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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1cubr 20101 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 20102 Lemma for dcubic1 20104 and dcubic2 20103: 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 20103* Reverse direction of dcubic 20105. 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 20104 Forward direction of dcubic 20105: 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 20105* Solutions to the depressed cubic, a special case of cubic 20108. (The definitions of  M ,  N ,  G ,  T here differ from mcubic 20106 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 20106* Solutions to a monic cubic equation, a special case of cubic 20108. (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 20107* 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 20108* The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3663 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 20109 Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12254, 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 20110 Lemma for dquart 20112. (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 20111 Lemma for dquart 20112. (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 20112 Solve a depressed quartic equation. To eliminate  S, which is the square root of a solution  M to the resolvent cubic equation, apply cubic 20108 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 20113 Closure lemmas for quart 20120. (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 20114 Lemma for quart1 20115. (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 20115 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 20116 Lemma for quart 20120. (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 20117 Closure lemmas for quart 20120. (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 20118 Closure lemmas for quart 20120. (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 20119 Closure lemmas for quart 20120. (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 20120 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 23059) 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 20121 The arcsine function.
 class arcsin
 
Syntaxcacos 20122 The arccosine function.
 class arccos
 
Syntaxcatan 20123 The arctangent function.
 class arctan
 
Definitiondf-asin 20124 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 20125 Define the arccosine function. See also remarks for df-asin 20124. 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 20126 Define the arctangent function. See also remarks for df-asin 20124. 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 20127 The argument to the logarithm in df-asin 20124 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0 )
 
Theoremasinlem2 20128 The argument to the logarithm in df-asin 20124 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 20129 Lemma for asinlem3 20130. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Im `  A )  <_  0 ) 
 ->  0  <_  ( Re
 `  ( ( _i 
 x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2
 ) ) ) ) ) )
 
Theoremasinlem3 20130 The argument to the logarithm in df-asin 20124 has nonnegative real part. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  0  <_  ( Re `  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
 
Theoremasinf 20131 Domain and range of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arcsin : CC --> CC
 
Theoremasincl 20132 Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  A )  e.  CC )
 
Theoremacosf 20133 Domain and range of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arccos : CC --> CC
 
Theoremacoscl 20134 Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arccos `  A )  e.  CC )
 
Theorematandm 20135 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 20136 This form of atandm 20135 is a bit more useful for showing that the logarithms in df-atan 20126 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 20137 A compact form of atandm 20135. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( A ^
 2 )  =/=  -u 1
 ) )
 
Theorematandm4 20138 A compact form of atandm 20135. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^ 2
 ) )  =/=  0
 ) )
 
Theorematanf 20139 Domain and range of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arctan : ( CC  \  { -u _i ,  _i } ) --> CC
 
Theorematancl 20140 Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  A )  e. 
 CC )
 
Theoremasinval 20141 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
 ) ) ) ) ) ) )
 
Theoremacosval 20142 Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arccos `  A )  =  ( ( pi  / 
 2 )  -  (arcsin `  A ) ) )
 
Theorematanval 20143 Value of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  / 
 2 )  x.  (
 ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
 
Theorematanre 20144 A real number is in the domain of the arctangent function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  RR  ->  A  e.  dom arctan )
 
Theoremasinneg 20145 The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  -u A )  =  -u (arcsin `  A ) )
 
Theoremacosneg 20146 The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  (arccos `  -u A )  =  ( pi  -  (arccos `  A ) ) )
 
Theoremefiasin 20147 The exponential of the arcsine function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  (arcsin `  A ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) )
 
Theoremsinasin 20148 The arcsine function is an inverse to  sin. This is the main property that justifies the notation arcsin or  sin
^ -u 1. Because  sin is not an injection, the other converse identity asinsin 20151 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  ( sin `  (arcsin `  A ) )  =  A )
 
Theoremcosacos 20149 The arccosine function is an inverse to  cos. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  ( cos `  (arccos `  A ) )  =  A )
 
Theoremasinsinlem 20150 Lemma for asinsin 20151. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  0  <  ( Re
 `  ( exp `  ( _i  x.  A ) ) ) )
 
Theoremasinsin 20151 The arcsine function composed with 
sin is equal to the identity. This plus sinasin 20148 allow us to view  sin and arcsin as inverse operations to each other. For ease of use, we have not defined precisely the correct domain of correctness of this identity; in addition to the main region described here it is also true for some points on the branch cuts, namely when  A  =  ( pi 
/  2 )  -  _i y for non-negative real  y and also symmetrically at  A  =  _i y  -  ( pi  / 
2 ). In particular, when restricted to reals this identity extends to the closed interval  [ -u (
pi  /  2 ) ,  ( pi  / 
2 ) ], not just the open interval (see reasinsin 20155). (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  (arcsin `  ( sin `  A ) )  =  A )
 
