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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitgulm2 19801* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( x  e.  S  |->  A )  e.  ( S -cn-> CC ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( x  e.  S  |->  A )  e.  L ^1 )   &    |-  ( ph  ->  ( k  e.  Z  |->  ( x  e.  S  |->  A ) ) ( ~~> u `  S ) ( x  e.  S  |->  B ) )   &    |-  ( ph  ->  ( vol `  S )  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  S  |->  B )  e.  L ^1  /\  ( k  e.  Z  |->  S. S A  _d x )  ~~>  S. S B  _d x ) )
 
13.2.3  Power series
 
Theorempserval 19802* Value of the function  G that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   =>    |-  ( X  e.  CC  ->  ( G `  X )  =  ( m  e.  NN0  |->  ( ( A `
  m )  x.  ( X ^ m ) ) ) )
 
Theorempserval2 19803* Value of the function  G that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   =>    |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  ( ( G `
  X ) `  N )  =  (
 ( A `  N )  x.  ( X ^ N ) ) )
 
Theorempsergf 19804* The sequence of terms in the infinite sequence defining a power series for fixed  X. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   =>    |-  ( ph  ->  ( G `  X ) :
 NN0
 --> CC )
 
Theoremradcnvlem1 19805* Lemma for radcnvlt1 19810, radcnvle 19812. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges absolutely at 
X, even if the terms in the sequence are multiplied by  n. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  ( abs `  Y ) )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e.  dom  ~~>  )   &    |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  ( ( G `
  X ) `  m ) ) ) )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H )  e.  dom  ~~>  )
 
Theoremradcnvlem2 19806* Lemma for radcnvlt1 19810, radcnvle 19812. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges absolutely at  X. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  ( abs `  Y ) )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  0 (  +  ,  ( abs  o.  ( G `
  X ) ) )  e.  dom  ~~>  )
 
Theoremradcnvlem3 19807* Lemma for radcnvlt1 19810, radcnvle 19812. If  X is a point closer to zero than  Y and the power series converges at 
Y, then it converges at  X. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  Y  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  ( abs `  Y ) )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  Y ) )  e.  dom  ~~>  )   =>    |-  ( ph  ->  seq  0 (  +  ,  ( G `  X ) )  e.  dom  ~~>  )
 
Theoremradcnv0 19808* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   =>    |-  ( ph  ->  0  e.  {
 r  e.  RR  |  seq  0 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } )
 
Theoremradcnvcl 19809* The radius of convergence  R of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   =>    |-  ( ph  ->  R  e.  ( 0 [,]  +oo ) )
 
Theoremradcnvlt1 19810* If  X is within the open disk of radius  R centered at zero, then the infinite series converges absolutely at  X, and also converges when the series is multiplied by  n. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   &    |-  H  =  ( m  e.  NN0  |->  ( m  x.  ( abs `  (
 ( G `  X ) `  m ) ) ) )   =>    |-  ( ph  ->  (  seq  0 (  +  ,  H )  e.  dom  ~~>  /\ 
 seq  0 (  +  ,  ( abs  o.  ( G `  X ) ) )  e.  dom  ~~>  ) )
 
Theoremradcnvlt2 19811* If  X is within the open disk of radius  R centered at zero, then the infinite series converges at  X. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   =>    |-  ( ph  ->  seq  0
 (  +  ,  ( G `  X ) )  e.  dom  ~~>  )
 
Theoremradcnvle 19812* If  X is a convergent point of the infinite series, then 
X is within the closed disk of radius  R centered at zero. Or, by contraposition, the series divergers at any point strictly more than  R from the origin. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  seq  0 (  +  ,  ( G `  X ) )  e.  dom  ~~>  )   =>    |-  ( ph  ->  ( abs `  X )  <_  R )
 
Theoremdvradcnv 19813* The radius of convergence of the (formal) derivative  H of the power series  G is at least as large as the radius of convergence of  G. (In fact they are equal, but we don't have as much use for the negative side of this claim.) (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( n  e.  NN0  |->  ( ( ( n  +  1 )  x.  ( A `  ( n  +  1 )
 ) )  x.  ( X ^ n ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  ( abs `  X )  <  R )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H )  e.  dom  ~~>  )
 
