HomeHome Metamath Proof Explorer
Theorem List (p. 204 of 323)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21811)
  Hilbert Space Explorer  Hilbert Space Explorer
(21812-23334)
  Users' Mathboxes  Users' Mathboxes
(23335-32225)
 

Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcxplt3d 20301 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 ) ) )
 
Theoremcxple3d 20302 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 ) ) )
 
Theoremcxpmuld 20303 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 ) )
 
Theoremdvcxp1 20304* 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
 ) ) ) ) )
 
Theoremdvcxp2 20305* 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 ) ) ) )
 
Theoremdvsqr 20306 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 ) ) ) )
 
Theoremcxpcn 20307* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  J  =  (
 TopOpen ` fld )   &    |-  K  =  ( Jt  D )   =>    |-  ( x  e.  D ,  y  e.  CC  |->  ( x  ^ c  y ) )  e.  (
 ( K  tX  J )  Cn  J )
 
Theoremcxpcn2 20308* Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  RR+ )   =>    |-  ( x  e.  RR+ ,  y  e.  CC  |->  ( x  ^ c  y ) )  e.  (
 ( K  tX  J )  Cn  J )
 
Theoremcxpcn3lem 20309* Lemma for cxpcn3 20310. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( `' Re " RR+ )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  ( 0 [,)  +oo ) )   &    |-  L  =  ( Jt  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 ) )
 
Theoremcxpcn3 20310* 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 ` fld )   &    |-  K  =  ( Jt  ( 0 [,)  +oo ) )   &    |-  L  =  ( Jt  D )   =>    |-  ( x  e.  (
 0 [,)  +oo ) ,  y  e.  D  |->  ( x  ^ c  y ) )  e.  (
 ( K  tX  L )  Cn  J )
 
Theoremresqrcn 20311 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
 
Theoremsqrcn 20312 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( sqr  |`  D )  e.  ( D -cn-> CC )
 
Theoremcxpaddlelem 20313 Lemma for cxpaddle 20314. (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 ) )
 
Theoremcxpaddle 20314 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 ) ) )
 
Theoremabscxpbnd 20315 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 ) ) ) )
 
Theoremroot1id 20316 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
 )
 
Theoremroot1eq1 20317 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 complexes.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( ( N  e.  NN  /\  K  e.  ZZ )  ->  ( ( (
 -u 1  ^ c  ( 2  /  N ) ) ^ K )  =  1  <->  N  ||  K ) )
 
Theoremroot1cj 20318 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 ) ) )
 
Theoremcxpeq 20319* 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 ) ) ) )
 
Theoremloglesqr 20320 An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( log `  ( A  +  1 )
 )  <_  ( sqr `  A ) )
 
13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords
 
Theoremangval 20321* 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 ) ) ) )
 
Theoremangcan 20322* 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 ) )
 
Theoremangneg 20323* 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 ) )
 
Theoremangvald 20324* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20321. (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 ) ) ) )
 
Theoremangcld 20325* 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 ) )
 
Theoremangrteqvd 20326* 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 ) )
 
Theoremcosangneg2d 20327* 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 ) ) )
 
Theoremangrtmuld 20328* 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
 ) } ) )
 
Theoremang180lem1 20329* Lemma for ang180 20334. Show that the "revolution number"  N is an integer, using efeq1 20109 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 )
 )
 
Theoremang180lem2 20330* Lemma for ang180 20334. 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 20145, 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 ) )
 
Theoremang180lem3 20331* Lemma for ang180 20334. Since ang180lem1 20329 shows that  N is an integer and ang180lem2 20330 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 ) } )
 
Theoremang180lem4 20332* Lemma for ang180 20334. 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 } )
 
Theoremang180lem5 20333* Lemma for ang180 20334: 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 } )
 
Theoremang180 20334* 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). (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 } )
 
Theoremlawcoslem1 20335 Lemma for Law of Cosines lawcos 20336. 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 ) ) ) ) ) ) )
 
Theoremlawcos 20336* Law of Cosines. 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 20334),  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 distinguished (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 20335 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 12635). The Pythagorean Theorem pythag 20337 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. (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 ) ) ) ) )
 
Theorempythag 20337* 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 20334),  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 distinguished (signed) right angle m/_ BCA. We use the law of cosines lawcos 20336 to prove this, along with simple trig facts like coshalfpi 20055 and cosneg 12635. (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 ) ) )
 
