P. 235 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulcxpd 23401	Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ B )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) )
Theorem	cxprecd 23402	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpcxpcld 23403	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A ^c B ) e. RR+ )
Theorem	logcxpd 23404	Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
Theorem	cxplt3d 23405	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )
Theorem	cxple3d 23406	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )
Theorem	cxpmuld 23407	Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) )
Theorem	dvcxp1 23408*	The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( A e. CC -> ( RR _D ( x e. RR+ |-> ( x ^c A ) ) ) = ( x e. RR+ |-> ( A x. ( x ^c ( A - 1 ) ) ) ) )
Theorem	dvcxp2 23409*	The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( A e. RR+ -> ( CC _D ( x e. CC |-> ( A ^c x ) ) ) = ( x e. CC |-> ( ( log ` A ) x. ( A ^c x ) ) ) )
Theorem	dvsqrt 23410	The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.)
|- ( RR _D ( x e. RR+ |-> ( sqr ` x ) ) ) = ( x e. RR+ |-> ( 1 / ( 2 x. ( sqr ` x ) ) ) )
Theorem	dvcncxp1 23411*	Derivative of complex power with respect to first argument on the complex plane. (Contributed by Brendan Leahy, 18-Dec-2018.)
|- D = ( CC \ ( -oo (,] 0 ) )   =>    |- ( A e. CC -> ( CC _D ( x e. D |-> ( x ^c A ) ) ) = ( x e. D |-> ( A x. ( x ^c ( A - 1 ) ) ) ) )
Theorem	dvcnsqrt 23412*	Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.)
|- D = ( CC \ ( -oo (,] 0 ) )   =>    |- ( CC _D ( x e. D |-> ( sqr ` x ) ) ) = ( x e. D |-> ( 1 / ( 2 x. ( sqr ` x ) ) ) )
Theorem	cxpcn 23413*	Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
|- D = ( CC \ ( -oo (,] 0 ) )   &    |- J = ( TopOpen ` ℂfld )   &    |- K = ( J ↾t D )   =>    |- ( x e. D , y e. CC |-> ( x ^c y ) ) e. ( ( K tX J ) Cn J )
Theorem	cxpcn2 23414*	Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
|- J = ( TopOpen ` ℂfld )   &    |- K = ( J ↾t RR+ )   =>    |- ( x e. RR+ , y e. CC |-> ( x ^c y ) ) e. ( ( K tX J ) Cn J )
Theorem	cxpcn3lem 23415*	Lemma for cxpcn3 23416. (Contributed by Mario Carneiro, 2-May-2016.)
|- D = ( `' Re " RR+ )   &    |- J = ( TopOpen ` ℂfld )   &    |- K = ( J ↾t ( 0 [,) +oo ) )   &    |- L = ( J ↾t D )   &    |- U = ( if ( ( Re ` A ) <_ 1 , ( Re ` A ) , 1 ) / 2 )   &    |- T = if ( U <_ ( E ^c ( 1 / U ) ) , U , ( E ^c ( 1 / U ) ) )   =>    |- ( ( A e. D /\ E e. RR+ ) -> E. d e. RR+ A. a e. ( 0 [,) +oo ) A. b e. D ( ( ( abs ` a ) < d /\ ( abs ` ( A - b ) ) < d ) -> ( abs ` ( a ^c b ) ) < E ) )
Theorem	cxpcn3 23416*	Extend continuity of the complex power function to a base of zero, as long as the exponent has strictly positive real part. (Contributed by Mario Carneiro, 2-May-2016.)