Theorempserulm 19814* If  S is a region contained in a circle of radius  M  <  R, then the sequence of partial sums of the infinite series converges uniformly on  S. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( i  e.  NN0  |->  ( y  e.  S  |->  (  seq  0 (  +  ,  ( G `  y
 ) ) `  i
 ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  M  <  R )   &    |-  ( ph  ->  S 
 C_  ( `' abs " ( 0 [,] M ) ) )   =>    |-  ( ph  ->  H ( ~~> u `  S ) F )
 
Theorempsercn2 19815* Since by pserulm 19814 the series converges uniformly, it is also continuous by ulmcn 19792. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  H  =  ( i  e.  NN0  |->  ( y  e.  S  |->  (  seq  0 (  +  ,  ( G `  y
 ) ) `  i
 ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  M  <  R )   &    |-  ( ph  ->  S 
 C_  ( `' abs " ( 0 [,] M ) ) )   =>    |-  ( ph  ->  F  e.  ( S -cn-> CC ) )
 
Theorempsercnlem2 19816* Lemma for psercn 19818. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  ( ( ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  <  M  /\  M  <  R ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  (
 a  e.  ( 0 ( ball `  ( abs  o. 
 -  ) ) M )  /\  ( 0 ( ball `  ( abs  o. 
 -  ) ) M )  C_  ( `' abs " ( 0 [,]
 M ) )  /\  ( `' abs " ( 0 [,] M ) ) 
 C_  S ) )
 
Theorempsercnlem1 19817* Lemma for psercn 19818. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  <  M  /\  M  <  R ) )
 
Theorempsercn 19818* An infinite series converges to a continuous function on the open disk of radius  R, where  R is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   =>    |-  ( ph  ->  F  e.  ( S -cn-> CC )
 )
 
Theorempserdvlem1 19819* Lemma for pserdv 19821. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  (
 ( ( ( abs `  a )  +  M )  /  2 )  e.  RR+  /\  ( abs `  a
 )  <  ( (
 ( abs `  a )  +  M )  /  2
 )  /\  ( (
 ( abs `  a )  +  M )  /  2
 )  <  R )
 )
 
Theorempserdvlem2 19820* Lemma for pserdv 19821. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   &    |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M )  /  2 ) )   =>    |-  ( ( ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( y  e.  B  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1
 ) ) )  x.  ( y ^ k
 ) ) ) )
 
Theorempserdv 19821* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   &    |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M )  /  2 ) )   =>    |-  ( ph  ->  ( CC  _D  F )  =  ( y  e.  S  |->  sum_ k  e.  NN0  ( (
 ( k  +  1 )  x.  ( A `
  ( k  +  1 ) ) )  x.  ( y ^
 k ) ) ) )
 
Theorempserdv2 19822* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  F  =  ( y  e.  S  |->  sum_ j  e.  NN0  ( ( G `  y ) `  j ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  ,  ( G `  r
 ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  S  =  ( `' abs " (
 0 [,) R ) )   &    |-  M  =  if ( R  e.  RR ,  (
 ( ( abs `  a
 )  +  R ) 
 /  2 ) ,  ( ( abs `  a
 )  +  1 ) )   &    |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M )  /  2 ) )   =>    |-  ( ph  ->  ( CC  _D  F )  =  ( y  e.  S  |->  sum_ k  e.  NN  ( ( k  x.  ( A `
  k ) )  x.  ( y ^
 ( k  -  1
 ) ) ) ) )
 
Theoremabelthlem1 19823* Lemma for abelth 19833. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   =>    |-  ( ph  ->  1  <_  sup ( { r  e.  RR  |  seq  0
 (  +  ,  (
 ( z  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( z ^ n ) ) ) ) `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )
 
Theoremabelthlem2 19824* Lemma for abelth 19833. The peculiar region  S, known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing  1. Indeed, except for  1 itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   =>    |-  ( ph  ->  ( 1  e.  S  /\  ( S  \  { 1 } )  C_  (
 0 ( ball `  ( abs  o.  -  ) ) 1 ) ) )
 
