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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremasincl 20001 Closure for the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  A )  e.  CC )
 
Theoremacosf 20002 Domain and range of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arccos : CC --> CC
 
Theoremacoscl 20003 Closure for the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arccos `  A )  e.  CC )
 
Theorematandm 20004 Since the property is a little lengthy, we abbreviate  A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i as  A  e.  dom arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i ) )
 
Theorematandm2 20005 This form of atandm 20004 is a bit more useful for showing that the logarithms in df-atan 19995 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
 
Theorematandm3 20006 A compact form of atandm 20004. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( A ^
 2 )  =/=  -u 1
 ) )
 
Theorematandm4 20007 A compact form of atandm 20004. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^ 2
 ) )  =/=  0
 ) )
 
Theorematanf 20008 Domain and range of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |- arctan : ( CC  \  { -u _i ,  _i } ) --> CC
 
Theorematancl 20009 Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  A )  e. 
 CC )
 
Theoremasinval 20010 Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  A )  =  ( -u _i  x.  ( log `  ( ( _i 
 x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2
 ) ) ) ) ) ) )
 
Theoremacosval 20011 Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  (arccos `  A )  =  ( ( pi  / 
 2 )  -  (arcsin `  A ) ) )
 
Theorematanval 20012 Value of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  / 
 2 )  x.  (
 ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
 
Theorematanre 20013 A real number is in the domain of the arctangent function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  RR  ->  A  e.  dom arctan )
 
Theoremasinneg 20014 The arcsine function is odd. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  (arcsin `  -u A )  =  -u (arcsin `  A ) )
 
Theoremacosneg 20015 The negative symmetry relation of the arccosine. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  (arccos `  -u A )  =  ( pi  -  (arccos `  A ) ) )
 
Theoremefiasin 20016 The exponential of the arcsine function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  (arcsin `  A ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) )
 
Theoremsinasin 20017 The arcsine function is an inverse to  sin. This is the main property that justifies the notation arcsin or  sin
^ -u 1. Because  sin is not an injection, the other converse identity asinsin 20020 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  ( sin `  (arcsin `  A ) )  =  A )
 
Theoremcosacos 20018 The arccosine function is an inverse to  cos. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( A  e.  CC  ->  ( cos `  (arccos `  A ) )  =  A )
 
Theoremasinsinlem 20019 Lemma for asinsin 20020. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  0  <  ( Re
 `  ( exp `  ( _i  x.  A ) ) ) )
 
Theoremasinsin 20020 The arcsine function composed with 
sin is equal to the identity. This plus sinasin 20017 allow us to view  sin and arcsin as inverse operations to each other. For ease of use, we have not defined precisely the correct domain of correctness of this identity; in addition to the main region described here it is also true for some points on the branch cuts, namely when  A  =  ( pi 
/  2 )  -  _i y for non-negative real  y and also symmetrically at  A  =  _i y  -  ( pi  / 
2 ). In particular, when restricted to reals this identity extends to the closed interval  [ -u (
pi  /  2 ) ,  ( pi  / 
2 ) ], not just the open interval (see reasinsin 20024). (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  (arcsin `  ( sin `  A ) )  =  A )
 
Theoremacoscos 20021 The arccosine function is an inverse to  cos. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  (
 0 (,) pi ) ) 
 ->  (arccos `  ( cos `  A ) )  =  A )
 
Theoremasin1 20022 The arcsine of  1 is  pi  / 
2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arcsin `  1 )  =  ( pi  /  2
 )
 
Theoremacos1 20023 The arcsine of  1 is  pi  / 
2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arccos `  1 )  =  0
 
Theoremreasinsin 20024 The arcsine function composed with 
sin is equal to the identity. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) [,] ( pi  / 
 2 ) )  ->  (arcsin `  ( sin `  A ) )  =  A )
 
Theoremasinsinb 20025 Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( Re `  B )  e.  ( -u ( pi  /  2 ) (,) ( pi  /  2
 ) ) )  ->  ( (arcsin `  A )  =  B  <->  ( sin `  B )  =  A )
 )
 