|- D = ( `' Re " RR+ )   &    |- J = ( TopOpen ` ℂfld )   &    |- K = ( J ↾t ( 0 [,) +oo ) )   &    |- L = ( J ↾t D )   =>    |- ( x e. ( 0 [,) +oo ) , y e. D |-> ( x ^c y ) ) e. ( ( K tX L ) Cn J )
Theorem	resqrtcn 23417	Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( sqr |` ( 0 [,) +oo ) ) e. ( ( 0 [,) +oo ) -cn-> RR )
Theorem	sqrtcn 23418	Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
|- D = ( CC \ ( -oo (,] 0 ) )   =>    |- ( sqr |` D ) e. ( D -cn-> CC )
Theorem	cxpaddlelem 23419	Lemma for cxpaddle 23420. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A <_ 1 )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> B <_ 1 )   =>    |- ( ph -> A <_ ( A ^c B ) )
Theorem	cxpaddle 23420	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ B )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> C <_ 1 )   =>    |- ( ph -> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) )
Theorem	abscxpbnd 23421	Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> 0 <_ ( Re ` B ) )   &    |- ( ph -> M e. RR )   &    |- ( ph -> ( abs ` A ) <_ M )   =>    |- ( ph -> ( abs ` ( A ^c B ) ) <_ ( ( M ^c ( Re ` B ) ) x. ( exp ` ( ( abs ` B ) x. pi ) ) ) )
Theorem	root1id 23422	Property of an N-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( N e. NN -> ( ( -u 1 ^c ( 2 / N ) ) ^ N ) = 1 )
Theorem	root1eq1 23423	The only powers of an N-th root of unity that equal 1 are the multiples of N. In other words, -u 1 ^c ( 2 / N ) has order N in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complex numbers.) (Contributed by Mario Carneiro, 28-Apr-2016.)
|- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ K ) = 1 <-> N || K ) )
Theorem	root1cj 23424	Within the N-th roots of unity, the conjugate of the K-th root is the N - K-th root. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( ( N e. NN /\ K e. ZZ ) -> ( * ` ( ( -u 1 ^c ( 2 / N ) ) ^ K ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ ( N - K ) ) )
Theorem	cxpeq 23425*	Solve an equation involving an N-th power. The expression -u 1 ^c ( 2 / N ) = exp ( 2 pi _i / N ) is a way to write the primitive N-th root of unity with the smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( ( A e. CC /\ N e. NN /\ B e. CC ) -> ( ( A ^ N ) = B <-> E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( -u 1 ^c ( 2 / N ) ) ^ n ) ) ) )
Theorem	loglesqrt 23426	An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( log ` ( A + 1 ) ) <_ ( sqr ` A ) )
Theorem	logreclem 23427	Symmetry of the natural logarithm range by negation. Lemma for logrec 23428. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
|- ( ( A e. ran log /\ -. ( Im ` A ) = pi ) -> -u A e. ran log )
Theorem	logrec 23428	Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) )
14.3.5  Logarithms to an arbitrary base Define "log using an arbitrary base" function and then prove some of its properties. Note that logb is generalized to an arbitrary base and arbitrary parameter in CC, but it doesn't accept infinities as arguments, unlike log. Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log". There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations): ( B logb X ) where B is the base and X is the argument of the logarithm function. An alternative would be to support the notational form ( ( logb ` B ) ` X ); that looks a little more like traditional notation. Such a function ( logb ` B ) for a fixed base can be obtained in Metamath (without the need for a new definition) by the curry function: (curry logb ` B ), see logbmpt 23453, logbf 23454 and logbfval 23455.
Syntax	clogb 23429	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 23430*	Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as ( B logb X ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition is according to Wikipedia "Complex logarithm": https://en.wikipedia.org/wiki/Complex_logarithm#Logarithms_to_other_bases (10-Jun-2020). (Contributed by David A. Wheeler, 21-Jan-2017.)
|- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
Theorem	logbval 23431	Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
Theorem	logbcl 23432	General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) e. CC )
Theorem	logbid1 23433	General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.)
