HomeHome Metamath Proof Explorer
Theorem List (p. 201 of 315)
< 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
(1-21459)
  Hilbert Space Explorer  Hilbert Space Explorer
(21460-22982)
  Users' Mathboxes  Users' Mathboxes
(22983-31404)
 

Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcxpmul2z 20001 Generalize cxpmul2 19999 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 20002 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 20003 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 20004 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 20005 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 20006 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 20007 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 20008 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 20009 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 20010 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 20011 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 20012 Lemma for cxpsqr 20013. (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 20013 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 20014 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  ( A  e.  RR+  ->  ( log `  ( sqr `  A ) )  =  ( ( log `  A )  /  2 ) )
 
Theoremcxp0d 20015 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 20016 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 20017 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 20018 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 20019 Product of exponents law for complex exponentiation. Variation on cxpmul 19998 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 20020 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 20021 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 20022 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 20023 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 20024 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 20025 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 20026 Generalize cxpmul2 19999 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 20027 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 20028 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 20029 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 20030 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 20031 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 20032 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 20033 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 20034 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 20035 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 20036 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 20037 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 20038 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 20039 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 20040 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 20041 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 20042 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 20043 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 20044 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 20045* 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 20046* 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 20047 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 20048* 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 20049* 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 20050* Lemma for cxpcn3 20051. (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 20051* 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 20052 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
 
Theoremsqrcn 20053 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   =>    |-  ( sqr  |`  D )  e.  ( D -cn-> CC )
 
Theoremcxpaddlelem 20054 Lemma for cxpaddle 20055. (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 20055 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 20056 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 20057 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 20058 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 20059 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 20060* 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 20061 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 20062* 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 20063* 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 20064* 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 20065* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20062. (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 20066* 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 20067* 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 20068* 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 20069* 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 20070* Lemma for ang180 20075. Show that the "revolution number"  N is an integer, using efeq1 19854 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 20071* Lemma for ang180 20075. 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 19890, 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 20072* Lemma for ang180 20075. Since ang180lem1 20070 shows that  N is an integer and ang180lem2 20071 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 20073* Lemma for ang180 20075. 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 20074* Lemma for ang180 20075: 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 20075* 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 20076 Lemma for Law of Cosines lawcos 20077. 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 20077* 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 20075),  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 20076 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 12390). The Pythagorean Theorem pythag 20078 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 20078* 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 20075),  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 20077 to prove this, along with simple trig facts like coshalfpi 19800 and cosneg 12390. (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 20079 Symmetry of the natural logarithm range by negation. Lemma for logrec 20080. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
 |-  ( ( A  e.  ran 
 log  /\  -.  ( Im
 `  A )  =  pi )  ->  -u A  e.  ran  log )
 
Theoremlogrec 20080 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 20081 Lemma for isosctr 20084. (Contributed by Saveliy Skresanov, 30-Dec-2016.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A ) 
 ->  ( Im `  ( log `  ( 1  -  A ) ) )  =/=  pi )
 
Theoremisosctrlem2 20082 Lemma for isosctr 20084. 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 20083* Lemma for isosctr 20084. 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 20084* 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 20085* 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 20077 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 20086 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 20087 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 20088 Equivalence used in the proof of angpieqvd 20091. (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 20089* Equivalence used in angpieqvd 20091. (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 20090* 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 20091* 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 20092* If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20085 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 20093* 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 20092, where P = B, and using angrtmuld 20069 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 20094 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 20093 and the Pythagorean theorem (pythag 20078) 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 20095 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 20096 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 20094 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 20095. (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 20097* 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 20096 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 20098 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 20099 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 20100 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 ) )
    < 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 31401-31404
  Copyright terms: Public domain < Previous  Next >