P. 230 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 22901-23000 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	mcubic 22901*	Solutions to a monic cubic equation, a special case of cubic 22903. (Contributed by Mario Carneiro, 24-Apr-2015.)
; $/\$

Theorem	cubic2 22902*	The solution to the general cubic equation, for arbitrary choices and of the square and cube roots. (Contributed by Mario Carneiro, 23-Apr-2015.)
; $/\$

Theorem	cubic 22903*	The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4078 to convert the existential quantifier to a triple disjunction. This is Metamath 100 proof #37. (Contributed by Mario Carneiro, 26-Apr-2015.)
;

Theorem	binom4 22904	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 13596, but it turns out to be just as much work to put it into this form after clearing all the sums and calculating binomial coefficients.) (Contributed by Mario Carneiro, 6-May-2015.)
$/\$

Theorem	dquartlem1 22905	Lemma for dquart 22907. (Contributed by Mario Carneiro, 6-May-2015.)
$\/$

Theorem	dquartlem2 22906	Lemma for dquart 22907. (Contributed by Mario Carneiro, 6-May-2015.)


Theorem	dquart 22907	Solve a depressed quartic equation. To eliminate , which is the square root of a solution to the resolvent cubic equation, apply cubic 22903 or one of its variants. (Contributed by Mario Carneiro, 6-May-2015.)
$\/$ $\/$ $\/$

Theorem	quart1cl 22908	Closure lemmas for quart 22915. (Contributed by Mario Carneiro, 7-May-2015.)
; ;; $/\$ $/\$

Theorem	quart1lem 22909	Lemma for quart1 22910. (Contributed by Mario Carneiro, 6-May-2015.)
; ;; ;;

Theorem	quart1 22910	Depress a quartic equation. (Contributed by Mario Carneiro, 6-May-2015.)
; ;;

Theorem	quartlem1 22911	Lemma for quart 22915. (Contributed by Mario Carneiro, 6-May-2015.)
; ; ; $/\$ ;

Theorem	quartlem2 22912	Closure lemmas for quart 22915. (Contributed by Mario Carneiro, 7-May-2015.)
; ;; ; ; ; $/\$ $/\$

Theorem	quartlem3 22913	Closure lemmas for quart 22915. (Contributed by Mario Carneiro, 7-May-2015.)
; ;; ; ; ; $/\$ $/\$

Theorem	quartlem4 22914	Closure lemmas for quart 22915. (Contributed by Mario Carneiro, 7-May-2015.)
; ;; ; ; ; $/\$ $/\$

Theorem	quart 22915	The quartic equation, writing out all roots using square and cube root functions so that only direct substitutions remain, and we can actually claim to have a "quartic equation". Naturally, this theorem is ridiculously long (see quartfull 28182) if all the substitutions are performed. This is Metamath 100 proof #46. (Contributed by Mario Carneiro, 6-May-2015.)
; ;; ; ; ; $\/$ $\/$ $\/$

14.3.7 Inverse trigonometric functions

Syntax	casin 22916	The arcsine function.
arcsin

Syntax	cacos 22917	The arccosine function.
arccos

Syntax	catan 22918	The arctangent function.
arctan

Definition	df-asin 22919	Define the arcsine function. Because is not a one-to-one function, the literal inverse is not a function. Rather than attempt to find the right domain on which to restrict in order to get a total function, we just define it in terms of , which we already know is total (except at ). There are branch points at and (at which the function is defined), and branch cuts along the real line not between and , which is to say . (Contributed by Mario Carneiro, 31-Mar-2015.)
arcsin

Definition	df-acos 22920	Define the arccosine function. See also remarks for df-asin 22919. Since we define arccos in terms of arcsin, it shares the same branch points and cuts, namely . (Contributed by Mario Carneiro, 31-Mar-2015.)
arccos arcsin

Definition	df-atan 22921	Define the arctangent function. See also remarks for df-asin 22919. Unlike arcsin and arccos, this function is not defined everywhere, because for all . For all other , there is a formula for arctan in terms of , and we take that as the definition. Branch points are at ; branch cuts are on the pure imaginary axis not between and , which is to say . (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan $\$

Theorem	asinlem 22922	The argument to the logarithm in df-asin 22919 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	asinlem2 22923	The argument to the logarithm in df-asin 22919 has the property that replacing with in the expression gives the reciprocal. (Contributed by Mario Carneiro, 1-Apr-2015.)