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A logb A ) = 1 )
Theorem	logb1 23434	The logarithm of 1 to an arbitrary base B is 0. Property 1(b) of [Cohen4] p. 361. See log1 23263. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> ( B logb 1 ) = 0 )
Theorem	elogb 23435	The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using _e as the base in logb is the same as log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
|- ( A e. ( CC \ { 0 } ) -> ( _e logb A ) = ( log ` A ) )
Theorem	logbchbase 23436	Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( A logb X ) = ( ( B logb X ) / ( B logb A ) ) )
Theorem	relogbval 23437	Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
Theorem	relogbcl 23438	Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. RR+ /\ X e. RR+ /\ B =/= 1 ) -> ( B logb X ) e. RR )
Theorem	relogbzcl 23439	Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) e. RR )
Theorem	relogbreexp 23440	Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) )
Theorem	relogbzexp 23441	Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. ZZ ) -> ( B logb ( C ^ N ) ) = ( N x. ( B logb C ) ) )
Theorem	relogbmul 23442	The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. C ) ) = ( ( B logb A ) + ( B logb C ) ) )
Theorem	relogbmulexp 23443	The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( B logb ( A x. ( C ^c E ) ) ) = ( ( B logb A ) + ( E x. ( B logb C ) ) ) )
Theorem	relogbdiv 23444	The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( ( B logb A ) - ( B logb C ) ) )
Theorem	relogbexp 23445	Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
|- ( ( B e. RR+ /\ B =/= 1 /\ M e. ZZ ) -> ( B logb ( B ^ M ) ) = M )
Theorem	nnlogbexp 23446	Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ M e. ZZ ) -> ( B logb ( B ^ M ) ) = M )
Theorem	logbrec 23447	Logarithm of a reciprocal changes sign. See logrec 23428. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = -u ( B logb A ) )
Theorem	logbleb 23448	The general logarithm function is monotone/increasing. See logleb 23280. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ /\ Y e. RR+ ) -> ( X <_ Y <-> ( B logb X ) <_ ( B logb Y ) ) )
Theorem	logblt 23449	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 23277. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ /\ Y e. RR+ ) -> ( X < Y <-> ( B logb X ) < ( B logb Y ) ) )
Theorem	relogbcxp 23450	Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
|- ( ( B e. ( RR+ \ { 1 } ) /\ X e. RR ) -> ( B logb ( B ^c X ) ) = X )
Theorem	cxplogb 23451	Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B ^c ( B logb X ) ) = X )
Theorem	relogbcxpb 23452	The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( B e. RR+ /\ B =/= 1 ) /\ X e. RR+ /\ Y e. RR ) -> ( ( B logb X ) = Y <-> ( B ^c Y ) = X ) )
Theorem	logbmpt 23453*	The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.)
|- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> (curry logb ` B ) = ( y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` B ) ) ) )
Theorem	logbf 23454	The general logarithm to a fixed base regarded as function. (Contributed by AV, 11-Jun-2020.)
|- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> (curry logb ` B ) : ( CC \ { 0 } ) --> CC )
Theorem	logbfval 23455	The general logarithm of a complex number to a fixed base. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( (curry logb ` B ) ` X ) = ( B logb X ) )
Theorem	relogbf 23456	The general logarithm to a real base greater than 1 regarded as function restricted to the positive integers. Property in [Cohen4] p. 349. (Contributed by AV, 12-Jun-2020.)
|- ( ( B e. RR+ /\ 1 < B ) -> ( (curry logb ` B ) |` RR+ ) : RR+ --> RR )
Theorem	logblog 23457	The general logarithm to the base being Euler's constant regarded as function is the natural logarithm. (Contributed by AV, 12-Jun-2020.)
|- (curry logb ` _e ) = log
14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords
Theorem	angval 23458*	Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range ( - pi , pi ]. To convert from the geometry notation, m A B C, the measure of the angle with legs A B, C B where C is more counterclockwise for positive angles, is represented by ( ( C - B ) F ( A - B ) ). (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A F B ) = ( Im ` ( log ` ( B / A ) ) ) )
Theorem	angcan 23459*	Cancel a constant multiplier in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. A ) F ( C x. B ) ) = ( A F B ) )
Theorem	angneg 23460*	Cancel a negative sign in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( -u A F -u B ) = ( A F B ) )
Theorem	angvald 23461*	The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 23458. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> X e. CC )   &    |- ( ph -> X =/= 0 )   &    |- ( ph -> Y e. CC )   &    |- ( ph -> Y =/= 0 )   =>    |- ( ph -> ( X F Y ) = ( Im ` ( log ` ( Y / X ) ) ) )
