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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorematandmcj 20201 The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  dom arctan  ->  ( * `  A )  e.  dom arctan )
 
Theorematancj 20202 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 20199 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 20203 The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
 
Theoremefiatan 20204 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 20205 Lemma for atanlogadd 20206. (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 20206 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 20207 Lemma for atanlogsub 20208. (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 20208 A variation on atanlogadd 20206, 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 20209 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 20210 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 20211 The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A ) )  =  A )
 
Theorematandmtan 20212 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 20213 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 20214 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 20215 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 20216 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 20217 Lemma for atanbnd 20218. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( A  e.  RR+  ->  (arctan `  A )  e.  ( -u ( pi  / 
 2 ) (,) ( pi  /  2 ) ) )
 
Theorematanbnd 20218 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 20219 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 20220 The arctangent of  1 is  pi  /  4. (Contributed by Mario Carneiro, 2-Apr-2015.)
 |-  (arctan `  1 )  =  ( pi  /  4
 )
 
Theorembndatandm 20221 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 20222* 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 20223* 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 20224* 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 20225* 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 20226* 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 20227* 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 20228* 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 20229* 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 20230* 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 20231* 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 20232 Lemma for leibpi 20234. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( N  e.  NN0  /\  ( -.  N  =  0  /\  -.  2  ||  N ) )  ->  ( N  e.  NN  /\  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 )
 
Theoremleibpilem2 20233* 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 20234 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 12153). (2) Using leibpilem2 20233 to rewrite the series as a power series, it is the  x  =  1 special case of the Taylor series for arctan (atantayl2 20230). (3) Although we cannot directly plug  x  =  1 into atantayl2 20230, Abel's theorem (abelth2 19814) 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 20228) 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 20235 The Leibniz formula for  pi. This version of leibpi 20234 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 20236 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 20237* Bound the error term in the series of log2cnv 20236. (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 ) ) ) ) )
 
13.3.8  The Birthday Problem
 
Theoremlog2ublem1 20238 Lemma for log2ub 20241. The proof of log2ub 20241, which is simply the evaluation of log2tlbnd 20237 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 20239* Lemma for log2ub 20241. (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 20240 Lemma for log2ub 20241. 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 20241  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 20242* Lemma for birthday 20245. (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 20243* 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 20244* 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 20245* 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 )
 
13.3.9  Areas in R^2
 
Syntaxcarea 20246 Area of regions in the complex plane.
 class area
 
Definitiondf-area 20247* 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 20248* 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 20249 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 20250 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 20251* Rewrite df-area 20247 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  (
 s " { x }
 ) )  _d x )
 
Theoremareaf 20252 Area meaurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |- area : dom area
 --> ( 0 [,)  +oo )
 
Theoremareacl 20253 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 20254 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 20255* 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 )
 
13.3.10  More miscellaneous converging sequences
 
Theoremrlimcnp 20256* 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 20257* 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 20258* 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 20259* 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 20260* 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 20261). (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 20261* 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 20262* 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 20263 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( n  e.  RR+  |->  ( 1  /  ( sqr `  n ) ) )  ~~> r  0
 
Theoremrlimcxp 20264* 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 )
 
Theoremo1cxp 20265* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  0 
 <_  ( Re `  C ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B 
 ^ c  C ) )  e.  O ( 1 ) )
 
Theoremcxp2limlem 20266* A linear factor grows slower than any exponential with base greater than  1. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  ( n  e.  RR+  |->  ( n  /  ( A  ^ c  n ) ) )  ~~> r  0 )
 
Theoremcxp2lim 20267* Any power grows slower than any exponential with base greater than  1. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B ) 
 ->  ( n  e.  RR+  |->  ( ( n  ^ c  A )  /  ( B  ^ c  n ) ) )  ~~> r  0 )
 
Theoremcxploglim 20268* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( ( log `  n )  /  ( n  ^ c  A ) ) )  ~~> r  0 )
 
Theoremcxploglim2 20269* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  RR+ )  ->  ( n  e.  RR+  |->  ( ( ( log `  n )  ^ c  A )  /  ( n  ^ c  B ) ) )  ~~> r  0 )
 