Theoremacoscosb 20026 Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( Re `  B )  e.  ( 0 (,) pi ) )  ->  ( (arccos `  A )  =  B  <->  ( cos `  B )  =  A )
 )
 
Theoremasinbnd 20027 The arcsine function has range within a vertical strip of the complex plane with real part between  -u pi  /  2 and  pi  /  2. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( Re `  (arcsin `  A ) )  e.  ( -u ( pi  / 
 2 ) [,] ( pi  /  2 ) ) )
 
Theoremacosbnd 20028 The arccosine function has range within a vertical strip of the complex plane with real part between  0 and  pi. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( Re `  (arccos `  A ) )  e.  ( 0 [,] pi ) )
 
Theoremasinrebnd 20029 Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u 1 [,] 1 ) 
 ->  (arcsin `  A )  e.  ( -u ( pi  / 
 2 ) [,] ( pi  /  2 ) ) )
 
Theoremasinrecl 20030 The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u 1 [,] 1 ) 
 ->  (arcsin `  A )  e.  RR )
 
Theoremacosrecl 20031 The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  ( -u 1 [,] 1 ) 
 ->  (arccos `  A )  e.  RR )
 
Theoremcosasin 20032 The cosine of the arcsine of  A is  sqr ( 1  -  A ^ 2 ). (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( cos `  (arcsin `  A ) )  =  ( sqr `  (
 1  -  ( A ^ 2 ) ) ) )
 
Theoremsinacos 20033 The sine of the arccosine of  A is  sqr ( 1  -  A ^ 2 ). (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  CC  ->  ( sin `  (arccos `  A ) )  =  ( sqr `  (
 1  -  ( A ^ 2 ) ) ) )
 
Theorematandmneg 20034 The domain of the arctangent function is closed under negatives. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  -u A  e.  dom arctan )
 
Theorematanneg 20035 The arctangent function is odd. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  (arctan `  -u A )  =  -u (arctan `  A )
 )
 
Theorematan0 20036 The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  (arctan `  0 )  =  0
 
Theorematandmcj 20037 The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  ( * `  A )  e.  dom arctan )
 
Theorematancj 20038 The arctangent function distributes under conjugation. (The condition that  Re ( A )  =/=  0 is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 20035 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between  -u 1 and  1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0
 )  ->  ( A  e.  dom arctan  /\  ( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) ) )
 
Theorematanrecl 20039 The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
 
Theoremefiatan 20040 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A )
 ) )  =  ( ( sqr `  (
 1  +  ( _i 
 x.  A ) ) )  /  ( sqr `  ( 1  -  ( _i  x.  A ) ) ) ) )
 
Theorematanlogaddlem 20041 Lemma for atanlogadd 20042. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  0  <_  ( Re `  A ) )  ->  ( ( log `  (
 1  +  ( _i 
 x.  A ) ) )  +  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
 
Theorematanlogadd 20042 The rule  sqr ( z w )  =  ( sqr z ) ( sqr w ) is not always true on the complexes, but it is true when the arguments of  z and  w sum to within the interval  ( -u pi ,  pi ], so there are some cases such as this one with  z  =  1  +  _i A and  w  =  1  -  _i A which are true unconditionally. This result can also be stated as " sqr ( 1  +  z )  +  sqr ( 1  -  z
) is analytic". (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( ( log `  (
 1  +  ( _i 
 x.  A ) ) )  +  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
 
Theorematanlogsublem 20043 Lemma for atanlogsub 20044. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  0  <  ( Re
 `  A ) ) 
 ->  ( Im `  (
 ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A ) ) ) ) )  e.  ( -u pi (,) pi ) )
 
Theorematanlogsub 20044 A variation on atanlogadd 20042, to show that  sqr ( 1  +  _i z )  /  sqr ( 1  -  _i z )  =  sqr ( ( 1  +  _i z )  /  ( 1  -  _i z ) ) under more limited conditions. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  ( Re `  A )  =/=  0 )  ->  ( ( log `  (
 1  +  ( _i 
 x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
 
Theoremefiatan2 20045 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A )
 ) )  =  ( ( 1  +  ( _i  x.  A ) ) 
 /  ( sqr `  (
 1  +  ( A ^ 2 ) ) ) ) )
 
