HomeHome Intuitionistic Logic Explorer
Theorem List (p. 153 of 160)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempicn 15201  pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  pi  e.  CC
 
Theorempipos 15202  pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  0  <  pi
 
Theorempirp 15203  pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  pi  e.  RR+
 
Theoremnegpicn 15204  -u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u pi  e.  CC
 
Theoremsinhalfpilem 15205 Lemma for sinhalfpi 15210 and coshalfpi 15211. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( ( sin `  ( pi  /  2 ) )  =  1  /\  ( cos `  ( pi  / 
 2 ) )  =  0 )
 
Theoremhalfpire 15206  pi  /  2 is real. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( pi  /  2
 )  e.  RR
 
Theoremneghalfpire 15207  -u pi  / 
2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u ( pi  /  2
 )  e.  RR
 
Theoremneghalfpirx 15208  -u pi  / 
2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u ( pi  /  2
 )  e.  RR*
 
Theorempidiv2halves 15209 Adding  pi  /  2 to itself gives  pi. See 2halves 9265. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ( pi  / 
 2 )  +  ( pi  /  2 ) )  =  pi
 
Theoremsinhalfpi 15210 The sine of  pi  /  2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  ( pi  /  2 ) )  =  1
 
Theoremcoshalfpi 15211 The cosine of  pi  /  2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  ( pi  /  2 ) )  =  0
 
Theoremcosneghalfpi 15212 The cosine of  -u pi  /  2 is zero. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( cos `  -u ( pi  /  2 ) )  =  0
 
Theoremefhalfpi 15213 The exponential of  _i pi  /  2 is  _i. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( pi  / 
 2 ) ) )  =  _i
 
Theoremcospi 15214 The cosine of  pi is  -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  pi )  =  -u 1
 
Theoremefipi 15215 The exponential of  _i  x.  pi is  -u 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
 |-  ( exp `  ( _i  x.  pi ) )  =  -u 1
 
Theoremeulerid 15216 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( ( exp `  ( _i  x.  pi ) )  +  1 )  =  0
 
Theoremsin2pi 15217 The sine of  2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  (
 2  x.  pi ) )  =  0
 
Theoremcos2pi 15218 The cosine of  2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( cos `  (
 2  x.  pi ) )  =  1
 
Theoremef2pi 15219 The exponential of  2 pi _i is  1. (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( exp `  ( _i  x.  ( 2  x.  pi ) ) )  =  1
 
Theoremef2kpi 15220 If  K is an integer, then 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 15221 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 15222 Lemma for sinper 15223 and cosper 15224. (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 15223 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 15224 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 15225 If  K is an integer, then 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 15226 If  K is an integer, then 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 15227 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 15228 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 15229 Sine of a number less  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( sin `  ( A  -  pi ) )  =  -u ( sin `  A ) )
 
Theoremcosmpi 15230 Cosine of a number less  pi. (Contributed by Paul Chapman, 15-Mar-2008.)
 |-  ( A  e.  CC  ->  ( cos `  ( A  -  pi ) )  =  -u ( cos `  A ) )
 
Theoremsinppi 15231 Sine of a number plus  pi. (Contributed by NM, 10-Aug-2008.)
 |-  ( A  e.  CC  ->  ( sin `  ( A  +  pi )
 )  =  -u ( sin `  A ) )
 
Theoremcosppi 15232 Cosine of a number plus  pi. (Contributed by NM, 18-Aug-2008.)
 |-  ( A  e.  CC  ->  ( cos `  ( A  +  pi )
 )  =  -u ( cos `  A ) )
 
Theoremefimpi 15233 The exponential function at  _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 15234 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 15235 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 15236 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 15237 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 15238 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 11997, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. This is Metamath 100 proof #95. (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 15239 Lemma for sincosq1sgn 15240. (Contributed by Paul Chapman, 24-Jan-2008.)
 |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  ( pi 
 /  2 ) ) 
 ->  0  <  ( sin `  A ) )
 
Theoremsincosq1sgn 15240 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 15241 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 15242 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 15243 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 ) ) )
 
Theoremsinq12gt0 15244 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 ) )
 
Theoremsinq34lt0t 15245 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 15246 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 ) )
 
Theoremcosq23lt0 15247 The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)
 |-  ( A  e.  (
 ( pi  /  2
 ) (,) ( 3  x.  ( pi  /  2
 ) ) )  ->  ( cos `  A )  <  0 )
 