Theorem	asinlem3a 22924	Lemma for asinlem3 22925. (Contributed by Mario Carneiro, 1-Apr-2015.)
$/\$

Theorem	asinlem3 22925	The argument to the logarithm in df-asin 22919 has nonnegative real part. (Contributed by Mario Carneiro, 1-Apr-2015.)


Theorem	asinf 22926	Domain and range of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arcsin

Theorem	asincl 22927	Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arcsin

Theorem	acosf 22928	Domain and range of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arccos

Theorem	acoscl 22929	Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arccos

Theorem	atandm 22930	Since the property is a little lengthy, we abbreviate $/\$ $/\$ as arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan $/\$ $/\$

Theorem	atandm2 22931	This form of atandm 22930 is a bit more useful for showing that the logarithms in df-atan 22921 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan $/\$ $/\$

Theorem	atandm3 22932	A compact form of atandm 22930. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan $/\$

Theorem	atandm4 22933	A compact form of atandm 22930. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan $/\$

Theorem	atanf 22934	Domain and range of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan $\$

Theorem	atancl 22935	Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan arctan

Theorem	asinval 22936	Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arcsin

Theorem	acosval 22937	Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arccos arcsin

Theorem	atanval 22938	Value of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan arctan

Theorem	atanre 22939	A real number is in the domain of the arctangent function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan

Theorem	asinneg 22940	The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.)
arcsin arcsin

Theorem	acosneg 22941	The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.)
arccos arccos

Theorem	efiasin 22942	The exponential of the arcsine function. (Contributed by Mario Carneiro, 31-Mar-2015.)
arcsin

Theorem	sinasin 22943	The arcsine function is an inverse to . This is the main property that justifies the notation arcsin or . Because is not an injection, the other converse identity asinsin 22946 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.)
arcsin

Theorem	cosacos 22944	The arccosine function is an inverse to . (Contributed by Mario Carneiro, 1-Apr-2015.)
arccos

Theorem	asinsinlem 22945	Lemma for asinsin 22946. (Contributed by Mario Carneiro, 2-Apr-2015.)
$/\$

Theorem	asinsin 22946	The arcsine function composed with is equal to the identity. This plus sinasin 22943 allow us to view and arcsin as inverse operations to each other. For ease of use, we have not defined precisely the correct domain of correctness of this identity; in addition to the main region described here it is also true for some points on the branch cuts, namely when for nonnegative real and also symmetrically at . In particular, when restricted to reals this identity extends to the closed interval , not just the open interval (see reasinsin 22950). (Contributed by Mario Carneiro, 2-Apr-2015.)
$/\$ arcsin

Theorem	acoscos 22947	The arccosine function is an inverse to . (Contributed by Mario Carneiro, 2-Apr-2015.)
$/\$ arccos

Theorem	asin1 22948	The arcsine of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arcsin

Theorem	acos1 22949	The arcsine of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arccos

Theorem	reasinsin 22950	The arcsine function composed with is equal to the identity. (Contributed by Mario Carneiro, 2-Apr-2015.)
arcsin

Theorem	asinsinb 22951	Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
$/\$ $/\$ arcsin

Theorem	acoscosb 22952	Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
$/\$ $/\$ arccos

Theorem	asinbnd 22953	The arcsine function has range within a vertical strip of the complex plane with real part between and . (Contributed by Mario Carneiro, 2-Apr-2015.)
arcsin

Theorem	acosbnd 22954	The arccosine function has range within a vertical strip of the complex plane with real part between and . (Contributed by Mario Carneiro, 2-Apr-2015.)
arccos

Theorem	asinrebnd 22955	Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
arcsin

Theorem	asinrecl 22956	The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
arcsin

Theorem	acosrecl 22957	The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
arccos

Theorem	cosasin 22958	The cosine of the arcsine of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arcsin

Theorem	sinacos 22959	The sine of the arccosine of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arccos

Theorem	atandmneg 22960	The domain of the arctangent function is closed under negatives. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theorem	atanneg 22961	The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan arctan

Theorem	atan0 22962	The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan

Theorem	atandmcj 22963	The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan arctan

Theorem	atancj 22964	The arctangent function distributes under conjugation. (The condition that is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 22961 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between and , though.) (Contributed by Mario Carneiro, 31-Mar-2015.)
$/\$ arctan $/\$ arctan arctan

Theorem	atanrecl 22965	The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan

Theorem	efiatan 22966	Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan arctan