Theoremlogreclem 20338 Symmetry of the natural logarithm range by negation. Lemma for logrec 20339. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
 |-  ( ( A  e.  ran 
 log  /\  -.  ( Im
 `  A )  =  pi )  ->  -u A  e.  ran  log )
 
Theoremlogrec 20339 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 )
 ) )
 
Theoremisosctrlem1 20340 Lemma for isosctr 20343. (Contributed by Saveliy Skresanov, 30-Dec-2016.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A ) 
 ->  ( Im `  ( log `  ( 1  -  A ) ) )  =/=  pi )
 
Theoremisosctrlem2 20341 Lemma for isosctr 20343. 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 ) ) ) ) )
 
Theoremisosctrlem3 20342* Lemma for isosctr 20343. 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 ) )
 
Theoremisosctr 20343* Isosceles triangle theorem. (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 ) ) )
 
Theoremssscongptld 20344* 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 20336 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 ) ) ) )
 
Theoremaffineequiv 20345 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 ) ) ) )
 
Theoremaffineequiv2 20346 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 ) ) ) )
 
Theoremangpieqvdlem 20347 Equivalence used in the proof of angpieqvd 20350. (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 ) ) )
 
Theoremangpieqvdlem2 20348* Equivalence used in angpieqvd 20350. (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 ) )
 
Theoremangpined 20349* 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 ) )
 
Theoremangpieqvd 20350* 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 ) ) ) )
 
Theoremchordthmlem 20351* If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20344 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 ) }
 )
 
Theoremchordthmlem2 20352* 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 20351, where P = B, and using angrtmuld 20328 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 ) }
 )
 
Theoremchordthmlem3 20353 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 20352 and the Pythagorean theorem (pythag 20337) 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 ) ) )
 
Theoremchordthmlem4 20354 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 ) ) )
 
Theoremchordthmlem5 20355 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 20353 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 20354. (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 ) ) )
 
Theoremchordthm 20356* 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 20355 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. (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 ) ) ) )
 
13.3.6  Solutions of quadratic, cubic, and quartic equations
 
Theoremquad2 20357 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 ) ) ) ) )
 
Theoremquad 20358 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 ) ) ) ) )
 
Theorem1cubrlem 20359 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 ) )
 
Theorem1cubr 20360 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 ) )
 
Theoremdcubic1lem 20361 Lemma for dcubic1 20363 and dcubic2 20362: 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 ) )
 
Theoremdcubic2 20362* Reverse direction of dcubic 20364. 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 ) ) ) ) )
 
Theoremdcubic1 20363 Forward direction of dcubic 20364: 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 )
 
Theoremdcubic 20364* Solutions to the depressed cubic, a special case of cubic 20367. (The definitions of  M ,  N ,  G ,  T here differ from mcubic 20365 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 ) ) ) ) ) )
 
Theoremmcubic 20365* Solutions to a monic cubic equation, a special case of cubic 20367. (Contributed by Mario Carneiro, 24-Apr-2015.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  T  e.  CC )   &    |-  ( ph  ->  ( T ^
 3 )  =  ( ( N  +  G )  /  2 ) )   &    |-  ( ph  ->  G  e.  CC )   &    |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )   &    |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  C ) ) )   &    |-  ( ph  ->  N  =  ( ( ( 2  x.  ( B ^ 3
 ) )  -  (
 9  x.  ( B  x.  C ) ) )  +  (; 2 7  x.  D ) ) )   &    |-  ( ph  ->  T  =/=  0
 )   =>    |-  ( ph  ->  (
 ( ( ( X ^ 3 )  +  ( B  x.  ( X ^ 2 ) ) )  +  ( ( C  x.  X )  +  D ) )  =  0  <->  E. r  e.  CC  ( ( r ^
 3 )  =  1 
 /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  /  (
 r  x.  T ) ) )  /  3
 ) ) ) )
 
