P. 143 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14201-14300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reefiso 14201	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+ )
Theorem	reapef 14202	Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( exp ` A ) # ( exp ` B ) ) )
10.1.2  Properties of pi = 3.14159...
Theorem	pilem1 14203	Lemma for pire , pigt2lt4 and sinpi . (Contributed by Mario Carneiro, 9-May-2014.)
|- ( A e. ( RR+ i^i ( `' sin " { 0 } ) ) <-> ( A e. RR+ /\ ( sin ` A ) = 0 ) )
Theorem	cosz12 14204	Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
|- E. p e. ( 1 (,) 2 ) ( cos ` p ) = 0
Theorem	sin0pilem1 14205*	Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
|- E. p e. ( 1 (,) 2 ) ( ( cos ` p ) = 0 /\ A. x e. ( p (,) ( 2 x. p ) ) 0 < ( sin ` x ) )
Theorem	sin0pilem2 14206*	Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
|- E. q e. ( 2 (,) 4 ) ( ( sin ` q ) = 0 /\ A. x e. ( 0 (,) q ) 0 < ( sin ` x ) )
Theorem	pilem3 14207	Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
Theorem	pigt2lt4 14208	pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- ( 2 < pi /\ pi < 4 )
Theorem	sinpi 14209	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` pi ) = 0
Theorem	pire 14210	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	picn 14211	pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- pi e. CC
Theorem	pipos 14212	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- 0 < pi
Theorem	pirp 14213	pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- pi e. RR+
Theorem	negpicn 14214	-u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u pi e. CC
Theorem	sinhalfpilem 14215	Lemma for sinhalfpi 14220 and coshalfpi 14221. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( ( sin ` ( pi / 2 ) ) = 1 /\ ( cos ` ( pi / 2 ) ) = 0 )
Theorem	halfpire 14216	pi / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
|- ( pi / 2 ) e. RR
Theorem	neghalfpire 14217	-u pi / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR
Theorem	neghalfpirx 14218	-u pi / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR*
Theorem	pidiv2halves 14219	Adding pi / 2 to itself gives pi. See 2halves 9148. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
Theorem	sinhalfpi 14220	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( pi / 2 ) ) = 1
Theorem	coshalfpi 14221	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( pi / 2 ) ) = 0
Theorem	cosneghalfpi 14222	The cosine of -u pi / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
|- ( cos ` -u ( pi / 2 ) ) = 0
Theorem	efhalfpi 14223	The exponential of _i pi / 2 is _i. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( pi / 2 ) ) ) = _i
Theorem	cospi 14224	The cosine of pi is -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` pi ) = -u 1
Theorem	efipi 14225	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
Theorem	eulerid 14226	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- ( ( exp ` ( _i x. pi ) ) + 1 ) = 0
Theorem	sin2pi 14227	The sine of 2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( 2 x. pi ) ) = 0
Theorem	cos2pi 14228	The cosine of 2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( 2 x. pi ) ) = 1
Theorem	ef2pi 14229	The exponential of 2 pi _i is 1. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( 2 x. pi ) ) ) = 1
Theorem	ef2kpi 14230	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 )
Theorem	efper 14231	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 ) )
Theorem	sinperlem 14232	Lemma for sinper 14233 and cosper 14234. (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 ) )
Theorem	sinper 14233	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 ) )
Theorem	cosper 14234	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 ) )
Theorem	sin2kpi 14235	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 )
Theorem	cos2kpi 14236	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 )
Theorem	sin2pim 14237	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 ) )
Theorem	cos2pim 14238	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 ) )
Theorem	sinmpi 14239	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( sin ` ( A - pi ) ) = -u ( sin ` A ) )
Theorem	cosmpi 14240	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( cos ` ( A - pi ) ) = -u ( cos ` A ) )
Theorem	sinppi 14241	Sine of a number plus pi. (Contributed by NM, 10-Aug-2008.)
|- ( A e. CC -> ( sin ` ( A + pi ) ) = -u ( sin ` A ) )
Theorem	cosppi 14242	Cosine of a number plus pi. (Contributed by NM, 18-Aug-2008.)