Theoremcoseq0q4123 15248 Location of the zeroes of cosine in  ( -u (
pi  /  2 ) (,) ( 3  x.  ( pi  /  2
) ) ). (Contributed by Jim Kingdon, 14-Mar-2024.)
 |-  ( A  e.  ( -u ( pi  /  2
 ) (,) ( 3  x.  ( pi  /  2
 ) ) )  ->  ( ( cos `  A )  =  0  <->  A  =  ( pi  /  2 ) ) )
 
Theoremcoseq00topi 15249 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 15250 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 15251 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  (
 0 (,) ( pi  / 
 2 ) )  ->  ( tan `  A )  e.  RR+ )
 
Theoremtangtx 15252 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 ) )
 
Theoremsincosq1eq 15253 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 15254 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 15255 The tangent of  pi  /  4. (Contributed by Mario Carneiro, 5-Apr-2015.)
 |-  ( tan `  ( pi  /  4 ) )  =  1
 
Theoremsincos6thpi 15256 The sine and cosine of  pi  /  6. (Contributed by Paul Chapman, 25-Jan-2008.) (Revised by Wolf Lammen, 24-Sep-2020.)
 |-  ( ( sin `  ( pi  /  6 ) )  =  ( 1  / 
 2 )  /\  ( cos `  ( pi  / 
 6 ) )  =  ( ( sqr `  3
 )  /  2 )
 )
 
Theoremsincos3rdpi 15257 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 ) )
 
Theorempigt3 15258  pi is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
 |-  3  <  pi
 
Theorempige3 15259  pi is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  3  <_  pi
 
Theoremabssinper 15260 The absolute value of sine has period  pi. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( abs `  ( sin `  ( A  +  ( K  x.  pi ) ) ) )  =  ( abs `  ( sin `  A ) ) )
 
Theoremsinkpi 15261 The sine of an integer multiple of 
pi is 0. (Contributed by NM, 11-Aug-2008.)
 |-  ( K  e.  ZZ  ->  ( sin `  ( K  x.  pi ) )  =  0 )
 
Theoremcoskpi 15262 The absolute value of the cosine of an integer multiple of  pi is 1. (Contributed by NM, 19-Aug-2008.)
 |-  ( K  e.  ZZ  ->  ( abs `  ( cos `  ( K  x.  pi ) ) )  =  1 )
 
Theoremcosordlem 15263 Cosine is decreasing over the closed interval from  0 to  pi. (Contributed by Mario Carneiro, 10-May-2014.)
 |-  ( ph  ->  A  e.  ( 0 [,] pi ) )   &    |-  ( ph  ->  B  e.  ( 0 [,]
 pi ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( cos `  B )  < 
 ( cos `  A )
 )
 
Theoremcosq34lt1 15264 Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.)
 |-  ( A  e.  ( pi [,) ( 2  x.  pi ) )  ->  ( cos `  A )  <  1 )
 
Theoremcos02pilt1 15265 Cosine is less than one between zero and  2  x.  pi. (Contributed by Jim Kingdon, 19-Mar-2024.)
 |-  ( A  e.  (
 0 (,) ( 2  x.  pi ) )  ->  ( cos `  A )  <  1 )
 
Theoremcos0pilt1 15266 Cosine is between minus one and one on the open interval between zero and  pi. (Contributed by Jim Kingdon, 7-May-2024.)
 |-  ( A  e.  (
 0 (,) pi )  ->  ( cos `  A )  e.  ( -u 1 (,) 1
 ) )
 
Theoremcos11 15267 Cosine is one-to-one over the closed interval from  0 to  pi. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)
 |-  ( ( A  e.  ( 0 [,] pi )  /\  B  e.  (
 0 [,] pi ) ) 
 ->  ( A  =  B  <->  ( cos `  A )  =  ( cos `  B ) ) )
 
Theoremioocosf1o 15268 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.)
 |-  ( cos  |`  ( 0 (,) pi ) ) : ( 0 (,)
 pi ) -1-1-onto-> ( -u 1 (,) 1
 )
 
Theoremnegpitopissre 15269 The interval  ( -u pi (,] pi ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( -u pi (,] pi )  C_  RR
 
11.2.3  The natural logarithm on complex numbers
 
Syntaxclog 15270 Extend class notation with the natural logarithm function on complex numbers.
 class  log
 
Syntaxccxp 15271 Extend class notation with the complex power function.
 class  ^c
 