Theorem	angcld 23462*	The (signed) angle between two vectors is in ( -u pi (,] pi ). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> X e. CC )   &    |- ( ph -> X =/= 0 )   &    |- ( ph -> Y e. CC )   &    |- ( ph -> Y =/= 0 )   =>    |- ( ph -> ( X F Y ) e. ( -u pi (,] pi ) )
Theorem	angrteqvd 23463*	Two vectors are at a right angle iff their quotient is purely imaginary. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> X e. CC )   &    |- ( ph -> X =/= 0 )   &    |- ( ph -> Y e. CC )   &    |- ( ph -> Y =/= 0 )   =>    |- ( ph -> ( ( X F Y ) e. { ( pi / 2 ) , -u ( pi / 2 ) } <-> ( Re ` ( Y / X ) ) = 0 ) )
Theorem	cosangneg2d 23464*	The cosine of the angle between X and -u Y is the negative of that between X and Y. If A, B and C are collinear points, this implies that the cosines of DBA and DBC sum to zero, i.e., that DBA and DBC are supplementary. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> X e. CC )   &    |- ( ph -> X =/= 0 )   &    |- ( ph -> Y e. CC )   &    |- ( ph -> Y =/= 0 )   =>    |- ( ph -> ( cos ` ( X F -u Y ) ) = -u ( cos ` ( X F Y ) ) )
Theorem	angrtmuld 23465*	Perpendicularity of two vectors does not change under rescaling the second. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> X e. CC )   &    |- ( ph -> Y e. CC )   &    |- ( ph -> Z e. CC )   &    |- ( ph -> X =/= 0 )   &    |- ( ph -> Y =/= 0 )   &    |- ( ph -> Z =/= 0 )   &    |- ( ph -> ( Z / Y ) e. RR )   =>    |- ( ph -> ( ( X F Y ) e. { ( pi / 2 ) , -u ( pi / 2 ) } <-> ( X F Z ) e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )
Theorem	ang180lem1 23466*	Lemma for ang180 23471. Show that the "revolution number" N is an integer, using efeq1 23206 to show that since the product of the three arguments A , 1 / ( 1 - A ) , ( A - 1 ) / A is -u 1, the sum of the logarithms must be an integer multiple of 2 pi _i away from pi _i = log ( -u 1 ). (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) )   &    |- N = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) )   =>    |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N e. ZZ /\ ( T / _i ) e. RR ) )
Theorem	ang180lem2 23467*	Lemma for ang180 23471. Show that the revolution number N is strictly between -u 2 and 1. Both bounds are established by iterating using the bounds on the imaginary part of the logarithm, logimcl 23247, but the resulting bound gives only N <_ 1 for the upper bound. The case N = 1 is not ruled out here, but it is in some sense an "edge case" that can only happen under very specific conditions; in particular we show that all the angle arguments A , 1 / ( 1 - A ) , ( A - 1 ) / A must lie on the negative real axis, which is a contradiction because clearly if A is negative then the other two are positive real. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) )   &    |- N = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) )   =>    |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 2 < N /\ N < 1 ) )
Theorem	ang180lem3 23468*	Lemma for ang180 23471. Since ang180lem1 23466 shows that N is an integer and ang180lem2 23467 shows that N is strictly between -u 2 and 1, it follows that N e. { -u 1 , 0 }, and these two cases correspond to the two possible values for T. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) )   &    |- N = ( ( ( T / _i ) / ( 2 x. pi ) ) - ( 1 / 2 ) )   =>    |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. { -u ( _i x. pi ) , ( _i x. pi ) } )
Theorem	ang180lem4 23469*	Lemma for ang180 23471. Reduce the statement to one variable. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( ( 1 - A ) F 1 ) + ( A F ( A - 1 ) ) ) + ( 1 F A ) ) e. { -u pi , pi } )
Theorem	ang180lem5 23470*	Lemma for ang180 23471: Reduce the statement to two variables. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ A =/= B ) -> ( ( ( ( A - B ) F A ) + ( B F ( B - A ) ) ) + ( A F B ) ) e. { -u pi , pi } )
Theorem	ang180 23471*	The sum of angles m A B C + m B C A + m C A B in a triangle adds up to either pi or -u pi, i.e. 180 degrees. (The sign is due to the two possible orientations of vertex arrangement and our signed notion of angle). This is Metamath 100 proof #27. (Contributed by Mario Carneiro, 23-Sep-2014.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( A =/= B /\ B =/= C /\ A =/= C ) ) -> ( ( ( ( C - B ) F ( A - B ) ) + ( ( A - C ) F ( B - C ) ) ) + ( ( B - A ) F ( C - A ) ) ) e. { -u pi , pi } )
Theorem	lawcoslem1 23472	Lemma for lawcos 23473. Here we prove the law for a point at the origin and two distinct points U and V, using an expanded version of the signed angle expression on the complex plane. (Contributed by David A. Wheeler, 11-Jun-2015.)
