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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcxprec 19901 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC )  ->  ( ( 1  /  A )  ^ c  B )  =  ( 1  /  ( A  ^ c  B ) ) )
 
Theoremdivcxp 19902 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR+  /\  C  e.  CC )  ->  ( ( A  /  B ) 
 ^ c  C )  =  ( ( A 
 ^ c  C ) 
 /  ( B  ^ c  C ) ) )
 
Theoremcxpmul 19903 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  C  e.  CC )  ->  ( A  ^ c  ( B  x.  C ) )  =  (
 ( A  ^ c  B )  ^ c  C ) )
 
Theoremcxpmul2 19904 Product of exponents law for complex exponentiation. Variation on cxpmul 19903 with more general conditions on  A and  B when  C is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  NN0 )  ->  ( A  ^ c  ( B  x.  C ) )  =  (
 ( A  ^ c  B ) ^ C ) )
 
Theoremcxproot 19905 The complex power function allows us to write n-th roots via the idiom  A  ^ c 
( 1  /  N
). (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A 
 ^ c  ( 1 
 /  N ) ) ^ N )  =  A )
 
Theoremcxpmul2z 19906 Generalize cxpmul2 19904 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  C  e.  ZZ ) )  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A  ^ c  B ) ^ C ) )
 
Theoremabscxp 19907 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC )  ->  ( abs `  ( A  ^ c  B ) )  =  ( A 
 ^ c  ( Re
 `  B ) ) )
 
Theoremabscxp2 19908 Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A )  ^ c  B )
 )
 
Theoremcxplt 19909 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <  C  <->  ( A  ^ c  B )  <  ( A  ^ c  C ) ) )
 
Theoremcxple 19910 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <_  C  <->  ( A  ^ c  B )  <_  ( A  ^ c  C ) ) )
 
Theoremcxplea 19911 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  ( B  e.  RR  /\  C  e.  RR )  /\  B  <_  C )  ->  ( A  ^ c  B ) 
 <_  ( A  ^ c  C ) )
 
Theoremcxple2 19912 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( A  <_  B  <->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) )
 
Theoremcxplt2 19913 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  ( A  <  B  <->  ( A  ^ c  C )  <  ( B  ^ c  C ) ) )
 
Theoremcxple2a 19914 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( 0 
 <_  A  /\  0  <_  C )  /\  A  <_  B )  ->  ( A  ^ c  C )  <_  ( B  ^ c  C ) )
 
Theoremcxplt3 19915 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( ( A  e.  RR+  /\  A  <  1 )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <  C  <->  ( A  ^ c  C )  <  ( A  ^ c  B ) ) )
 
Theoremcxple3 19916 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( ( A  e.  RR+  /\  A  <  1 )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( B  <_  C  <->  ( A  ^ c  C )  <_  ( A  ^ c  B ) ) )
 
Theoremcxpsqrlem 19917 Lemma for cxpsqr 19918. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( A  ^ c  ( 1 
 /  2 ) )  =  -u ( sqr `  A ) )  ->  ( _i 
 x.  ( sqr `  A ) )  e.  RR )
 
Theoremcxpsqr 19918 The complex exponential function with exponent  1  /  2 exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other  n-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( A  e.  CC  ->  ( A  ^ c  ( 1  /  2
 ) )  =  ( sqr `  A )
 )
 
Theoremlogsqr 19919 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  ( A  e.  RR+  ->  ( log `  ( sqr `  A ) )  =  ( ( log `  A )  /  2 ) )
 
Theoremcxp0d 19920 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  0 )  =  1 )
 
Theoremcxp1d 19921 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  1 )  =  A )
 
Theorem1cxpd 19922 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 1  ^ c  A )  =  1 )
 
Theoremcxpcld 19923 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  B )  e.  CC )
 
Theoremcxpmul2d 19924 Product of exponents law for complex exponentiation. Variation on cxpmul 19903 with more general conditions on  A and  B when  C is an integer. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  NN0 )   =>    |-  ( ph  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A 
 ^ c  B ) ^ C ) )
 
Theorem0cxpd 19925 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 0  ^ c  A )  =  0 )
 
Theoremcxpexpzd 19926 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  ^ c  B )  =  ( A ^ B ) )
 
Theoremcxpefd 19927 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
 
Theoremcxpne0d 19928 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  B )  =/=  0 )
 
Theoremcxpp1d 19929 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  +  1 ) )  =  ( ( A 
 ^ c  B )  x.  A ) )
 
Theoremcxpnegd 19930 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  -u B )  =  ( 1  /  ( A  ^ c  B ) ) )
 
Theoremcxpmul2zd 19931 Generalize cxpmul2 19904 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  ZZ )   =>    |-  ( ph  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A 
 ^ c  B ) ^ C ) )
 
