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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcxpp1d 23201 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 23202 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 23203 Generalize cxpmul2 23176 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 23204 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 23205 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 23206 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 23207 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 23208 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 23209 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 23210 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 23211 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 23212 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 23213 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 23214 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 23215 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 23216 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 23217 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 23218 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 23219 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 23220 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 23221 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 23222* 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 23223* 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 ) ) ) )
 
Theoremdvsqrt 23224 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 23225* 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 23226* 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 23227* Lemma for cxpcn3 23228. (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 23228* 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 )
 
Theoremresqrtcn 23229 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )
 
Theoremsqrtcn 23230 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  D  =  ( CC  \  ( -oo (,] 0
 ) )   =>    |-  ( sqr  |`  D )  e.  ( D -cn-> CC )
 
Theoremcxpaddlelem 23231 Lemma for cxpaddle 23232. (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 23232 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 23233 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 23234 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 23235 The only powers of an  N-th root of unity that equal 
1 are the multiples of  N. In other words,  -u 1  ^c 
( 2  /  N
) has order  N in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complex numbers.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  ( ( N  e.  NN  /\  K  e.  ZZ )  ->  ( ( (
 -u 1  ^c 
 ( 2  /  N ) ) ^ K )  =  1  <->  N  ||  K ) )
 
Theoremroot1cj 23236 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 23237* Solve an equation involving an  N-th power. The expression  -u 1  ^c  ( 2  /  N )  =  exp ( 2 pi _i 
/  N ) is a way to write the primitive  N-th root of unity with the smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( A  e.  CC  /\  N  e.  NN  /\  B  e.  CC )  ->  ( ( A ^ N )  =  B  <->  E. n  e.  ( 0
 ... ( N  -  1 ) ) A  =  ( ( B 
 ^c  ( 1 
 /  N ) )  x.  ( ( -u 1  ^c  ( 2 
 /  N ) ) ^ n ) ) ) )
 
Theoremloglesqrt 23238 An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( log `  ( A  +  1 )
 )  <_  ( sqr `  A ) )
 
Theoremlogreclem 23239 Symmetry of the natural logarithm range by negation. Lemma for logrec 23240. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
 |-  ( ( A  e.  ran 
 log  /\  -.  ( Im
 `  A )  =  pi )  ->  -u A  e.  ran  log )
 
Theoremlogrec 23240 Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  ( Im `  ( log `  A ) )  =/=  pi )  ->  ( log `  A )  =  -u ( log `  (
 1  /  A )
 ) )
 
14.3.5  Logarithms to an arbitrary base

Define "log using an arbitrary base" function and then prove some of its properties. Note that logb is generalized to an arbitrary base and arbitrary parameter in  CC, but it doesn't accept infinities as arguments, unlike  log.

Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations):  ( B logb  X ) where  B is the base and 
X is the argument of the logarithm function. An alternative would be to support the notational form  ( ( logb  `  B
) `  X ); that looks a little more like traditional notation. Such a function  ( logb  `  B ) for a fixed base can be obtained in Metamath (without the need for a new definition) by the curry function:  (curry logb  `  B ), see logbmpt 23265, logbf 23266 and logbfval 23267.

 
Syntaxclogb 23241 Extend class notation to include the logarithm generalized to an arbitrary base.
 class logb
 
Definitiondf-logb 23242* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as  ( B logb  X ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition is according to Wikipedia "Complex logarithm": https://en.wikipedia.org/wiki/Complex_logarithm#Logarithms_to_other_bases (10-Jun-2020). (Contributed by David A. Wheeler, 21-Jan-2017.)
 |- logb  =  ( x  e.  ( CC  \  { 0 ,  1 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( ( log `  y
 )  /  ( log `  x ) ) )
 
Theoremlogbval 23243 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  ( B logb  X )  =  ( ( log `  X )  /  ( log `  B )
 ) )
 
Theoremlogbcl 23244 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  ( B logb  X )  e.  CC )
 
Theoremlogbid1 23245 General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 ) 
 ->  ( A logb  A )  =  1 )
 
Theoremlogb1 23246 The logarithm of  1 to an arbitrary base  B is 0. Property 1(b) of [Cohen4] p. 361. See log1 23077. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
 |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 ) 
 ->  ( B logb  1 )  =  0 )
 
Theoremelogb 23247 The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using  _e as the base in logb is the same as  log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
 |-  ( A  e.  ( CC  \  { 0 } )  ->  ( _e logb  A )  =  ( log `  A ) )
 
Theoremlogbchbase 23248 Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  /\  X  e.  ( CC  \  { 0 } ) )  ->  ( A logb  X )  =  ( ( B logb  X ) 
 /  ( B logb  A ) ) )
 