Theoremabelthlem3 19825* Lemma for abelth 19833. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   =>    |-  ( ( ph  /\  X  e.  S ) 
 ->  seq  0 (  +  ,  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( X ^ n ) ) ) )  e.  dom  ~~>  )
 
Theoremabelthlem4 19826* Lemma for abelth 19833. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ph  ->  F : S
 --> CC )
 
Theoremabelthlem5 19827* Lemma for abelth 19833. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  ~~>  0 )   =>    |-  ( ( ph  /\  X  e.  ( 0 ( ball `  ( abs  o.  -  ) ) 1 ) )  ->  seq  0 (  +  ,  ( k  e.  NN0  |->  ( ( 
 seq  0 (  +  ,  A ) `  k
 )  x.  ( X ^ k ) ) ) )  e.  dom  ~~>  )
 
Theoremabelthlem6 19828* Lemma for abelth 19833. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  ~~>  0 )   &    |-  ( ph  ->  X  e.  ( S  \  { 1 } )
 )   =>    |-  ( ph  ->  ( F `  X )  =  ( ( 1  -  X )  x.  sum_ n  e.  NN0  ( (  seq  0 (  +  ,  A ) `  n )  x.  ( X ^ n ) ) ) )
 
Theoremabelthlem7a 19829* Lemma for abelth 19833. (Contributed by Mario Carneiro, 8-May-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  ~~>  0 )   &    |-  ( ph  ->  X  e.  ( S  \  { 1 } )
 )   =>    |-  ( ph  ->  ( X  e.  CC  /\  ( abs `  ( 1  -  X ) )  <_  ( M  x.  (
 1  -  ( abs `  X ) ) ) ) )
 
Theoremabelthlem7 19830* Lemma for abelth 19833. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  ~~>  0 )   &    |-  ( ph  ->  X  e.  ( S  \  { 1 } )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A. k  e.  ( ZZ>= `  N ) ( abs `  (  seq  0 (  +  ,  A ) `
  k ) )  <  R )   &    |-  ( ph  ->  ( abs `  (
 1  -  X ) )  <  ( R 
 /  ( sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( abs `  (  seq  0 (  +  ,  A ) `  n ) )  +  1
 ) ) )   =>    |-  ( ph  ->  ( abs `  ( F `  X ) )  < 
 ( ( M  +  1 )  x.  R ) )
 
Theoremabelthlem8 19831* Lemma for abelth 19833. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  ~~>  0 )   =>    |-  ( ( ph  /\  R  e.  RR+ )  ->  E. w  e.  RR+  A. y  e.  S  ( ( abs `  (
 1  -  y ) )  <  w  ->  ( abs `  ( ( F `  1 )  -  ( F `  y ) ) )  <  R ) )
 
Theoremabelthlem9 19832* Lemma for abelth 19833. By adjusting the constant term, we can assume that the entire series converges to 
0. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ( ph  /\  R  e.  RR+ )  ->  E. w  e.  RR+  A. y  e.  S  ( ( abs `  (
 1  -  y ) )  <  w  ->  ( abs `  ( ( F `  1 )  -  ( F `  y ) ) )  <  R ) )
 
Theoremabelth 19833* Abel's theorem. If the power series  sum_ n  e.  NN0 A
( n ) ( x ^ n ) is convergent at  1, then it is equal to the limit from "below", along a Stolz angle  S (note that the  M  =  1 case of a Stolz angle is the real line  [ 0 ,  1 ]). (Continuity on  S  \  { 1 } follows more generally from psercn 19818.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  0  <_  M )   &    |-  S  =  { z  e.  CC  |  ( abs `  ( 1  -  z
 ) )  <_  ( M  x.  ( 1  -  ( abs `  z )
 ) ) }   &    |-  F  =  ( x  e.  S  |->  sum_
 n  e.  NN0  (
 ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ph  ->  F  e.  ( S -cn-> CC ) )
 
Theoremabelth2 19834* Abel's theorem, restricted to the 
[ 0 ,  1 ] interval. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ph  ->  A : NN0 --> CC )   &    |-  ( ph  ->  seq  0 (  +  ,  A )  e.  dom  ~~>  )   &    |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  sum_ n  e.  NN0  ( ( A `  n )  x.  ( x ^ n ) ) )   =>    |-  ( ph  ->  F  e.  ( ( 0 [,] 1 ) -cn-> CC ) )
 