|- ( ph -> U e. CC )   &    |- ( ph -> V e. CC )   &    |- ( ph -> U =/= 0 )   &    |- ( ph -> V =/= 0 )   =>    |- ( ph -> ( ( abs ` ( U - V ) ) ^ 2 ) = ( ( ( ( abs ` U ) ^ 2 ) + ( ( abs ` V ) ^ 2 ) ) - ( 2 x. ( ( ( abs ` U ) x. ( abs ` V ) ) x. ( ( Re ` ( U / V ) ) / ( abs ` ( U / V ) ) ) ) ) ) )
Theorem	lawcos 23473*	Law of cosines (also known as the Al-Kashi theorem or the generalized Pythagorean theorem, or the cosine formula or cosine rule). Given three distinct points A, B, and C, prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where F is the signed angle construct (as used in ang180 23471), X is the distance of line segment BC, Y is the distance of line segment AC, Z is the distance of line segment AB, and O is the signed angle m/_ BCA on the complex plane. We translate triangle ABC to move C to the origin (C-C), B to U=(B-C), and A to V=(A-C), then use lemma lawcoslem1 23472 to prove this algebraically simpler case. The metamath convention is to use a signed angle; in this case the sign doesn't matter because we use the cosine of the angle (see cosneg 14089). The Pythagorean theorem pythag 23474 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. This is Metamath 100 proof #94. (Contributed by David A. Wheeler, 12-Jun-2015.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- X = ( abs ` ( B - C ) )   &    |- Y = ( abs ` ( A - C ) )   &    |- Z = ( abs ` ( A - B ) )   &    |- O = ( ( B - C ) F ( A - C ) )   =>    |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( A =/= C /\ B =/= C ) ) -> ( Z ^ 2 ) = ( ( ( X ^ 2 ) + ( Y ^ 2 ) ) - ( 2 x. ( ( X x. Y ) x. ( cos ` O ) ) ) ) )
Theorem	pythag 23474*	Pythagorean theorem. Given three distinct points A, B, and C that form a right triangle (with the right angle at C), prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where F is the signed angle construct (as used in ang180 23471), X is the distance of line segment BC, Y is the distance of line segment AC, Z is the distance of line segment AB (the hypotenuse), and O is the signed right angle m/_ BCA. We use the law of cosines lawcos 23473 to prove this, along with simple trigonometry facts like coshalfpi 23152 and cosneg 14089. (Contributed by David A. Wheeler, 13-Jun-2015.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- X = ( abs ` ( B - C ) )   &    |- Y = ( abs ` ( A - C ) )   &    |- Z = ( abs ` ( A - B ) )   &    |- O = ( ( B - C ) F ( A - C ) )   =>    |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( A =/= C /\ B =/= C ) /\ O e. { ( pi / 2 ) , -u ( pi / 2 ) } ) -> ( Z ^ 2 ) = ( ( X ^ 2 ) + ( Y ^ 2 ) ) )
Theorem	isosctrlem1 23475	Lemma for isosctr 23478. (Contributed by Saveliy Skresanov, 30-Dec-2016.)
|- ( ( A e. CC /\ ( abs ` A ) = 1 /\ -. 1 = A ) -> ( Im ` ( log ` ( 1 - A ) ) ) =/= pi )
Theorem	isosctrlem2 23476	Lemma for isosctr 23478. Corresponds to the case where one vertex is at 0, another at 1 and the third lies on the unit circle. (Contributed by Saveliy Skresanov, 31-Dec-2016.)