Theoremacoscos 20152 The arccosine function is an inverse to  cos. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  (
 0 (,) pi ) ) 
 ->  (arccos `  ( cos `  A ) )  =  A )
 
Theoremasin1 20153 The arcsine of  1 is  pi  / 
2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arcsin `  1 )  =  ( pi  /  2
 )
 
Theoremacos1 20154 The arcsine of  1 is  pi  / 
2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arccos `  1 )  =  0
 
Theoremreasinsin 20155 The arcsine function composed with 
sin is equal to the identity. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) )  ->  (arcsin `  ( sin `  A ) )  =  A )
 
Theoremasinsinb 20156 Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( Re `  B )  e.  ( -u ( pi  /  2 ) (,) ( pi  /  2
 ) ) )  ->  ( (arcsin `  A )  =  B  <->  ( sin `  B )  =  A )
 )
 
Theoremacoscosb 20157 Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( Re `  B )  e.  ( 0 (,) pi ) )  ->  ( (arccos `  A )  =  B  <->  ( cos `  B )  =  A )
 )
 
Theoremasinbnd 20158 The arcsine function has range within a vertical strip of the complex plane with real part between  -u pi  /  2 and  pi  /  2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( Re `  (arcsin `  A ) )  e.  ( -u ( pi  / 
 2 ) [,] ( pi  /  2 ) ) )
 
Theoremacosbnd 20159 The arccosine function has range within a vertical strip of the complex plane with real part between  0 and  pi. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( Re `  (arccos `  A ) )  e.  ( 0 [,] pi ) )
 
Theoremasinrebnd 20160 Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u 1 [,] 1 ) 
 ->  (arcsin `  A )  e.  ( -u ( pi  / 
 2 ) [,] ( pi  /  2 ) ) )
 
Theoremasinrecl 20161 The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u 1 [,] 1 ) 
 ->  (arcsin `  A )  e.  RR )
 
Theoremacosrecl 20162 The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u 1 [,] 1 ) 
 ->  (arccos `  A )  e.  RR )
 
Theoremcosasin 20163 The cosine of the arcsine of  A is  sqr ( 1  -  A ^ 2 ). (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( cos `  (arcsin `  A ) )  =  ( sqr `  (
 1  -  ( A ^ 2 ) ) ) )
 
Theoremsinacos 20164 The sine of the arccosine of  A is  sqr ( 1  -  A ^ 2 ). (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( sin `  (arccos `  A ) )  =  ( sqr `  (
 1  -  ( A ^ 2 ) ) ) )
 
Theorematandmneg 20165 The domain of the arctangent function is closed under negatives. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  -u A  e.  dom arctan )
 
Theorematanneg 20166 The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  -u A )  =  -u (arctan `  A )
 )
 
Theorematan0 20167 The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  (arctan `  0 )  =  0
 
Theorematandmcj 20168 The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  ( * `  A )  e.  dom arctan )
 
Theorematancj 20169 The arctangent function distributes under conjugation. (The condition that  Re ( A )  =/=  0 is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 20166 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between  -u 1 and  1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0
 )  ->  ( A  e.  dom arctan  /\  ( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) ) )
 
Theorematanrecl 20170 The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
 
Theoremefiatan 20171 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A )
 ) )  =  ( ( sqr `  (
 1  +  ( _i 
 x.  A ) ) )  /  ( sqr `  ( 1  -  ( _i  x.  A ) ) ) ) )
 
Theorematanlogaddlem 20172 Lemma for atanlogadd 20173. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  0  <_  ( Re `  A ) )  ->  ( ( log `  (
 1  +  ( _i 
 x.  A ) ) )  +  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
 
Theorematanlogadd 20173 The rule  sqr ( z w )  =  ( sqr z ) ( sqr w ) is not always true on the complexes, but it is true when the arguments of  z and  w sum to within the interval  ( -u pi ,  pi ], so there are some cases such as this one with  z  =  1  +  _i A and  w  =  1  -  _i A which are true unconditionally. This result can also be stated as " sqr ( 1  +  z )  +  sqr ( 1  -  z
) is analytic". (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( ( log `  (
 1  +  ( _i 
 x.  A ) ) )  +  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
 
Theorematanlogsublem 20174 Lemma for atanlogsub 20175. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  0  <  ( Re
 `  A ) ) 
 ->  ( Im `  (
 ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A ) ) ) ) )  e.  ( -u pi (,) pi ) )
 