Theoremcubic2 20366* The solution to the general cubic equation, for arbitrary choices  G and  T of the square and cube roots. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  T  e.  CC )   &    |-  ( ph  ->  ( T ^ 3 )  =  ( ( N  +  G )  / 
 2 ) )   &    |-  ( ph  ->  G  e.  CC )   &    |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^
 2 )  -  (
 4  x.  ( M ^ 3 ) ) ) )   &    |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C ) ) ) )   &    |-  ( ph  ->  N  =  ( ( ( 2  x.  ( B ^ 3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C ) ) )  +  (; 2 7  x.  ( ( A ^ 2 )  x.  D ) ) ) )   &    |-  ( ph  ->  T  =/=  0 )   =>    |-  ( ph  ->  ( ( ( ( A  x.  ( X ^
 3 ) )  +  ( B  x.  ( X ^ 2 ) ) )  +  ( ( C  x.  X )  +  D ) )  =  0  <->  E. r  e.  CC  ( ( r ^
 3 )  =  1 
 /\  X  =  -u ( ( ( B  +  ( r  x.  T ) )  +  ( M  /  (
 r  x.  T ) ) )  /  (
 3  x.  A ) ) ) ) )
 
Theoremcubic 20367* The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3779 to convert the existential quantifier to a triple disjunction. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  R  =  { 1 ,  ( ( -u 1  +  ( _i  x.  ( sqr `  3
 ) ) )  / 
 2 ) ,  (
 ( -u 1  -  ( _i  x.  ( sqr `  3
 ) ) )  / 
 2 ) }   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  T  =  ( ( ( N  +  ( sqr `  G )
 )  /  2 )  ^ c  ( 1  /  3 ) ) )   &    |-  ( ph  ->  G  =  ( ( N ^ 2 )  -  ( 4  x.  ( M ^ 3 ) ) ) )   &    |-  ( ph  ->  M  =  ( ( B ^ 2 )  -  ( 3  x.  ( A  x.  C ) ) ) )   &    |-  ( ph  ->  N  =  ( ( ( 2  x.  ( B ^ 3 ) )  -  ( ( 9  x.  A )  x.  ( B  x.  C ) ) )  +  (; 2 7  x.  ( ( A ^ 2 )  x.  D ) ) ) )   &    |-  ( ph  ->  M  =/=  0 )   =>    |-  ( ph  ->  ( ( ( ( A  x.  ( X ^
 3 ) )  +  ( B  x.  ( X ^ 2 ) ) )  +  ( ( C  x.  X )  +  D ) )  =  0  <->  E. r  e.  R  X  =  -u ( ( ( B  +  (
 r  x.  T ) )  +  ( M 
 /  ( r  x.  T ) ) ) 
 /  ( 3  x.  A ) ) ) )
 
Theorembinom4 20368 Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12496, 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^
 4 )  =  ( ( ( A ^
 4 )  +  (
 4  x.  ( ( A ^ 3 )  x.  B ) ) )  +  ( ( 6  x.  ( ( A ^ 2 )  x.  ( B ^
 2 ) ) )  +  ( ( 4  x.  ( A  x.  ( B ^ 3 ) ) )  +  ( B ^ 4 ) ) ) ) )
 
Theoremdquartlem1 20369 Lemma for dquart 20371. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  S  e.  CC )   &    |-  ( ph  ->  M  =  ( ( 2  x.  S ) ^
 2 ) )   &    |-  ( ph  ->  M  =/=  0
 )   &    |-  ( ph  ->  I  e.  CC )   &    |-  ( ph  ->  ( I ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( B  / 
 2 ) )  +  ( ( C  / 
 4 )  /  S ) ) )   =>    |-  ( ph  ->  ( ( ( ( X ^ 2 )  +  ( ( M  +  B )  /  2
 ) )  +  (
 ( ( ( M 
 /  2 )  x.  X )  -  ( C  /  4 ) ) 
 /  S ) )  =  0  <->  ( X  =  ( -u S  +  I
 )  \/  X  =  ( -u S  -  I
 ) ) ) )
 
Theoremdquartlem2 20370 Lemma for dquart 20371. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  S  e.  CC )   &    |-  ( ph  ->  M  =  ( ( 2  x.  S ) ^
 2 ) )   &    |-  ( ph  ->  M  =/=  0
 )   &    |-  ( ph  ->  I  e.  CC )   &    |-  ( ph  ->  ( I ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( B  / 
 2 ) )  +  ( ( C  / 
 4 )  /  S ) ) )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  (
 ( ( M ^
 3 )  +  (
 ( 2  x.  B )  x.  ( M ^
 2 ) ) )  +  ( ( ( ( B ^ 2
 )  -  ( 4  x.  D ) )  x.  M )  +  -u ( C ^ 2
 ) ) )  =  0 )   =>    |-  ( ph  ->  (
 ( ( ( M  +  B )  / 
 2 ) ^ 2
 )  -  ( ( ( C ^ 2
 )  /  4 )  /  M ) )  =  D )
 