Theorem	atanlogaddlem 22967	Lemma for atanlogadd 22968. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan $/\$

Theorem	atanlogadd 22968	The rule is not always true on the complex numbers, but it is true when the arguments of and sum to within the interval , so there are some cases such as this one with and which are true unconditionally. This result can also be stated as " is analytic". (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan

Theorem	atanlogsublem 22969	Lemma for atanlogsub 22970. (Contributed by Mario Carneiro, 4-Apr-2015.)
arctan $/\$

Theorem	atanlogsub 22970	A variation on atanlogadd 22968, to show that under more limited conditions. (Contributed by Mario Carneiro, 4-Apr-2015.)
arctan $/\$

Theorem	efiatan2 22971	Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theorem	2efiatan 22972	Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan arctan

Theorem	tanatan 22973	The arctangent function is an inverse to . (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan arctan

Theorem	atandmtan 22974	The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.)
$/\$ arctan

Theorem	cosatan 22975	The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theorem	cosatanne0 22976	The arctangent function has range contained in the domain of the tangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theorem	atantan 22977	The arctangent function is an inverse to . (Contributed by Mario Carneiro, 5-Apr-2015.)
$/\$ arctan

Theorem	atantanb 22978	Relationship between tangent and arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan $/\$ $/\$ arctan

Theorem	atanbndlem 22979	Lemma for atanbnd 22980. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorem	atanbnd 22980	The arctangent function is bounded by on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorem	atanord 22981	The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
$/\$ arctan arctan

Theorem	atan1 22982	The arctangent of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan

Theorem	bndatandm 22983	A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
$/\$ arctan

Theorem	atans 22984*	The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$ $/\$

Theorem	atans2 22985*	It suffices to show that and are in the continuity domain of to show that is in the continuity domain of arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$ $/\$ $/\$

Theorem	atansopn 22986*	The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$ ℂ_fld

Theorem	atansssdm 22987*	The domain of continuity of the arctangent is a subset of the actual domain of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$ arctan

Theorem	ressatans 22988*	The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$

Theorem	dvatan 22989*	The derivative of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$ arctan

Theorem	atancn 22990*	The arctangent is a continuous function. (Contributed by Mario Carneiro, 7-Apr-2015.)
$\$ arctan

Theorem	atantayl 22991*	The Taylor series for arctan. (Contributed by Mario Carneiro, 1-Apr-2015.)
$/\$ arctan

Theorem	atantayl2 22992*	The Taylor series for arctan. (Contributed by Mario Carneiro, 1-Apr-2015.)
$/\$ arctan

Theorem	atantayl3 22993*	The Taylor series for arctan. (Contributed by Mario Carneiro, 7-Apr-2015.)
$/\$ arctan

Theorem	leibpilem1 22994	Lemma for leibpi 22996. (Contributed by Mario Carneiro, 7-Apr-2015.)
$/\$ $/\$ $/\$

Theorem	leibpilem2 22995*	The Leibniz formula for . (Contributed by Mario Carneiro, 7-Apr-2015.)
$\/$

Theorem	leibpi 22996	The Leibniz formula for . This proof depends on three main facts: (1) the series is convergent, because it is an alternating series (iseralt 13458). (2) Using leibpilem2 22995 to rewrite the series as a power series, it is the special case of the Taylor series for arctan (atantayl2 22992). (3) Although we cannot directly plug into atantayl2 22992, Abel's theorem (abelth2 22566) says that the limit along any sequence converging to , such as , of the power series converges to the power series extended to , and then since arctan is continuous at (atancn 22990) we get the desired result. This is Metamath 100 proof #26. (Contributed by Mario Carneiro, 7-Apr-2015.)


Theorem	leibpisum 22997	The Leibniz formula for . This version of leibpi 22996 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.)


Theorem	log2cnv 22998	Using the Taylor series for arctan , produce a rapidly convergent series for . (Contributed by Mario Carneiro, 7-Apr-2015.)


Theorem	log2tlbnd 22999*	Bound the error term in the series of log2cnv 22998. (Contributed by Mario Carneiro, 7-Apr-2015.)


14.3.8 The Birthday Problem

Theorem	log2ublem1 23000	Lemma for log2ub 23003. The proof of log2ub 23003, which is simply the evaluation of log2tlbnd 22999 for , takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator (usually a large power of ) and work with the closest approximations of the form for some integer instead. It turns out that for our purposes it is sufficient to take , which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)

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