Theorem2efiatan 20046 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1
 ) )
 
Theoremtanatan 20047 The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A ) )  =  A )
 
Theorematandmtan 20048 The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  e.  dom arctan )
 
Theoremcosatan 20049 The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( cos `  (arctan `  A ) )  =  (
 1  /  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) ) )
 
Theoremcosatanne0 20050 The arctangent function has range contained in the domain of the tangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( cos `  (arctan `  A ) )  =/=  0
 )
 
Theorematantan 20051 The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ( -u ( pi  /  2
 ) (,) ( pi  / 
 2 ) ) ) 
 ->  (arctan `  ( tan `  A ) )  =  A )
 
Theorematantanb 20052 Relationship between tangent and arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  dom arctan  /\  B  e.  CC  /\  ( Re `  B )  e.  ( -u ( pi  /  2 ) (,) ( pi  /  2
 ) ) )  ->  ( (arctan `  A )  =  B  <->  ( tan `  B )  =  A )
 )
 
Theorematanbndlem 20053 Lemma for atanbnd 20054. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( A  e.  RR+  ->  (arctan `  A )  e.  ( -u ( pi  / 
 2 ) (,) ( pi  /  2 ) ) )
 
Theorematanbnd 20054 The arctangent function is bounded by  pi  /  2 on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( A  e.  RR  ->  (arctan `  A )  e.  ( -u ( pi  / 
 2 ) (,) ( pi  /  2 ) ) )
 
Theorematanord 20055 The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 (arctan `  A )  < 
 (arctan `  B ) ) )
 
Theorematan1 20056 The arctangent of  1 is  pi  /  4. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arctan `  1 )  =  ( pi  /  4
 )
 
Theorembndatandm 20057 A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  A  e.  dom arctan )
 
Theorematans 20058* The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  +  ( A ^ 2
 ) )  e.  D ) )
 
Theorematans2 20059* It suffices to show that  1  -  _i A and  1  +  _i A are in the continuity domain of  log to show that  A is in the continuity domain of arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  e.  D  /\  ( 1  +  ( _i  x.  A ) )  e.  D ) )
 
Theorematansopn 20060* The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  S  e.  ( TopOpen ` fld )
 
Theorematansssdm 20061* The domain of continuity of the arctangent is a subset of the actual domain of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  S  C_ 
 dom arctan
 
Theoremressatans 20062* The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  RR  C_  S
 
Theoremdvatan 20063* The derivative of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  ( CC  _D  (arctan  |`  S ) )  =  ( x  e.  S  |->  ( 1 
 /  ( 1  +  ( x ^ 2
 ) ) ) )
 
Theorematancn 20064* The arctangent is a continuous function. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  D  =  ( CC  \  (  -oo (,] 0
 ) )   &    |-  S  =  {
 y  e.  CC  |  ( 1  +  (
 y ^ 2 ) )  e.  D }   =>    |-  (arctan  |`  S )  e.  ( S -cn-> CC )
 
Theorematantayl 20065* The Taylor series for arctan ( A ). (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( _i  x.  (
 ( -u _i ^ n )  -  ( _i ^ n ) ) ) 
 /  2 )  x.  ( ( A ^ n )  /  n ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  < 
 1 )  ->  seq  1
 (  +  ,  F ) 
 ~~>  (arctan `  A )
 )
 
Theorematantayl2 20066* The Taylor series for arctan ( A ). (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  if (
 2  ||  n , 
 0 ,  ( (
 -u 1 ^ (
 ( n  -  1
 )  /  2 )
 )  x.  ( ( A ^ n ) 
 /  n ) ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  1 (  +  ,  F )  ~~>  (arctan `  A )
 )
 
Theorematantayl3 20067* The Taylor series for arctan ( A ). (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( (
 -u 1 ^ n )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) )   =>    |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  ~~>  (arctan `  A )
 )
 
Theoremleibpilem1 20068 Lemma for leibpi 20070. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( N  e.  NN0  /\  ( -.  N  =  0  /\  -.  2  ||  N ) )  ->  ( N  e.  NN  /\  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 )
 