Theoremdivsqrsumlem 20270* Lemma for divsqrsum 20272 and divsqrsum2 20273. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   =>    |-  ( F : RR+ --> RR 
 /\  F  e.  dom  ~~> r 
 /\  ( ( F  ~~> r  L  /\  A  e.  RR+ )  ->  ( abs `  ( ( F `  A )  -  L ) )  <_  ( 1 
 /  ( sqr `  A ) ) ) )
 
Theoremdivsqrsumf 20271* The function  F used in divsqrsum 20272 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   =>    |-  F : RR+ --> RR
 
Theoremdivsqrsum 20272* The sum  sum_ n  <_  x ( 1  /  sqr n ) is asymptotic to  2 sqr x  +  L with a finite limit  L. (In fact, this limit is  zeta ( 1  /  2 )  ~~  -u 1 period 4 6 ....) (Contributed by Mario Carneiro, 9-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   =>    |-  F  e.  dom  ~~> r
 
Theoremdivsqrsum2 20273* A bound on the distance of the sum  sum_ n  <_  x (
1  /  sqr n
) from its asymptotic value  2 sqr x  +  L. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   =>    |-  ( ( ph  /\  A  e.  RR+ )  ->  ( abs `  ( ( F `
  A )  -  L ) )  <_  ( 1  /  ( sqr `  A ) ) )
 
Theoremdivsqrsumo1 20274* The sum  sum_ n  <_  x ( 1  /  sqr n ) has the asymptotic expansion  2 sqr x  +  L  +  O
( 1  /  sqr x ), for some  L. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  F  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 1 
 /  ( sqr `  n ) )  -  (
 2  x.  ( sqr `  x ) ) ) )   &    |-  ( ph  ->  F  ~~> r  L )   =>    |-  ( ph  ->  (
 y  e.  RR+  |->  ( ( ( F `  y
 )  -  L )  x.  ( sqr `  y
 ) ) )  e.  O ( 1 ) )
 
13.3.11  Inequality of arithmetic and geometric means
 
Theoremcvxcl 20275* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D ) )  ->  ( x [,] y )  C_  D )   =>    |-  ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) ) 
 ->  ( ( T  x.  X )  +  (
 ( 1  -  T )  x.  Y ) )  e.  D )
 
Theoremscvxcvx 20276* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  (
 ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  x  < 
 y )  /\  t  e.  ( 0 (,) 1
 ) )  ->  ( F `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
 ) ) )  < 
 ( ( t  x.  ( F `  x ) )  +  (
 ( 1  -  t
 )  x.  ( F `
  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) ) 
 ->  ( F `  (
 ( T  x.  X )  +  ( (
 1  -  T )  x.  Y ) ) )  <_  ( ( T  x.  ( F `  X ) )  +  ( ( 1  -  T )  x.  ( F `  Y ) ) ) )
 
Theoremjensenlem1 20277* Lemma for jensen 20279. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  T : A --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  X : A --> D )   &    |-  ( ph  ->  0  <  (fld  gsumg  T ) )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  (
 0 [,] 1 ) ) )  ->  ( F `  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) ) )  <_  (
 ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y
 ) ) ) )   &    |-  ( ph  ->  -.  z  e.  B )   &    |-  ( ph  ->  ( B  u.  { z } )  C_  A )   &    |-  S  =  (fld  gsumg  ( T  |`  B ) )   &    |-  L  =  (fld  gsumg  ( T  |`  ( B  u.  {
 z } ) ) )   =>    |-  ( ph  ->  L  =  ( S  +  ( T `  z ) ) )
 
