (0 < (sin` 1) /\ 0 < (cos` 1))
Theorem	sincos2sgn 7689	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (0 < (sin` 2) /\ (cos` 2) < 0)
Theorem	sin4lt0 7690	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
|- (sin` 4) < 0
Theorem	absefi 7691	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	absef 7692	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	absefib 7693	A number is real iff its imaginary exponential has absolute value one.
|- (A e. CC -> (A e. RR <-> (abs` (exp` (i x. A))) = 1))
Theorem	efieq1re 7694	A number whose imaginary exponential is one is real.
|- ((A e. CC /\ (exp` (i x. A)) = 1) -> A e. RR)
Theorem	demoivre 7695	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 7696	Shorter proof of demoivre 7695 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 7697	Lemma for acdc3 7698. 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 7698	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	acdc4lem1 7699	Lemma for acdc4 (future).
Theorem	acdc2lem1 7700	Lemma for acdc2 7702.