Theoremrelogbval 23249 Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( ( B  e.  ( ZZ>= `  2 )  /\  X  e.  RR+ )  ->  ( B logb  X )  =  ( ( log `  X )  /  ( log `  B ) ) )
 
Theoremrelogbcl 23250 Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
 |-  ( ( B  e.  RR+  /\  X  e.  RR+  /\  B  =/=  1 )  ->  ( B logb  X )  e.  RR )
 
Theoremrelogbzcl 23251 Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.)
 |-  ( ( B  e.  ( ZZ>= `  2 )  /\  X  e.  RR+ )  ->  ( B logb  X )  e. 
 RR )
 
Theoremrelogbreexp 23252 Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  C  e.  RR+  /\  E  e.  RR )  ->  ( B logb 
 ( C  ^c  E ) )  =  ( E  x.  ( B logb  C ) ) )
 
Theoremrelogbzexp 23253 Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  C  e.  RR+  /\  N  e.  ZZ )  ->  ( B logb 
 ( C ^ N ) )  =  ( N  x.  ( B logb  C ) ) )
 
Theoremrelogbmul 23254 The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  ( A  e.  RR+  /\  C  e.  RR+ )
 )  ->  ( B logb  ( A  x.  C ) )  =  ( ( B logb  A )  +  ( B logb  C ) ) )
 
Theoremrelogbmulexp 23255 The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  ( A  e.  RR+  /\  C  e.  RR+  /\  E  e.  RR ) )  ->  ( B logb  ( A  x.  ( C  ^c  E ) ) )  =  ( ( B logb  A )  +  ( E  x.  ( B logb  C ) ) ) )
 
Theoremrelogbdiv 23256 The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  ( A  e.  RR+  /\  C  e.  RR+ )
 )  ->  ( B logb  ( A  /  C ) )  =  ( ( B logb  A )  -  ( B logb  C ) ) )
 
Theoremrelogbexp 23257 Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
 |-  ( ( B  e.  RR+  /\  B  =/=  1  /\  M  e.  ZZ )  ->  ( B logb  ( B ^ M ) )  =  M )
 
Theoremnnlogbexp 23258 Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
 |-  ( ( B  e.  ( ZZ>= `  2 )  /\  M  e.  ZZ )  ->  ( B logb  ( B ^ M ) )  =  M )
 
Theoremlogbrec 23259 Logarithm of a reciprocal changes sign. See logrec 23240. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( ( B  e.  ( ZZ>= `  2 )  /\  A  e.  RR+ )  ->  ( B logb  ( 1  /  A ) )  =  -u ( B logb  A ) )
 
Theoremlogbleb 23260 The general logarithm function is monotone/increasing. See logleb 23094. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
 |-  ( ( B  e.  ( ZZ>= `  2 )  /\  X  e.  RR+  /\  Y  e.  RR+ )  ->  ( X  <_  Y  <->  ( B logb  X ) 
 <_  ( B logb  Y ) ) )
 
Theoremlogblt 23261 The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 23091. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
 |-  ( ( B  e.  ( ZZ>= `  2 )  /\  X  e.  RR+  /\  Y  e.  RR+ )  ->  ( X  <  Y  <->  ( B logb  X )  <  ( B logb  Y ) ) )
 
Theoremrelogbcxp 23262 Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
 |-  ( ( B  e.  ( RR+  \  { 1 } )  /\  X  e.  RR )  ->  ( B logb  ( B  ^c  X ) )  =  X )
 
Theoremcxplogb 23263 Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
 |-  ( ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  ( B  ^c  ( B logb  X ) )  =  X )
 
Theoremrelogbcxpb 23264 The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.)
 |-  ( ( ( B  e.  RR+  /\  B  =/=  1 )  /\  X  e.  RR+  /\  Y  e.  RR )  ->  ( ( B logb  X )  =  Y  <->  ( B  ^c  Y )  =  X ) )
 
Theoremlogbmpt 23265* The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.)
 |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 ) 
 ->  (curry logb  `  B )  =  ( y  e.  ( CC  \  { 0 } )  |->  ( ( log `  y
 )  /  ( log `  B ) ) ) )
 
Theoremlogbf 23266 The general logarithm to a fixed base regarded as function. (Contributed by AV, 11-Jun-2020.)
 |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 ) 
 ->  (curry logb  `  B ) : ( CC  \  { 0 } ) --> CC )
 
Theoremlogbfval 23267 The general logarithm of a complex number to a fixed base. (Contributed by AV, 11-Jun-2020.)
 |-  ( ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  /\  X  e.  ( CC  \  { 0 } ) )  ->  ( (curry logb  `  B ) `  X )  =  ( B logb  X ) )
 