Definitiondf-relog 15272 Define the natural logarithm function. Defining the logarithm on complex numbers is similar to square root - there are ways to define it but they tend to make use of excluded middle. Therefore, we merely define logarithms on positive reals. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Jim Kingdon, 14-May-2024.)
 |- 
 log  =  `' ( exp  |`  RR )
 
Definitiondf-rpcxp 15273* Define the power function on complex numbers. Because df-relog 15272 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.)
 |- 
 ^c  =  ( x  e.  RR+ ,  y  e.  CC  |->  ( exp `  (
 y  x.  ( log `  x ) ) ) )
 
Theoremdfrelog 15274 The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
 
Theoremrelogf1o 15275 The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( log  |`  RR+ ) : RR+
 -1-1-onto-> RR
 
Theoremrelogcl 15276 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
 
Theoremreeflog 15277 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR+  ->  ( exp `  ( log `  A ) )  =  A )
 
Theoremrelogef 15278 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( A  e.  RR  ->  ( log `  ( exp `  A ) )  =  A )
 
Theoremrelogeftb 15279 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR )  ->  ( ( log `  A )  =  B  <->  ( exp `  B )  =  A )
 )
 
Theoremlog1 15280 The natural logarithm of  1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( log `  1
 )  =  0
 
Theoremloge 15281 The natural logarithm of  _e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( log `  _e )  =  1
 
Theoremrelogoprlem 15282 Lemma for relogmul 15283 and relogdiv 15284. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2"). (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( ( log `  A )  e.  CC  /\  ( log `  B )  e.  CC )  ->  ( exp `  (
 ( log `  A ) F ( log `  B ) ) )  =  ( ( exp `  ( log `  A ) ) G ( exp `  ( log `  B ) ) ) )   &    |-  ( ( ( log `  A )  e.  RR  /\  ( log `  B )  e.  RR )  ->  ( ( log `  A ) F ( log `  B )
 )  e.  RR )   =>    |-  (
 ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( log `  ( A G B ) )  =  ( ( log `  A ) F ( log `  B )
 ) )
 
Theoremrelogmul 15283 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( log `  ( A  x.  B ) )  =  ( ( log `  A )  +  ( log `  B ) ) )
 
Theoremrelogdiv 15284 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( log `  ( A  /  B ) )  =  ( ( log `  A )  -  ( log `  B ) ) )
 
Theoremreexplog 15285 Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  =  ( exp `  ( N  x.  ( log `  A ) ) ) )
 
Theoremrelogexp 15286 The natural logarithm of positive 
A raised to an integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and integer powers  N. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( log `  ( A ^ N ) )  =  ( N  x.  ( log `  A )
 ) )
 
Theoremrelogiso 15287 The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( log  |`  RR+ )  Isom  <  ,  <  ( RR+
 ,  RR )
 
Theoremlogltb 15288 The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <  B  <->  ( log `  A )  <  ( log `  B ) ) )
 
Theoremlogleb 15289 Natural logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
 
Theoremlogrpap0b 15290 The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.)
 |-  ( A  e.  RR+  ->  ( A #  1  <->  ( log `  A ) #  0 ) )
 
Theoremlogrpap0 15291 The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.)
 |-  ( ( A  e.  RR+  /\  A #  1 )  ->  ( log `  A ) #  0 )
 
Theoremlogrpap0d 15292 Deduction form of logrpap0 15291. (Contributed by Jim Kingdon, 3-Jul-2024.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  A #  1 )   =>    |-  ( ph  ->  ( log `  A ) #  0 )
 
Theoremrplogcl 15293 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  ( log `  A )  e.  RR+ )
 
Theoremlogge0 15294 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  0  <_  ( log `  A ) )
 
Theoremlogdivlti 15295 The  log x  /  x function is strictly decreasing on the reals greater than  _e. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  _e  <_  A )  /\  A  <  B )  ->  ( ( log `  B )  /  B )  <  ( ( log `  A )  /  A ) )
 
Theoremrelogcld 15296 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( log `  A )  e. 
 RR )
 
Theoremreeflogd 15297 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( exp `  ( log `  A ) )  =  A )
 
Theoremrelogmuld 15298 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( log `  ( A  x.  B ) )  =  ( ( log `  A )  +  ( log `  B ) ) )
 
Theoremrelogdivd 15299 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( log `  ( A  /  B ) )  =  ( ( log `  A )  -  ( log `  B ) ) )
 
Theoremlogled 15300 Natural logarithm preserves  <_. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( log `  A )  <_  ( log `  B ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-15956
  Copyright terms: Public domain < Previous  Next >