Theoremleibpilem2 20069* The Leibniz formula for  pi. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( (
 -u 1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  G  =  ( k  e.  NN0  |->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u 1 ^ ( ( k  -  1 )  / 
 2 ) )  /  k ) ) )   &    |-  A  e.  _V   =>    |-  (  seq  0 (  +  ,  F )  ~~>  A 
 <-> 
 seq  0 (  +  ,  G )  ~~>  A )
 
Theoremleibpi 20070 The Leibniz formula for  pi. This proof depends on three main facts: (1) the series  F is convergent, because it is an alternating series (iseralt 12034). (2) Using leibpilem2 20069 to rewrite the series as a power series, it is the  x  =  1 special case of the Taylor series for arctan (atantayl2 20066). (3) Although we cannot directly plug  x  =  1 into atantayl2 20066, Abel's theorem (abelth2 19650) says that the limit along any sequence converging to  1, such as 
1  -  1  /  n, of the power series converges to the power series extended to  1, and then since arctan is continuous at  1 (atancn 20064) we get the desired result. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( (
 -u 1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )   =>    |-  seq  0 (  +  ,  F )  ~~>  ( pi  / 
 4 )
 
Theoremleibpisum 20071 The Leibniz formula for  pi. This version of leibpi 20070 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |- 
 sum_ n  e.  NN0  (
 ( -u 1 ^ n )  /  ( ( 2  x.  n )  +  1 ) )  =  ( pi  /  4
 )
 
Theoremlog2cnv 20072 Using the Taylor series for arctan ( _i  /  3
), produce a rapidly convergent series for  log 2. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  F  =  ( n  e.  NN0  |->  ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) )   =>    |-  seq  0 (  +  ,  F )  ~~>  ( log `  2
 )
 
Theoremlog2tlbnd 20073* Bound the error term in the series of log2cnv 20072. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( N  e.  NN0  ->  ( ( log `  2
 )  -  sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( 2  /  (
 ( 3  x.  (
 ( 2  x.  n )  +  1 )
 )  x.  ( 9 ^ n ) ) ) )  e.  (
 0 [,] ( 3  /  ( ( 4  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
 9 ^ N ) ) ) ) )
 
13.3.7  The Birthday Problem
 
Theoremlog2ublem1 20074 Lemma for log2ub 20077. The proof of log2ub 20077, which is simply the evaluation of log2tlbnd 20073 for  N  =  4, takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator  d (usually a large power of  10) and work with closest approximations of the form  n  /  d for some integer  n instead. It turns out that for our purposes it is sufficient to take  d  =  ( 3 ^ 7 )  x.  5  x.  7, which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  A )  <_  B   &    |-  A  e.  RR   &    |-  D  e.  NN0   &    |-  E  e.  NN   &    |-  B  e.  NN0   &    |-  F  e.  NN0   &    |-  C  =  ( A  +  ( D  /  E ) )   &    |-  ( B  +  F )  =  G   &    |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  D )  <_  ( E  x.  F )   =>    |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  C )  <_  G
 
Theoremlog2ublem2 20075* Lemma for log2ub 20077. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  sum_ n  e.  ( 0 ...
 K ) ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) ) 
 <_  ( 2  x.  B )   &    |-  B  e.  NN0   &    |-  F  e.  NN0   &    |-  N  e.  NN0   &    |-  ( N  -  1
 )  =  K   &    |-  ( B  +  F )  =  G   &    |-  M  e.  NN0   &    |-  ( M  +  N )  =  3   &    |-  ( ( 5  x.  7 )  x.  ( 9 ^ M ) )  =  (
 ( ( 2  x.  N )  +  1 )  x.  F )   =>    |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  sum_ n  e.  ( 0 ...
 N ) ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) ) 
 <_  ( 2  x.  G )
 
Theoremlog2ublem3 20076 Lemma for log2ub 20077. In decimal, this is a proof that the first four terms of the series for 
log 2 is less than  5 3
0 5 6  / 
7 6 5 4 5. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7
 ) )  x.  sum_ n  e.  ( 0 ... 3 ) ( 2 
 /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) ) 
 <_ ;;;; 5 3 0 5 6
 