Theoremrelogbf 23268 The general logarithm to a real base greater than 1 regarded as function restricted to the positive integers. Property in [Cohen4] p. 349. (Contributed by AV, 12-Jun-2020.)
 |-  ( ( B  e.  RR+  /\  1  <  B ) 
 ->  ( (curry logb  `  B )  |`  RR+ ) : RR+ --> RR )
 
Theoremlogblog 23269 The general logarithm to the base being Euler's constant regarded as function is the natural logarithm. (Contributed by AV, 12-Jun-2020.)
 |-  (curry logb  `  _e )  =  log
 
14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords
 
Theoremangval 23270* 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 23271* 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 23272* 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 23273* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 23270. (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 23274* 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 23275* 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 23276* 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 23277* 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 23278* Lemma for ang180 23283. Show that the "revolution number"  N is an integer, using efeq1 23020 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 23279* Lemma for ang180 23283. 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 23061, 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 23280* Lemma for ang180 23283. Since ang180lem1 23278 shows that  N is an integer and ang180lem2 23279 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 23281* Lemma for ang180 23283. 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 23282* Lemma for ang180 23283: 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 23283* The sum of angles  m A B C  +  m B C A  +  m C A B in a triangle adds up to either  pi or  -u pi, i.e. 180 degrees. (The sign is due to the two possible orientations of vertex arrangement and our signed notion of angle). This is Metamath 100 proof #27. (Contributed by Mario Carneiro, 23-Sep-2014.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C ) )  ->  (
 ( ( ( C  -  B ) F ( A  -  B ) )  +  (
 ( A  -  C ) F ( B  -  C ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  e. 
 { -u pi ,  pi } )
 
Theoremlawcoslem1 23284 Lemma for lawcos 23285. 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 23285* Law of cosines (also known as the Al-Kashi theorem or the generalized Pythagorean theorem, or the cosine formula or cosine rule). Given three distinct points A, B, and C, prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where  F is the signed angle construct (as used in ang180 23283),  X is the distance of line segment BC,  Y is the distance of line segment AC,  Z is the distance of line segment AB, and  O is the signed angle m/_ BCA on the complex plane. We translate triangle ABC to move C to the origin (C-C), B to U=(B-C), and A to V=(A-C), then use lemma lawcoslem1 23284 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 13903). The Pythagorean theorem pythag 23286 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. This is Metamath 100 proof #94. (Contributed by David A. Wheeler, 12-Jun-2015.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   &    |-  X  =  ( abs `  ( B  -  C ) )   &    |-  Y  =  ( abs `  ( A  -  C ) )   &    |-  Z  =  ( abs `  ( A  -  B ) )   &    |-  O  =  ( ( B  -  C ) F ( A  -  C ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C ) )  ->  ( Z ^ 2 )  =  ( ( ( X ^ 2 )  +  ( Y ^
 2 ) )  -  ( 2  x.  (
 ( X  x.  Y )  x.  ( cos `  O ) ) ) ) )
 
Theorempythag 23286* 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 23283),  X is the distance of line segment BC,  Y is the distance of line segment AC,  Z is the distance of line segment AB (the hypotenuse), and  O is the signed right angle m/_ BCA. We use the law of cosines lawcos 23285 to prove this, along with simple trigonometry facts like coshalfpi 22966 and cosneg 13903. (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 ) ) )
 
Theoremisosctrlem1 23287 Lemma for isosctr 23290. (Contributed by Saveliy Skresanov, 30-Dec-2016.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A ) 
 ->  ( Im `  ( log `  ( 1  -  A ) ) )  =/=  pi )
 
Theoremisosctrlem2 23288 Lemma for isosctr 23290. 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 23289* Lemma for isosctr 23290. 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 23290* Isosceles triangle theorem. This is Metamath 100 proof #65. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im `  ( log `  ( y  /  x ) ) ) )   =>    |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C  /\  A  =/=  B )  /\  ( abs `  ( A  -  C ) )  =  ( abs `  ( B  -  C ) ) ) 
 ->  ( ( C  -  A ) F ( B  -  A ) )  =  ( ( A  -  B ) F ( C  -  B ) ) )
 
Theoremssscongptld 23291* 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 23285 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 23292 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 23293 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 23294 Equivalence used in the proof of angpieqvd 23297. (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 23295* Equivalence used in angpieqvd 23297. (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 23296* 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 23297* 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 23298* If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 23291 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 23299* 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 23298, where P = B, and using angrtmuld 23277 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 23300 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 23299 and the Pythagorean theorem (pythag 23286) 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 ) ) )
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