Theoremjensenlem2 20278* Lemma for jensen 20279. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  T : A --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  X : A --> D )   &    |-  ( ph  ->  0  <  (fld  gsumg  T ) )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  (
 0 [,] 1 ) ) )  ->  ( F `  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) ) )  <_  (
 ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y
 ) ) ) )   &    |-  ( ph  ->  -.  z  e.  B )   &    |-  ( ph  ->  ( B  u.  { z } )  C_  A )   &    |-  S  =  (fld  gsumg  ( T  |`  B ) )   &    |-  L  =  (fld  gsumg  ( T  |`  ( B  u.  {
 z } ) ) )   &    |-  ( ph  ->  S  e.  RR+ )   &    |-  ( ph  ->  ( (fld 
 gsumg  ( ( T  o F  x.  X )  |`  B ) )  /  S )  e.  D )   &    |-  ( ph  ->  ( F `  ( (fld  gsumg  ( ( T  o F  x.  X )  |`  B ) )  /  S ) )  <_  ( (fld 
 gsumg  ( ( T  o F  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )   =>    |-  ( ph  ->  ( ( (fld 
 gsumg  ( ( T  o F  x.  X )  |`  ( B  u.  { z } ) ) ) 
 /  L )  e.  D  /\  ( F `
  ( (fld  gsumg  ( ( T  o F  x.  X )  |`  ( B  u.  { z } ) ) ) 
 /  L ) ) 
 <_  ( (fld 
 gsumg  ( ( T  o F  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) ) 
 /  L ) ) )
 
Theoremjensen 20279* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  ( ph  ->  D  C_ 
 RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  ( a  e.  D  /\  b  e.  D ) )  ->  ( a [,] b
 )  C_  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  T : A --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  X : A --> D )   &    |-  ( ph  ->  0  <  (fld  gsumg  T ) )   &    |-  ( ( ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  (
 0 [,] 1 ) ) )  ->  ( F `  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) ) )  <_  (
 ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y
 ) ) ) )   =>    |-  ( ph  ->  ( (
 (fld  gsumg  ( T  o F  x.  X ) )  /  (fld  gsumg  T ) )  e.  D  /\  ( F `  (
 (fld  gsumg  ( T  o F  x.  X ) )  /  (fld  gsumg  T ) ) )  <_  ( (fld 
 gsumg  ( T  o F  x.  ( F  o.  X ) ) )  /  (fld  gsumg  T ) ) ) )
 
Theoremamgmlem 20280 Lemma for amgm 20281. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  M  =  (mulGrp ` fld )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  F : A --> RR+ )   =>    |-  ( ph  ->  (
 ( M  gsumg 
 F )  ^ c  ( 1  /  ( # `
  A ) ) )  <_  ( (fld  gsumg 
 F )  /  ( # `
  A ) ) )
 
Theoremamgm 20281 Inequality of arithmetic and geometric means. Here  ( M  gsumg  F ) calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements  F ( x ) ,  x  e.  A together), and  (fld 
gsumg  F ) calculates the group sum in the additive group (i.e. the sum of the elements). (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  M  =  (mulGrp ` fld )   =>    |-  ( ( A  e.  Fin  /\  A  =/=  (/)  /\  F : A --> ( 0 [,)  +oo ) )  ->  (
 ( M  gsumg 
 F )  ^ c  ( 1  /  ( # `
  A ) ) )  <_  ( (fld  gsumg 
 F )  /  ( # `
  A ) ) )
 
13.3.12  Euler-Mascheroni constant
 
Syntaxcem 20282 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)
 class  gamma
 
Definitiondf-em 20283 Define the Euler-Macheroni constant,  gamma  = 0.577... . This is the limit of the series  sum_ k  e.  ( 1 ... m ) ( 1  /  k
)  -  ( log `  m ), with a proof that the limit exists in emcl 20292. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  =  sum_ k  e. 
 NN  ( ( 1 
 /  k )  -  ( log `  ( 1  +  ( 1  /  k
 ) ) ) )
 
Theoremlogdifbnd 20284 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)
 |-  ( A  e.  RR+  ->  ( ( log `  ( A  +  1 )
 )  -  ( log `  A ) )  <_  ( 1  /  A ) )
 
Theorememcllem1 20285* Lemma for emcl 20292. The series  F and 
G are sequences of real numbers that approach 
gamma from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   =>    |-  ( F : NN
 --> RR  /\  G : NN
 --> RR )
 