13.3  Basic trigonometry
 
13.3.1  The exponential, sine, and cosine functions (cont.)
 
Theoremefcn 19835 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |- 
 exp  e.  ( CC -cn-> CC )
 
Theoremsincn 19836 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 sin  e.  ( CC -cn-> CC )
 
Theoremcoscn 19837 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 cos  e.  ( CC -cn-> CC )
 
Theoremreeff1olem 19838* Lemma for reeff1o 19839. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  RR  ( exp `  x )  =  U )
 
Theoremreeff1o 19839 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( exp  |`  RR ) : RR
 -1-1-onto-> RR+
 
Theoremreefiso 19840 The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)
 |-  ( exp  |`  RR )  Isom  <  ,  <  ( RR ,  RR+ )
 
Theoremefcvx 19841 The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
 ) )  ->  ( exp `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) ) )  < 
 ( ( T  x.  ( exp `  A )
 )  +  ( ( 1  -  T )  x.  ( exp `  B ) ) ) )
 
Theoremreefgim 19842 The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  R  =  (flds  RR )   &    |-  P  =  ( (mulGrp ` fld )s  RR+ )   =>    |-  ( exp  |`  RR )  e.  ( R GrpIso  P )
 
13.3.2  Properties of pi = 3.14159...
 
Theorempilem1 19843 Lemma for pire 19848, pigt2lt4 19846 and sinpi 19847. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( A  e.  ( RR+ 
 i^i  ( `' sin " { 0 } )
 ) 
 <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 )
 )
 
Theorempilem2 19844 Lemma for pire 19848, pigt2lt4 19846 and sinpi 19847. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ph  ->  A  e.  ( 2 (,) 4
 ) )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( sin `  A )  =  0 )   &    |-  ( ph  ->  ( sin `  B )  =  0 )   &    |-  ( ph  ->  pi  <  A )   =>    |-  ( ph  ->  (
 ( pi  +  A )  /  2 )  <_  B )
 
Theorempilem3 19845 Lemma for pire 19848, pigt2lt4 19846 and sinpi 19847. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 |-  ( pi  e.  (
 2 (,) 4 )  /\  ( sin `  pi )  =  0 )
 
Theorempigt2lt4 19846  pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( 2  <  pi  /\  pi  <  4 )
 
Theoremsinpi 19847 The sine of  pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  pi )  =  0
 
Theorempire 19848  pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  pi  e.  RR
 
Theorempipos 19849  pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  0  <  pi
 
Theoremsinhalfpilem 19850 Lemma for sinhalfpi 19852 and coshalfpi 19853. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( ( sin `  ( pi  /  2 ) )  =  1  /\  ( cos `  ( pi  / 
 2 ) )  =  0 )
 
Theoremhalfpire 19851  pi  /  2 is real. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( pi  /  2
 )  e.  RR
 
Theoremsinhalfpi 19852 The sine of  pi  /  2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  ( pi  /  2 ) )  =  1
 
Theoremcoshalfpi 19853 The cosine of  pi  /  2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  ( pi  /  2 ) )  =  0
 
Theoremcosneghalfpi 19854 The cosine of  -u pi  /  2 is zero. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( cos `  -u ( pi  /  2 ) )  =  0
 
Theoremefhalfpi 19855 The exponential of  _i pi  /  2 is  _i. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( pi  / 
 2 ) ) )  =  _i
 
Theoremcospi 19856 The cosine of  pi is  -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  pi )  =  -u 1
 
Theoremefipi 19857 The exponential of  _i pi. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( exp `  ( _i  x.  pi ) )  =  -u 1
 
Theoremeulerid 19858 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( ( exp `  ( _i  x.  pi ) )  +  1 )  =  0
 
Theoremsin2pi 19859 The sine of  2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  (
 2  x.  pi ) )  =  0
 
Theoremcos2pi 19860 The cosine of  2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  (
 2  x.  pi ) )  =  1
 
Theoremef2pi 19861 The exponential of  2 pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( 2  x.  pi ) ) )  =  1
 