Theoremlog2ub 20077  log 2 is less than  2 5 3  /  3
6 5. If written in decimal, this is because  log 2  = 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( log `  2
 )  <  (;; 2 5 3  / ;; 3 6 5 )
 
Theorembirthdaylem1 20078* Lemma for birthday 20081. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   =>    |-  ( T  C_  S  /\  S  e.  Fin  /\  ( N  e.  NN  ->  S  =/=  (/) ) )
 
Theorembirthdaylem2 20079* For general  N and  K, count the fraction of injective functions from  1 ... K to  1 ... N. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   =>    |-  (
 ( N  e.  NN  /\  K  e.  ( 0
 ... N ) ) 
 ->  ( ( # `  T )  /  ( # `  S ) )  =  ( exp `  sum_ k  e.  (
 0 ... ( K  -  1 ) ) ( log `  ( 1  -  ( k  /  N ) ) ) ) )
 
Theorembirthdaylem3 20080* For general  N and  K, upper-bound the fraction of injective functions from  1 ... K to  1 ... N. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   =>    |-  (
 ( K  e.  NN0  /\  N  e.  NN )  ->  ( ( # `  T )  /  ( # `  S ) )  <_  ( exp `  -u ( ( ( ( K ^ 2
 )  -  K ) 
 /  2 )  /  N ) ) )
 
Theorembirthday 20081* The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for  K  =  2 3 and  N  =  3 6 5, fewer than half of the set of all functions from  1 ... K to  1 ... N are injective. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  S  =  { f  |  f : ( 1
 ... K ) --> ( 1
 ... N ) }   &    |-  T  =  { f  |  f : ( 1 ...
 K ) -1-1-> ( 1
 ... N ) }   &    |-  K  = ; 2 3   &    |-  N  = ;; 3 6 5   =>    |-  ( ( # `  T )  /  ( # `  S ) )  <  ( 1 
 /  2 )
 
13.3.8  Areas in R^2
 
Syntaxcarea 20082 Area of regions in the complex plane.
 class area
 
Definitiondf-area 20083* Define the area of a subset of  RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area  =  ( s  e.  {
 t  e.  ~P ( RR  X.  RR )  |  ( A. x  e. 
 RR  ( t " { x } )  e.  ( `' vol " RR )  /\  ( x  e. 
 RR  |->  ( vol `  (
 t " { x }
 ) ) )  e.  L ^1 ) }  |->  S. RR ( vol `  ( s " { x } ) )  _d x )
 
Theoremdmarea 20084* The domain of the area function is the set of finitely measurable subsets of  RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( A  e.  dom area  <->  ( A  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  ( x  e. 
 RR  |->  ( vol `  ( A " { x }
 ) ) )  e.  L ^1 ) )
 
Theoremareambl 20085 The fibers of a measurable region are finitely meaurable subsets of  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ( S  e.  dom area  /\  A  e.  RR )  ->  ( ( S " { A } )  e. 
 dom  vol  /\  ( vol `  ( S " { A } ) )  e. 
 RR ) )
 
Theoremareass 20086 A measurable region is a subset of 
RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  ->  S  C_  ( RR  X.  RR ) )
 
Theoremdfarea 20087* Rewrite df-area 20083 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  (
 s " { x }
 ) )  _d x )
 
Theoremareaf 20088 Area meaurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area : dom area
 --> ( 0 [,)  +oo )
 
Theoremareacl 20089 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  ->  (area `  S )  e. 
 RR )
 
Theoremareage0 20090 The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  -> 
 0  <_  (area `  S ) )
 
Theoremareaval 20091* The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( S  e.  dom area  ->  (area `  S )  =  S. RR ( vol `  ( S " { x } ) )  _d x )
 