Theoremdquart 20371 Solve a depressed quartic equation. To eliminate  S, which is the square root of a solution  M to the resolvent cubic equation, apply cubic 20367 or one of its variants. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  S  e.  CC )   &    |-  ( ph  ->  M  =  ( ( 2  x.  S ) ^
 2 ) )   &    |-  ( ph  ->  M  =/=  0
 )   &    |-  ( ph  ->  I  e.  CC )   &    |-  ( ph  ->  ( I ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( B  / 
 2 ) )  +  ( ( C  / 
 4 )  /  S ) ) )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  (
 ( ( M ^
 3 )  +  (
 ( 2  x.  B )  x.  ( M ^
 2 ) ) )  +  ( ( ( ( B ^ 2
 )  -  ( 4  x.  D ) )  x.  M )  +  -u ( C ^ 2
 ) ) )  =  0 )   &    |-  ( ph  ->  J  e.  CC )   &    |-  ( ph  ->  ( J ^
 2 )  =  ( ( -u ( S ^
 2 )  -  ( B  /  2 ) )  -  ( ( C 
 /  4 )  /  S ) ) )   =>    |-  ( ph  ->  ( (
 ( ( X ^
 4 )  +  ( B  x.  ( X ^
 2 ) ) )  +  ( ( C  x.  X )  +  D ) )  =  0  <->  ( ( X  =  ( -u S  +  I )  \/  X  =  ( -u S  -  I
 ) )  \/  ( X  =  ( S  +  J )  \/  X  =  ( S  -  J ) ) ) ) )
 
Theoremquart1cl 20372 Closure lemmas for quart 20379. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   =>    |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC ) )
 
Theoremquart1lem 20373 Lemma for quart1 20374. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  =  ( X  +  ( A 
 /  4 ) ) )   =>    |-  ( ph  ->  D  =  ( ( ( ( A ^ 4 ) 
 / ;; 2 5 6 )  +  ( P  x.  ( ( A 
 /  4 ) ^
 2 ) ) )  +  ( ( Q  x.  ( A  / 
 4 ) )  +  R ) ) )
 
Theoremquart1 20374 Depress a quartic equation. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  =  ( X  +  ( A 
 /  4 ) ) )   =>    |-  ( ph  ->  (
 ( ( X ^
 4 )  +  ( A  x.  ( X ^
 3 ) ) )  +  ( ( B  x.  ( X ^
 2 ) )  +  ( ( C  x.  X )  +  D ) ) )  =  ( ( ( Y ^ 4 )  +  ( P  x.  ( Y ^ 2 ) ) )  +  ( ( Q  x.  Y )  +  R ) ) )
 
Theoremquartlem1 20375 Lemma for quart 20379. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  P  e.  CC )   &    |-  ( ph  ->  Q  e.  CC )   &    |-  ( ph  ->  R  e.  CC )   &    |-  ( ph  ->  U  =  ( ( P ^
 2 )  +  (; 1 2  x.  R ) ) )   &    |-  ( ph  ->  V  =  ( ( -u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R ) ) ) )   =>    |-  ( ph  ->  ( U  =  ( (
 ( 2  x.  P ) ^ 2 )  -  ( 3  x.  (
 ( P ^ 2
 )  -  ( 4  x.  R ) ) ) )  /\  V  =  ( ( ( 2  x.  ( ( 2  x.  P ) ^
 3 ) )  -  ( 9  x.  (
 ( 2  x.  P )  x.  ( ( P ^ 2 )  -  ( 4  x.  R ) ) ) ) )  +  (; 2 7  x.  -u ( Q ^ 2 ) ) ) ) )
 
Theoremquartlem2 20376 Closure lemmas for quart 20379. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  E  =  -u ( A  /  4
 ) )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   &    |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R ) ) )   &    |-  ( ph  ->  V  =  ( ( -u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R ) ) ) )   &    |-  ( ph  ->  W  =  ( sqr `  (
 ( V ^ 2
 )  -  ( 4  x.  ( U ^
 3 ) ) ) ) )   =>    |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC ) )
 