|- ( A e. CC -> ( cos ` ( A + pi ) ) = -u ( cos ` A ) )
Theorem	efimpi 14243	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 ) ) )
Theorem	sinhalfpip 14244	The sine of pi / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( sin ` ( ( pi / 2 ) + A ) ) = ( cos ` A ) )
Theorem	sinhalfpim 14245	The sine of pi / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( sin ` ( ( pi / 2 ) - A ) ) = ( cos ` A ) )
Theorem	coshalfpip 14246	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 ) )
Theorem	coshalfpim 14247	The cosine of pi / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( cos ` ( ( pi / 2 ) - A ) ) = ( sin ` A ) )
Theorem	ptolemy 14248	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 11752, 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 ) ) ) )
Theorem	sincosq1lem 14249	Lemma for sincosq1sgn 14250. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( ( A e. RR /\ 0 < A /\ A < ( pi / 2 ) ) -> 0 < ( sin ` A ) )
Theorem	sincosq1sgn 14250	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 ) ) )
Theorem	sincosq2sgn 14251	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 ) )
Theorem	sincosq3sgn 14252	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 ) )
Theorem	sincosq4sgn 14253	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 ) ) )
Theorem	sinq12gt0 14254	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 ) )
Theorem	sinq34lt0t 14255	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 )
Theorem	cosq14gt0 14256	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 ) )
Theorem	cosq23lt0 14257	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 )
Theorem	coseq0q4123 14258	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 ) ) )
Theorem	coseq00topi 14259	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 ) ) )
Theorem	coseq0negpitopi 14260	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 ) } ) )
Theorem	tanrpcl 14261	Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
|- ( A e. ( 0 (,) ( pi / 2 ) ) -> ( tan ` A ) e. RR+ )
Theorem	tangtx 14262	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 ) )
Theorem	sincosq1eq 14263	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 ) ) ) )
Theorem	sincos4thpi 14264	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 ) ) )
Theorem	tan4thpi 14265	The tangent of pi / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
|- ( tan ` ( pi / 4 ) ) = 1
Theorem	sincos6thpi 14266	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 ) )
Theorem	sincos3rdpi 14267	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 ) )
Theorem	pigt3 14268	pi is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
|- 3 < pi
Theorem	pige3 14269	pi is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
|- 3 <_ pi
Theorem	abssinper 14270	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 ) ) )
Theorem	sinkpi 14271	The sine of an integer multiple of pi is 0. (Contributed by NM, 11-Aug-2008.)
|- ( K e. ZZ -> ( sin ` ( K x. pi ) ) = 0 )
Theorem	coskpi 14272	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 )
Theorem	cosordlem 14273	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 ) )
Theorem	cosq34lt1 14274	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 )
Theorem	cos02pilt1 14275	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 )
Theorem	cos0pilt1 14276	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 ) )
Theorem	cos11 14277	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 ) ) )
Theorem	ioocosf1o 14278	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 )
Theorem	negpitopissre 14279	The interval ( -u pi (,] pi ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
|- ( -u pi (,] pi ) C_ RR
10.1.3  The natural logarithm on complex numbers
Syntax	clog 14280	Extend class notation with the natural logarithm function on complex numbers.
class log
Syntax	ccxp 14281	Extend class notation with the complex power function.
class ^c
Definition	df-relog 14282	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 )
Definition	df-rpcxp 14283*	Define the power function on complex numbers. Because df-relog 14282 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 ) ) ) )
Theorem	dfrelog 14284	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 )
Theorem	relogf1o 14285	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
Theorem	relogcl 14286	Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( A e. RR+ -> ( log ` A ) e. RR )
Theorem	reeflog 14287	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( A e. RR+ -> ( exp ` ( log ` A ) ) = A )
Theorem	relogef 14288	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( A e. RR -> ( log ` ( exp ` A ) ) = A )
Theorem	relogeftb 14289	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 ) )
Theorem	log1 14290	The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( log ` 1 ) = 0
Theorem	loge 14291	The natural logarithm of _e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( log ` _e ) = 1
Theorem	relogoprlem 14292	Lemma for relogmul 14293 and relogdiv 14294. 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 ) ) )
Theorem	relogmul 14293	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 ) ) )
Theorem	relogdiv 14294	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 ) ) )
Theorem	reexplog 14295	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 ) ) ) )
Theorem	relogexp 14296	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 ) ) )
Theorem	relogiso 14297	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 )
Theorem	logltb 14298	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 ) ) )
Theorem	logleb 14299	Natural logarithm preserves <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A <_ B <-> ( log ` A ) <_ ( log ` B ) ) )
Theorem	logrpap0b 14300	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 ) )