|- ( ( A e. CC /\ ( abs ` A ) = 1 /\ -. 1 = A ) -> ( Im ` ( log ` ( 1 - A ) ) ) = ( Im ` ( log ` ( -u A / ( 1 - A ) ) ) ) )
Theorem	isosctrlem3 23477*	Lemma for isosctr 23478. Corresponds to the case where one vertex is at 0. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ B e. CC ) /\ ( A =/= 0 /\ B =/= 0 /\ A =/= B ) /\ ( abs ` A ) = ( abs ` B ) ) -> ( -u A F ( B - A ) ) = ( ( A - B ) F -u B ) )
Theorem	isosctr 23478*	Isosceles triangle theorem. This is Metamath 100 proof #65. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   =>    |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( A =/= C /\ B =/= C /\ A =/= B ) /\ ( abs ` ( A - C ) ) = ( abs ` ( B - C ) ) ) -> ( ( C - A ) F ( B - A ) ) = ( ( A - B ) F ( C - B ) ) )
Theorem	ssscongptld 23479*	If two triangles have equal sides, one angle in one triangle has the same cosine as the corresponding angle in the other triangle. This is a partial form of the SSS congruence theorem. This theorem is proven by using lawcos 23473 on both triangles to express one side in terms of the other two, and then equating these expressions and reducing this algebraically to get an equality of cosines of angles. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> E e. CC )   &    |- ( ph -> G e. CC )   &    |- ( ph -> A =/= B )   &    |- ( ph -> B =/= C )   &    |- ( ph -> D =/= E )   &    |- ( ph -> E =/= G )   &    |- ( ph -> ( abs ` ( A - B ) ) = ( abs ` ( D - E ) ) )   &    |- ( ph -> ( abs ` ( B - C ) ) = ( abs ` ( E - G ) ) )   &    |- ( ph -> ( abs ` ( C - A ) ) = ( abs ` ( G - D ) ) )   =>    |- ( ph -> ( cos ` ( ( A - B ) F ( C - B ) ) ) = ( cos ` ( ( D - E ) F ( G - E ) ) ) )
Theorem	affineequiv 23480	Equivalence between two ways of expressing B as an affine combination of A and C. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( C - B ) = ( D x. ( C - A ) ) ) )
Theorem	affineequiv2 23481	Equivalence between two ways of expressing B as an affine combination of A and C. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( B - A ) = ( ( 1 - D ) x. ( C - A ) ) ) )
Theorem	angpieqvdlem 23482	Equivalence used in the proof of angpieqvd 23485. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= B )   &    |- ( ph -> A =/= C )   =>    |- ( ph -> ( -u ( ( C - B ) / ( A - B ) ) e. RR+ <-> ( ( C - B ) / ( C - A ) ) e. ( 0 (,) 1 ) ) )
Theorem	angpieqvdlem2 23483*	Equivalence used in angpieqvd 23485. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= B )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> ( -u ( ( C - B ) / ( A - B ) ) e. RR+ <-> ( ( A - B ) F ( C - B ) ) = pi ) )
Theorem	angpined 23484*	If the angle at ABC is pi, then A is not equal to C. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= B )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> ( ( ( A - B ) F ( C - B ) ) = pi -> A =/= C ) )
Theorem	angpieqvd 23485*	The angle ABC is pi iff B is a nontrivial convex combination of A and C, i.e., iff B is in the interior of the segment AC. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= B )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> ( ( ( A - B ) F ( C - B ) ) = pi <-> E. w e. ( 0 (,) 1 ) B = ( ( w x. A ) + ( ( 1 - w ) x. C ) ) ) )
Theorem	chordthmlem 23486*	If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 23479 to observe that, since AMQ and BMQ have equal sides, the angles QMB and QMA must be equal. Since they are supplementary, both must be right. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> M = ( ( A + B ) / 2 ) )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> Q =/= M )   =>    |- ( ph -> ( ( Q - M ) F ( B - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
Theorem	chordthmlem2 23487*	If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then QMP is a right angle. This is proven by reduction to the special case chordthmlem 23486, where P = B, and using angrtmuld 23465 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. RR )   &    |- ( ph -> M = ( ( A + B ) / 2 ) )   &    |- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )   &    |- ( ph -> P =/= M )   &    |- ( ph -> Q =/= M )   =>    |- ( ph -> ( ( Q - M ) F ( P - M ) ) e. { ( pi / 2 ) , -u ( pi / 2 ) } )