Theoremquartlem3 20377 Closure lemmas for quart 20379. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  E  =  -u ( A  /  4
 ) )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   &    |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R ) ) )   &    |-  ( ph  ->  V  =  ( ( -u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R ) ) ) )   &    |-  ( ph  ->  W  =  ( sqr `  (
 ( V ^ 2
 )  -  ( 4  x.  ( U ^
 3 ) ) ) ) )   &    |-  ( ph  ->  S  =  ( ( sqr `  M )  /  2
 ) )   &    |-  ( ph  ->  M  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T ) )  /  3
 ) )   &    |-  ( ph  ->  T  =  ( ( ( V  +  W ) 
 /  2 )  ^ c  ( 1  /  3
 ) ) )   &    |-  ( ph  ->  T  =/=  0
 )   =>    |-  ( ph  ->  ( S  e.  CC  /\  M  e.  CC  /\  T  e.  CC ) )
 
Theoremquartlem4 20378 Closure lemmas for quart 20379. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  E  =  -u ( A  /  4
 ) )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   &    |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R ) ) )   &    |-  ( ph  ->  V  =  ( ( -u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R ) ) ) )   &    |-  ( ph  ->  W  =  ( sqr `  (
 ( V ^ 2
 )  -  ( 4  x.  ( U ^
 3 ) ) ) ) )   &    |-  ( ph  ->  S  =  ( ( sqr `  M )  /  2
 ) )   &    |-  ( ph  ->  M  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T ) )  /  3
 ) )   &    |-  ( ph  ->  T  =  ( ( ( V  +  W ) 
 /  2 )  ^ c  ( 1  /  3
 ) ) )   &    |-  ( ph  ->  T  =/=  0
 )   &    |-  ( ph  ->  M  =/=  0 )   &    |-  ( ph  ->  I  =  ( sqr `  (
 ( -u ( S ^
 2 )  -  ( P  /  2 ) )  +  ( ( Q 
 /  4 )  /  S ) ) ) )   &    |-  ( ph  ->  J  =  ( sqr `  (
 ( -u ( S ^
 2 )  -  ( P  /  2 ) )  -  ( ( Q 
 /  4 )  /  S ) ) ) )   =>    |-  ( ph  ->  ( S  =/=  0  /\  I  e.  CC  /\  J  e.  CC ) )
 
Theoremquart 20379 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 24244) if all the substitutions are performed. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  E  =  -u ( A  /  4
 ) )   &    |-  ( ph  ->  P  =  ( B  -  ( ( 3  / 
 8 )  x.  ( A ^ 2 ) ) ) )   &    |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  / 
 2 ) )  +  ( ( A ^
 3 )  /  8
 ) ) )   &    |-  ( ph  ->  R  =  ( ( D  -  (
 ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2
 )  x.  B ) 
 / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
 4 ) ) ) ) )   &    |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R ) ) )   &    |-  ( ph  ->  V  =  ( ( -u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R ) ) ) )   &    |-  ( ph  ->  W  =  ( sqr `  (
 ( V ^ 2
 )  -  ( 4  x.  ( U ^
 3 ) ) ) ) )   &    |-  ( ph  ->  S  =  ( ( sqr `  M )  /  2
 ) )   &    |-  ( ph  ->  M  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T ) )  /  3
 ) )   &    |-  ( ph  ->  T  =  ( ( ( V  +  W ) 
 /  2 )  ^ c  ( 1  /  3
 ) ) )   &    |-  ( ph  ->  T  =/=  0
 )   &    |-  ( ph  ->  M  =/=  0 )   &    |-  ( ph  ->  I  =  ( sqr `  (
 ( -u ( S ^
 2 )  -  ( P  /  2 ) )  +  ( ( Q 
 /  4 )  /  S ) ) ) )   &    |-  ( ph  ->  J  =  ( sqr `  (
 ( -u ( S ^
 2 )  -  ( P  /  2 ) )  -  ( ( Q 
 /  4 )  /  S ) ) ) )   =>    |-  ( ph  ->  (
 ( ( ( X ^ 4 )  +  ( A  x.  ( X ^ 3 ) ) )  +  ( ( B  x.  ( X ^ 2 ) )  +  ( ( C  x.  X )  +  D ) ) )  =  0  <->  ( ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I
 ) )  \/  ( X  =  ( ( E  +  S )  +  J )  \/  X  =  ( ( E  +  S )  -  J ) ) ) ) )
 