Theoremef2kpi 19862 The exponential of  2 K pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( K  e.  ZZ  ->  ( exp `  (
 ( _i  x.  (
 2  x.  pi ) )  x.  K ) )  =  1 )
 
Theoremefper 19863 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  ( A  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  A ) )
 
Theoremsinperlem 19864 Lemma for sinper 19865 and cosper 19866. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( A  e.  CC  ->  ( F `  A )  =  ( (
 ( exp `  ( _i  x.  A ) ) O ( exp `  ( -u _i  x.  A ) ) )  /  D ) )   &    |-  ( ( A  +  ( K  x.  ( 2  x.  pi ) ) )  e. 
 CC  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( ( ( exp `  ( _i  x.  ( A  +  ( K  x.  (
 2  x.  pi ) ) ) ) ) O ( exp `  ( -u _i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) ) )  /  D ) )   =>    |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( F `
  A ) )
 
Theoremsinper 19865 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( sin `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( sin `  A ) )
 
Theoremcosper 19866 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( cos `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( cos `  A ) )
 
Theoremsin2kpi 19867 If  K is an integer, the sine of  2 K pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( K  e.  ZZ  ->  ( sin `  ( K  x.  ( 2  x.  pi ) ) )  =  0 )
 
Theoremcos2kpi 19868 If  K is an integer, the cosine of  2 K pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( K  e.  ZZ  ->  ( cos `  ( K  x.  ( 2  x.  pi ) ) )  =  1 )
 
Theoremsin2pim 19869 Sine of a number subtracted from  2  x.  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 ( 2  x.  pi )  -  A ) )  =  -u ( sin `  A ) )
 
Theoremcos2pim 19870 Cosine of a number subtracted from  2  x.  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 ( 2  x.  pi )  -  A ) )  =  ( cos `  A ) )
 
Theoremsinmpi 19871 Sine of a number less  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( sin `  ( A  -  pi ) )  =  -u ( sin `  A ) )
 
Theoremcosmpi 19872 Cosine of a number less  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( cos `  ( A  -  pi ) )  =  -u ( cos `  A ) )
 
Theoremsinppi 19873 Sine of a number plus  pi. (Contributed by NM, 10-Aug-2008.)
 |-  ( A  e.  CC  ->  ( sin `  ( A  +  pi )
 )  =  -u ( sin `  A ) )
 
Theoremcosppi 19874 Cosine of a complex number plus  pi. (Contributed by NM, 18-Aug-2008.)
 |-  ( A  e.  CC  ->  ( cos `  ( A  +  pi )
 )  =  -u ( cos `  A ) )
 
Theoremefimpi 19875 The exponential function of  _i times a real number less 
pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( A  -  pi ) ) )  =  -u ( exp `  ( _i  x.  A ) ) )
 
Theoremsinhalfpip 19876 The sine of  pi  /  2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 ( pi  /  2
 )  +  A ) )  =  ( cos `  A ) )
 
Theoremsinhalfpim 19877 The sine of  pi  /  2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( sin `  (
 ( pi  /  2
 )  -  A ) )  =  ( cos `  A ) )
 
Theoremcoshalfpip 19878 The cosine of  pi  /  2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 ( pi  /  2
 )  +  A ) )  =  -u ( sin `  A ) )
 
Theoremcoshalfpim 19879 The cosine of  pi  /  2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  CC  ->  ( cos `  (
 ( pi  /  2
 )  -  A ) )  =  ( sin `  A ) )
 
Theoremptolemy 19880 Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 12468, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. (Contributed by David A. Wheeler, 31-May-2015.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC )  /\  (
 ( A  +  B )  +  ( C  +  D ) )  =  pi )  ->  (
 ( ( sin `  A )  x.  ( sin `  B ) )  +  (
 ( sin `  C )  x.  ( sin `  D ) ) )  =  ( ( sin `  ( B  +  C )
 )  x.  ( sin `  ( A  +  C ) ) ) )
 
Theoremsincosq1lem 19881 Lemma for sincosq1sgn 19882. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  ( pi 
 /  2 ) ) 
 ->  0  <  ( sin `  A ) )
 