Theorememcllem2 20286* Lemma for emcl 20292. 
F is increasing, and  G is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   =>    |-  ( N  e.  NN  ->  ( ( F `
  ( N  +  1 ) )  <_  ( F `  N ) 
 /\  ( G `  N )  <_  ( G `
  ( N  +  1 ) ) ) )
 
Theorememcllem3 20287* Lemma for emcl 20292. The function  H is the difference between  F and  G. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   =>    |-  ( N  e.  NN  ->  ( H `  N )  =  ( ( F `  N )  -  ( G `  N ) ) )
 
Theorememcllem4 20288* Lemma for emcl 20292. The difference between series  F and  G tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   =>    |-  H 
 ~~>  0
 
Theorememcllem5 20289* Lemma for emcl 20292. The partial sums of the series  T, which is used in the definition df-em 20283, is in fact the same as  G. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   &    |-  T  =  ( n  e.  NN  |->  ( ( 1 
 /  n )  -  ( log `  ( 1  +  ( 1  /  n ) ) ) ) )   =>    |-  G  =  seq  1
 (  +  ,  T )
 
Theorememcllem6 20290* Lemma for emcl 20292. By the previous lemmas,  F and  G must approach a common limit, which is  gamma by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   &    |-  T  =  ( n  e.  NN  |->  ( ( 1 
 /  n )  -  ( log `  ( 1  +  ( 1  /  n ) ) ) ) )   =>    |-  ( F  ~~>  gamma  /\  G  ~~>  gamma
 )
 
Theorememcllem7 20291* Lemma for emcl 20292 and harmonicbnd 20293. Derive bounds on  gamma as  F ( 1 ) and  G ( 1 ). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)
 |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1 
 /  m )  -  ( log `  n )
 ) )   &    |-  G  =  ( n  e.  NN  |->  (
 sum_ m  e.  (
 1 ... n ) ( 1  /  m )  -  ( log `  ( n  +  1 )
 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )   &    |-  T  =  ( n  e.  NN  |->  ( ( 1 
 /  n )  -  ( log `  ( 1  +  ( 1  /  n ) ) ) ) )   =>    |-  ( gamma  e.  (
 ( 1  -  ( log `  2 ) ) [,] 1 )  /\  F : NN --> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2
 ) ) [,] gamma ) )
 
Theorememcl 20292 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  e.  ( ( 1  -  ( log `  2
 ) ) [,] 1
 )
 
Theoremharmonicbnd 20293* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( N  e.  NN  ->  ( sum_ m  e.  (
 1 ... N ) ( 1  /  m )  -  ( log `  N ) )  e.  ( gamma [,] 1 ) )
 
Theoremharmonicbnd2 20294* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( N  e.  NN  ->  ( sum_ m  e.  (
 1 ... N ) ( 1  /  m )  -  ( log `  ( N  +  1 )
 ) )  e.  (
 ( 1  -  ( log `  2 ) ) [,] gamma ) )
 
Theorememre 20295 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |- 
 gamma  e.  RR
 
Theorememgt0 20296 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  0  <  gamma
 
Theoremharmonicbnd3 20297* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( sum_ m  e.  (
 1 ... N ) ( 1  /  m )  -  ( log `  ( N  +  1 )
 ) )  e.  (
 0 [,] gamma ) )
 
Theoremharmoniclbnd 20298* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( A  e.  RR+  ->  ( log `  A )  <_ 
 sum_ m  e.  (
 1 ... ( |_ `  A ) ) ( 1 
 /  m ) )
 
Theoremharmonicubnd 20299* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ m  e.  (
 1 ... ( |_ `  A ) ) ( 1 
 /  m )  <_  ( ( log `  A )  +  1 )
 )
 
Theoremharmonicbnd4 20300* The asymptotic behavior of  sum_ m  <_  A ,  1  /  m  =  log A  +  gamma  +  O ( 1  /  A ). (Contributed by Mario Carneiro, 14-May-2016.)
 |-  ( A  e.  RR+  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1 
 /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  ( 1  /  A ) )
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