Theorem	chordthmlem3 23488	If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then PQ 2 = QM 2 + PM 2 . This follows from chordthmlem2 23487 and the Pythagorean theorem (pythag 23474) in the case where P and Q are unequal to M. If either P or Q equals M, the result is trivial. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. RR )   &    |- ( ph -> M = ( ( A + B ) / 2 ) )   &    |- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )   =>    |- ( ph -> ( ( abs ` ( P - Q ) ) ^ 2 ) = ( ( ( abs ` ( Q - M ) ) ^ 2 ) + ( ( abs ` ( P - M ) ) ^ 2 ) ) )
Theorem	chordthmlem4 23489	If P is on the segment AB and M is the midpoint of AB, then PA x. PB = BM 2 - PM 2 . If all lengths are reexpressed as fractions of AB, this reduces to the identity X x. ( 1 - X ) = ( 1 / 2 ) 2 - ( ( 1 / 2 ) - X ) 2 . (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> X e. ( 0 [,] 1 ) )   &    |- ( ph -> M = ( ( A + B ) / 2 ) )   &    |- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )   =>    |- ( ph -> ( ( abs ` ( P - A ) ) x. ( abs ` ( P - B ) ) ) = ( ( ( abs ` ( B - M ) ) ^ 2 ) - ( ( abs ` ( P - M ) ) ^ 2 ) ) )
Theorem	chordthmlem5 23490	If P is on the segment AB and AQ = BQ, then PA x. PB = BQ 2 - PQ 2 . This follows from two uses of chordthmlem3 23488 to show that PQ 2 = QM 2 + PM 2 and BQ 2 = QM 2 + BM 2 , so BQ 2 - PQ 2 = (QM 2 + BM 2 ) - (QM 2 + PM 2 ) = BM 2 - PM 2 , which equals PA x. PB by chordthmlem4 23489. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. ( 0 [,] 1 ) )   &    |- ( ph -> P = ( ( X x. A ) + ( ( 1 - X ) x. B ) ) )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )   =>    |- ( ph -> ( ( abs ` ( P - A ) ) x. ( abs ` ( P - B ) ) ) = ( ( ( abs ` ( B - Q ) ) ^ 2 ) - ( ( abs ` ( P - Q ) ) ^ 2 ) ) )
Theorem	chordthm 23491*	The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA x. PB and PC x. PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to pi. The result is proven by using chordthmlem5 23490 twice to show that PA x. PB and PC x. PD both equal BQ 2 - PQ 2 . This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. This is Metamath 100 proof #55. (Contributed by David Moews, 28-Feb-2017.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> P e. CC )   &    |- ( ph -> A =/= P )   &    |- ( ph -> B =/= P )   &    |- ( ph -> C =/= P )   &    |- ( ph -> D =/= P )   &    |- ( ph -> ( ( A - P ) F ( B - P ) ) = pi )   &    |- ( ph -> ( ( C - P ) F ( D - P ) ) = pi )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( B - Q ) ) )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( C - Q ) ) )   &    |- ( ph -> ( abs ` ( A - Q ) ) = ( abs ` ( D - Q ) ) )   =>    |- ( ph -> ( ( abs ` ( P - A ) ) x. ( abs ` ( P - B ) ) ) = ( ( abs ` ( P - C ) ) x. ( abs ` ( P - D ) ) ) )
Theorem	heron 23492*	Heron's formula gives the area of a triangle given only the side lengths. If points A, B, C form a triangle, then the area of the triangle, represented here as ( 1 / 2 ) x. X x. Y x. abs ( sin O ), is equal to the square root of S x. ( S - X ) x. ( S - Y ) x. ( S - Z ), where S = ( X + Y + Z ) / 2 is half the perimeter of the triangle. Based on work by Jon Pennant. This is Metamath 100 proof #57. (Contributed by Mario Carneiro, 10-Mar-2019.)
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )   &    |- X = ( abs ` ( B - C ) )   &    |- Y = ( abs ` ( A - C ) )   &    |- Z = ( abs ` ( A - B ) )   &    |- O = ( ( B - C ) F ( A - C ) )   &    |- S = ( ( ( X + Y ) + Z ) / 2 )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= C )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> ( ( ( 1 / 2 ) x. ( X x. Y ) ) x. ( abs ` ( sin ` O ) ) ) = ( sqr ` ( ( S x. ( S - X ) ) x. ( ( S - Y ) x. ( S - Z ) ) ) ) )
14.3.7  Solutions of quadratic, cubic, and quartic equations