13.3.7  Inverse trigonometric functions
 
Syntaxcasin 20380 The arcsine function.
 class arcsin
 
Syntaxcacos 20381 The arccosine function.
 class arccos
 
Syntaxcatan 20382 The arctangent function.
 class arctan
 
Definitiondf-asin 20383 Define the arcsine function. Because  sin is not a one-to-one function, the literal inverse  `' sin is not a function. Rather than attempt to find the right domain on which to restrict  sin in order to get a total function, we just define it in terms of  log, which we already know is total (except at  0). There are branch points at  -u 1 and  1 (at which the function is defined), and branch cuts along the real line not between  -u
1 and  1, which is to say  (  -oo ,  -u 1 )  u.  (
1 ,  +oo ). (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arcsin  =  ( x  e.  CC  |->  ( -u _i  x.  ( log `  ( ( _i 
 x.  x )  +  ( sqr `  ( 1  -  ( x ^ 2
 ) ) ) ) ) ) )
 
Definitiondf-acos 20384 Define the arccosine function. See also remarks for df-asin 20383. Since we define arccos in terms of arcsin, it shares the same branch points and cuts, namely  (  -oo ,  -u
1 )  u.  (
1 ,  +oo ). (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arccos  =  ( x  e.  CC  |->  ( ( pi  / 
 2 )  -  (arcsin `  x ) ) )
 
Definitiondf-atan 20385 Define the arctangent function. See also remarks for df-asin 20383. Unlike arcsin and arccos, this function is not defined everywhere, because  tan ( z )  =/=  pm _i for all  z  e.  CC. For all other  z, there is a formula for arctan ( z ) in terms of  log, and we take that as the definition. Branch points are at  pm _i; branch cuts are on the pure imaginary axis not between  -u _i and  _i, which is to say  { z  e.  CC  |  ( _i  x.  z )  e.  (  -oo ,  -u
1 )  u.  (
1 ,  +oo ) }. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arctan  =  ( x  e.  ( CC  \  { -u _i ,  _i } )  |->  ( ( _i  /  2
 )  x.  ( ( log `  ( 1  -  ( _i  x.  x ) ) )  -  ( log `  ( 1  +  ( _i  x.  x ) ) ) ) ) )
 
Theoremasinlem 20386 The argument to the logarithm in df-asin 20383 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0 )
 
Theoremasinlem2 20387 The argument to the logarithm in df-asin 20383 has the property that replacing  A with  -u A in the expression gives the reciprocal. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  ( ( ( _i 
 x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2
 ) ) ) )  x.  ( ( _i 
 x.  -u A )  +  ( sqr `  ( 1  -  ( -u A ^ 2
 ) ) ) ) )  =  1 )
 
Theoremasinlem3a 20388 Lemma for asinlem3 20389. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Im `  A )  <_  0 ) 
 ->  0  <_  ( Re
 `  ( ( _i 
 x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2
 ) ) ) ) ) )
 
Theoremasinlem3 20389 The argument to the logarithm in df-asin 20383 has nonnegative real part. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  0  <_  ( Re `  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
 
Theoremasinf 20390 Domain and range of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arcsin : CC --> CC
 
Theoremasincl 20391 Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  A )  e.  CC )
 
Theoremacosf 20392 Domain and range of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arccos : CC --> CC
 
Theoremacoscl 20393 Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arccos `  A )  e.  CC )
 
Theorematandm 20394 Since the property is a little lengthy, we abbreviate  A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i as  A  e.  dom arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i ) )
 
Theorematandm2 20395 This form of atandm 20394 is a bit more useful for showing that the logarithms in df-atan 20385 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
 
Theorematandm3 20396 A compact form of atandm 20394. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( A ^
 2 )  =/=  -u 1
 ) )
 
Theorematandm4 20397 A compact form of atandm 20394. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^ 2
 ) )  =/=  0
 ) )
 
Theorematanf 20398 Domain and range of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arctan : ( CC  \  { -u _i ,  _i } ) --> CC
 
Theorematancl 20399 Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  A )  e. 
 CC )
 
Theoremasinval 20400 Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  A )  =  ( -u _i  x.  ( log `  ( ( _i 
 x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2
 ) ) ) ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32225
  Copyright terms: Public domain < Previous  Next >