Theoremcxpaddd 19932 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  +  C ) )  =  ( ( A 
 ^ c  B )  x.  ( A  ^ c  C ) ) )
 
Theoremcxpsubd 19933 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  ^ c  ( B  -  C ) )  =  ( ( A 
 ^ c  B ) 
 /  ( A  ^ c  C ) ) )
 
Theoremcxpltd 19934 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  <  C  <->  ( A  ^ c  B )  <  ( A  ^ c  C ) ) )
 
Theoremcxpled 19935 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  <_  C  <->  ( A  ^ c  B )  <_  ( A  ^ c  C ) ) )
 
Theoremcxplead 19936 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  ( A  ^ c  B ) 
 <_  ( A  ^ c  C ) )
 
Theoremdivcxpd 19937 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  /  B )  ^ c  C )  =  ( ( A 
 ^ c  C ) 
 /  ( B  ^ c  C ) ) )
 
Theoremrecxpcld 19938 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  ^ c  B )  e.  RR )
 
Theoremcxpge0d 19939 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  0 
 <_  ( A  ^ c  B ) )
 
Theoremcxple2ad 19940 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( A  ^ c  C ) 
 <_  ( B  ^ c  C ) )
 
Theoremcxplt2d 19941 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  ^ c  C )  <  ( B  ^ c  C ) ) )
 
Theoremcxple2d 19942 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  ^ c  C )  <_  ( B  ^ c  C ) ) )
 
Theoremmulcxpd 19943 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  ^ c  C )  =  ( ( A 
 ^ c  C )  x.  ( B  ^ c  C ) ) )
 
Theoremcxprecd 19944 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( 1  /  A )  ^ c  B )  =  ( 1  /  ( A  ^ c  B ) ) )
 
Theoremrpcxpcld 19945 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  ^ c  B )  e.  RR+ )
 
Theoremlogcxpd 19946 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( log `  ( A  ^ c  B ) )  =  ( B  x.  ( log `  A )
 ) )
 
Theoremcxplt3d 19947 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 19948 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 19949 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 19950* 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 19951* 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 19952 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 19953* 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 19954* 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 19955* Lemma for cxpcn3 19956. (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 19956* 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 19957 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
 
Theoremsqrcn 19958 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( sqr  |`  D )  e.  ( D -cn-> CC )
 
Theoremcxpaddlelem 19959 Lemma for cxpaddle 19960. (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 19960 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 19961 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 19962 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 19963 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 19964 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 19965* 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 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 19966 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  Solutions of quardatic, cubic, and quartic equations
 
Theoremquad2 19967 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 19968 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 19969 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 19970 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 19971 Lemma for dcubic1 19973 and dcubic2 19972: 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 19972* Reverse direction of dcubic 19974. 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 19973 Forward direction of dcubic 19974: 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 19974* Solutions to the depressed cubic, a special case of cubic 19977. (The definitions of  M ,  N ,  G ,  T here differ from mcubic 19975 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 19975* Solutions to a monic cubic equation, a special case of cubic 19977. (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 19976* 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 19977* The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 3593 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 19978 Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12165, 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 19979 Lemma for dquart 19981. (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 19980 Lemma for dquart 19981. (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 19981 Solve a depressed quartic equation. To eliminate  S, which is the square root of a solution  M to the resolvent cubic equation, apply cubic 19977 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 19982 Closure lemmas for quart 19989. (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 19983 Lemma for quart1 19984. (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 19984 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 19985 Lemma for quart 19989. (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 19986 Closure lemmas for quart 19989. (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 19987 Closure lemmas for quart 19989. (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 19988 Closure lemmas for quart 19989. (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 19989 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 22857) 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 ) ) ) ) )
 
13.3.6  Inverse trigonometric functions
 
Syntaxcasin 19990 The arcsine function.
 class arcsin
 
Syntaxcacos 19991 The arccosine function.
 class arccos
 
Syntaxcatan 19992 The arctangent function.
 class arctan
 
Definitiondf-asin 19993 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 19994 Define the arccosine function. See also remarks for df-asin 19993. 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 19995 Define the arctangent function. See also remarks for df-asin 19993. 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 19996 The argument to the logarithm in df-asin 19993 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) )  =/=  0 )
 
Theoremasinlem2 19997 The argument to the logarithm in df-asin 19993 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 19998 Lemma for asinlem3 19999. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Im `  A )  <_  0 ) 
 ->  0  <_  ( Re
 `  ( ( _i 
 x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2
 ) ) ) ) ) )
 
Theoremasinlem3 19999 The argument to the logarithm in df-asin 19993 has nonnegative real part. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  0  <_  ( Re `  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
 
Theoremasinf 20000 Domain and range of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arcsin : CC --> CC
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