Theorem	quad2 23493	The quadratic equation, without specifying the particular branch D to the square root. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( ph -> A e. CC )   &    |- ( ph -> A =/= 0 )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> ( D ^ 2 ) = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) )   =>    |- ( ph -> ( ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 <-> ( X = ( ( -u B + D ) / ( 2 x. A ) ) \/ X = ( ( -u B - D ) / ( 2 x. A ) ) ) ) )
Theorem	quad 23494	The quadratic equation. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( ph -> A e. CC )   &    |- ( ph -> A =/= 0 )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> D = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) )   =>    |- ( ph -> ( ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 <-> ( X = ( ( -u B + ( sqr ` D ) ) / ( 2 x. A ) ) \/ X = ( ( -u B - ( sqr ` D ) ) / ( 2 x. A ) ) ) ) )
Theorem	1cubrlem 23495	The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( ( -u 1 ^c ( 2 / 3 ) ) = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) /\ ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) )
Theorem	1cubr 23496	The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- R = { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) }   =>    |- ( A e. R <-> ( A e. CC /\ ( A ^ 3 ) = 1 ) )
Theorem	dcubic1lem 23497	Lemma for dcubic1 23499 and dcubic2 23498: simplify the cubic equation under the substitution X = U - M / U. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ph -> P e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> T e. CC )   &    |- ( ph -> ( T ^ 3 ) = ( G - N ) )   &    |- ( ph -> G e. CC )   &    |- ( ph -> ( G ^ 2 ) = ( ( N ^ 2 ) + ( M ^ 3 ) ) )   &    |- ( ph -> M = ( P / 3 ) )   &    |- ( ph -> N = ( Q / 2 ) )   &    |- ( ph -> T =/= 0 )   &    |- ( ph -> U e. CC )   &    |- ( ph -> U =/= 0 )   &    |- ( ph -> X = ( U - ( M / U ) ) )   =>    |- ( ph -> ( ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 <-> ( ( ( U ^ 3 ) ^ 2 ) + ( ( Q x. ( U ^ 3 ) ) - ( M ^ 3 ) ) ) = 0 ) )
Theorem	dcubic2 23498*	Reverse direction of dcubic 23500. Given a solution U to the "substitution" quadratic equation X = U - M / U, show that X is in the desired form. (Contributed by Mario Carneiro, 25-Apr-2015.)
|- ( ph -> P e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> T e. CC )   &    |- ( ph -> ( T ^ 3 ) = ( G - N ) )   &    |- ( ph -> G e. CC )   &    |- ( ph -> ( G ^ 2 ) = ( ( N ^ 2 ) + ( M ^ 3 ) ) )   &    |- ( ph -> M = ( P / 3 ) )   &    |- ( ph -> N = ( Q / 2 ) )   &    |- ( ph -> T =/= 0 )   &    |- ( ph -> U e. CC )   &    |- ( ph -> U =/= 0 )   &    |- ( ph -> X = ( U - ( M / U ) ) )   &    |- ( ph -> ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 )   =>    |- ( ph -> E. r e. CC ( ( r ^ 3 ) = 1 /\ X = ( ( r x. T ) - ( M / ( r x. T ) ) ) ) )
Theorem	dcubic1 23499	Forward direction of dcubic 23500: the claimed formula produces solutions to the cubic equation. (Contributed by Mario Carneiro, 25-Apr-2015.)
|- ( ph -> P e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> T e. CC )   &    |- ( ph -> ( T ^ 3 ) = ( G - N ) )   &    |- ( ph -> G e. CC )   &    |- ( ph -> ( G ^ 2 ) = ( ( N ^ 2 ) + ( M ^ 3 ) ) )   &    |- ( ph -> M = ( P / 3 ) )   &    |- ( ph -> N = ( Q / 2 ) )   &    |- ( ph -> T =/= 0 )   &    |- ( ph -> X = ( T - ( M / T ) ) )   =>    |- ( ph -> ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 )
Theorem	dcubic 23500*	Solutions to the depressed cubic, a special case of cubic 23503. (The definitions of M , N , G , T here differ from mcubic 23501 by scale factors of -u 9, 5 4, 5 4 and -u 2 7 respectively, to simplify the algebra and presentation.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ph -> P e. CC )   &    |- ( ph -> Q e. CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> T e. CC )   &    |- ( ph -> ( T ^ 3 ) = ( G - N ) )   &    |- ( ph -> G e. CC )   &    |- ( ph -> ( G ^ 2 ) = ( ( N ^ 2 ) + ( M ^ 3 ) ) )   &    |- ( ph -> M = ( P / 3 ) )   &    |- ( ph -> N = ( Q / 2 ) )   &    |- ( ph -> T =/= 0 )   =>    |- ( ph -> ( ( ( X ^ 3 ) + ( ( P x. X ) + Q ) ) = 0 <-> E. r e. CC ( ( r ^ 3 ) = 1 /\ X = ( ( r x. T ) - ( M / ( r x. T ) ) ) ) ) )