Theorematanlogsub 20175 A variation on atanlogadd 20173, to show that  sqr ( 1  +  _i z )  /  sqr ( 1  -  _i z )  =  sqr ( ( 1  +  _i z )  /  ( 1  -  _i z ) ) under more limited conditions. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  ( Re `  A )  =/=  0 )  ->  ( ( log `  (
 1  +  ( _i 
 x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
 
Theoremefiatan2 20176 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A )
 ) )  =  ( ( 1  +  ( _i  x.  A ) ) 
 /  ( sqr `  (
 1  +  ( A ^ 2 ) ) ) ) )
 
Theorem2efiatan 20177 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1
 ) )
 
Theoremtanatan 20178 The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A ) )  =  A )
 
Theorematandmtan 20179 The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  e.  dom arctan )
 
Theoremcosatan 20180 The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( cos `  (arctan `  A ) )  =  (
 1  /  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) ) )
 
Theoremcosatanne0 20181 The arctangent function has range contained in the domain of the tangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( cos `  (arctan `  A ) )  =/=  0
 )
 
Theorematantan 20182 The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  (arctan `  ( tan `  A ) )  =  A )
 
Theorematantanb 20183 Relationship between tangent and arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  B  e.  CC  /\  ( Re `  B )  e.  ( -u ( pi  /  2 ) (,) ( pi  /  2
 ) ) )  ->  ( (arctan `  A )  =  B  <->  ( tan `  B )  =  A )
 )
 
Theorematanbndlem 20184 Lemma for atanbnd 20185. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( A  e.  RR+  ->  (arctan `  A )  e.  ( -u ( pi  / 
 2 ) (,) ( pi  /  2 ) ) )
 
Theorematanbnd 20185 The arctangent function is bounded by  pi  /  2 on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( A  e.  RR  ->  (arctan `  A )  e.  ( -u ( pi  / 
 2 ) (,) ( pi  /  2 ) ) )
 
Theorematanord 20186 The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 (arctan `  A )  < 
 (arctan `  B ) ) )
 
Theorematan1 20187 The arctangent of  1 is  pi  /  4. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arctan `  1 )  =  ( pi  /  4
 )
 
Theorembndatandm 20188 A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  A  e.  dom arctan )
 
Theorematans 20189* The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  +  ( A ^ 2
 ) )  e.  D ) )
 
Theorematans2 20190* It suffices to show that  1  -  _i A and  1  +  _i A are in the continuity domain of  log to show that  A is in the continuity domain of arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  e.  D  /\  ( 1  +  ( _i  x.  A ) )  e.  D ) )
 
Theorematansopn 20191* The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  S  e.  ( TopOpen ` fld )
 
Theorematansssdm 20192* The domain of continuity of the arctangent is a subset of the actual domain of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  S  C_ 
 dom arctan
 
Theoremressatans 20193* The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  RR  C_  S
 
Theoremdvatan 20194* The derivative of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  ( CC  _D  (arctan  |`  S ) )  =  ( x  e.  S  |->  ( 1 
 /  ( 1  +  ( x ^ 2
 ) ) ) )
 
Theorematancn 20195* The arctangent is a continuous function. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  (arctan  |`  S )  e.  ( S -cn-> CC )
 
Theorematantayl 20196* The Taylor series for arctan ( A ). (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( _i  x.  (
 ( -u _i ^ n )  -  ( _i ^ n ) ) ) 
 /  2 )  x.  ( ( A ^ n )  /  n ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  < 
 1 )  ->  seq  1
 (  +  ,  F ) 
 ~~>  (arctan `  A )
 )
 
Theorematantayl2 20197* The Taylor series for arctan ( A ). (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  if (
 2  ||  n , 
 0 ,  ( (
 -u 1 ^ (
 ( n  -  1
 )  /  2 )
 )  x.  ( ( A ^ n ) 
 /  n ) ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  1 (  +  ,  F )  ~~>  (arctan `  A )
 )
 
Theorematantayl3 20198* The Taylor series for arctan ( A ). (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( (
 -u 1 ^ n )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  ~~>  (arctan `  A )
 )
 
Theoremleibpilem1 20199 Lemma for leibpi 20201. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( N  e.  NN0  /\  ( -.  N  =  0  /\  -.  2  ||  N ) )  ->  ( N  e.  NN  /\  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 )
 
Theoremleibpilem2 20200* The Leibniz formula for  pi. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( (
 -u 1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  G  =  ( k  e.  NN0  |->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u 1 ^ ( ( k  -  1 )  / 
 2 ) )  /  k ) ) )   &    |-  A  e.  _V   =>    |-  (  seq  0 (  +  ,  F )  ~~>  A 
 <-> 
 seq  0 (  +  ,  G )  ~~>  A )
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