13.3.9  More miscellaneous converging sequences
 
Theoremrlimcnp 20092* Relate a limit of a real-valued sequence at infinity to the continuity of the function  S ( y )  =  R ( 1  /  y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  A  C_  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  0  e.  A )   &    |-  ( ph  ->  B  C_  RR+ )   &    |-  (
 ( ph  /\  x  e.  A )  ->  R  e.  CC )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  ( x  e.  A  <->  ( 1  /  x )  e.  B ) )   &    |-  ( x  =  0  ->  R  =  C )   &    |-  ( x  =  (
 1  /  y )  ->  R  =  S )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  A )   =>    |-  ( ph  ->  (
 ( y  e.  B  |->  S )  ~~> r  C  <->  ( x  e.  A  |->  R )  e.  ( ( K  CnP  J ) `  0 ) ) )
 
Theoremrlimcnp2 20093* Relate a limit of a real-valued sequence at infinity to the continuity of the function  S ( y )  =  R ( 1  /  y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  A  C_  ( 0 [,)  +oo ) )   &    |-  ( ph  ->  0  e.  A )   &    |-  ( ph  ->  B  C_  RR )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  S  e.  CC )   &    |-  (
 ( ph  /\  y  e.  RR+ )  ->  ( y  e.  B  <->  ( 1  /  y )  e.  A ) )   &    |-  ( y  =  ( 1  /  x )  ->  S  =  R )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  A )   =>    |-  ( ph  ->  (
 ( y  e.  B  |->  S )  ~~> r  C  <->  ( x  e.  A  |->  if ( x  =  0 ,  C ,  R ) )  e.  ( ( K  CnP  J ) `  0 ) ) )
 
Theoremrlimcnp3 20094* Relate a limit of a real-valued sequence at infinity to the continuity of the function  S ( y )  =  R ( 1  /  y ) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  y  e.  RR+ )  ->  S  e.  CC )   &    |-  (
 y  =  ( 1 
 /  x )  ->  S  =  R )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  (
 ( y  e.  RR+  |->  S )  ~~> r  C  <->  ( x  e.  ( 0 [,)  +oo )  |->  if ( x  =  0 ,  C ,  R ) )  e.  ( ( K  CnP  J ) `  0 ) ) )
 
Theoremxrlimcnp 20095* Relate a limit of a real-valued sequence at infinity to the continuity of the corresponding extended real function at  +oo. Since any  ~~> r limit can be written in the form on the left side of the implication, this shows that real limits are a special case of topological continuity at a point. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  ( ph  ->  A  =  ( B  u.  {  +oo } ) )   &    |-  ( ph  ->  B  C_  RR )   &    |-  ( ( ph  /\  x  e.  A )  ->  R  e.  CC )   &    |-  ( x  = 
 +oo  ->  R  =  C )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( (ordTop `  <_  )t  A )   =>    |-  ( ph  ->  (
 ( x  e.  B  |->  R )  ~~> r  C  <->  ( x  e.  A  |->  R )  e.  ( ( K  CnP  J ) `  +oo )
 ) )
 
Theoremefrlim 20096* The limit of the sequence  ( 1  +  A  /  k ) ^
k is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 20097). (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  S  =  ( 0 ( ball `  ( abs  o. 
 -  ) ) ( 1  /  ( ( abs `  A )  +  1 ) ) )   =>    |-  ( A  e.  CC  ->  ( k  e.  RR+  |->  ( ( 1  +  ( A  /  k
 ) )  ^ c  k ) )  ~~> r  ( exp `  A )
 )
 
Theoremdfef2 20097* The limit of the sequence  ( 1  +  A  /  k ) ^
k as  k goes to  +oo is  ( exp `  A
). This is another common definition of  _e. (Contributed by Mario Carneiro, 1-Mar-2015.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( F `  k )  =  ( ( 1  +  ( A  /  k
 ) ) ^ k
 ) )   =>    |-  ( ph  ->  F  ~~>  ( exp `  A )
 )
 
Theoremcxplim 20098* A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)
 |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1 
 /  ( n  ^ c  A ) ) )  ~~> r  0 )
 
Theoremsqrlim 20099 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( n  e.  RR+  |->  ( 1  /  ( sqr `  n ) ) )  ~~> r  0
 
Theoremrlimcxp 20100* Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ( ph  /\  n  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( n  e.  A  |->  B )  ~~> r  0 )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( n  e.  A  |->  ( B 
 ^ c  C ) )  ~~> r  0 )
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