Theoremsincosq1sgn 19882 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  ( 0  <  ( sin `  A )  /\  0  <  ( cos `  A ) ) )
 
Theoremsincosq2sgn 19883 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  (
 ( pi  /  2
 ) (,) pi )  ->  ( 0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
 
Theoremsincosq3sgn 19884 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  ( pi (,) ( 3  x.  ( pi  /  2
 ) ) )  ->  ( ( sin `  A )  <  0  /\  ( cos `  A )  < 
 0 ) )
 
Theoremsincosq4sgn 19885 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( A  e.  (
 ( 3  x.  ( pi  /  2 ) ) (,) ( 2  x.  pi ) )  ->  ( ( sin `  A )  <  0  /\  0  <  ( cos `  A ) ) )
 
Theoremcoseq00topi 19886 Location of the zeroes of cosine in 
( 0 [,] pi ). (Contributed by David Moews, 28-Feb-2017.)
 |-  ( A  e.  (
 0 [,] pi )  ->  ( ( cos `  A )  =  0  <->  A  =  ( pi  /  2 ) ) )
 
Theoremcoseq0negpitopi 19887 Location of the zeroes of cosine in 
( -u pi (,] pi ). (Contributed by David Moews, 28-Feb-2017.)
 |-  ( A  e.  ( -u pi (,] pi ) 
 ->  ( ( cos `  A )  =  0  <->  A  e.  { ( pi  /  2 ) ,  -u ( pi  /  2
 ) } ) )
 
Theoremtanrpcl 19888 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  ( tan `  A )  e.  RR+ )
 
Theoremtangtx 19889 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  A  <  ( tan `  A ) )
 
Theoremtanabsge 19890 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) )  ->  ( abs `  A )  <_  ( abs `  ( tan `  A ) ) )
 
Theoremsinq12gt0 19891 The sine of a number strictly between 
0 and  pi is positive. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  (
 0 (,) pi )  -> 
 0  <  ( sin `  A ) )
 
Theoremsinq12ge0 19892 The sine of a number between  0 and  pi is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  ( A  e.  (
 0 [,] pi )  -> 
 0  <_  ( sin `  A ) )
 
Theoremsinq34lt0t 19893 The sine of a number strictly between  pi and  2  x.  pi is negative. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  ( pi (,) ( 2  x.  pi ) )  ->  ( sin `  A )  <  0 )
 
Theoremcosq14gt0 19894 The cosine of a number strictly between  -u pi  /  2 and  pi  /  2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) )  -> 
 0  <  ( cos `  A ) )
 
Theoremcosq14ge0 19895 The cosine of a number between  -u pi  /  2 and  pi  /  2 is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) )  -> 
 0  <_  ( cos `  A ) )
 
Theoremsincosq1eq 19896 Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  +  B )  =  1 )  ->  ( sin `  ( A  x.  ( pi  / 
 2 ) ) )  =  ( cos `  ( B  x.  ( pi  / 
 2 ) ) ) )
 
Theoremsincos4thpi 19897 The sine and cosine of  pi  /  4. (Contributed by Paul Chapman, 25-Jan-2008.)
 |-  ( ( sin `  ( pi  /  4 ) )  =  ( 1  /  ( sqr `  2 )
 )  /\  ( cos `  ( pi  /  4
 ) )  =  ( 1  /  ( sqr `  2 ) ) )
 
Theoremtan4thpi 19898 The tangent of  pi  /  4. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( tan `  ( pi  /  4 ) )  =  1
 
Theoremsincos6thpi 19899 The sine and cosine of  pi  /  6. (Contributed by Paul Chapman, 25-Jan-2008.)
 |-  ( ( sin `  ( pi  /  6 ) )  =  ( 1  / 
 2 )  /\  ( cos `  ( pi  / 
 6 ) )  =  ( ( sqr `  3
 )  /  2 )
 )
 
Theoremsincos3rdpi 19900 The sine and cosine of  pi  /  3. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( ( sin `  ( pi  /  3 ) )  =  ( ( sqr `  3 )  /  2
 )  /\  ( cos `  ( pi  /  3
 ) )  =  ( 1  /  2 ) )
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