P. 155 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15401-15500   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-rpcxp 15401*	Define the power function on complex numbers. Because df-relog 15400 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 15402	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 15403	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 15404	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 15405	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 15406	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 15407	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 15408	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 15409	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 15410	Lemma for relogmul 15411 and relogdiv 15412. 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 15411	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 15412	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 15413	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 15414	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 15415	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 15416	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 15417	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 15418	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 ) )
Theorem	logrpap0 15419	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 )
Theorem	logrpap0d 15420	Deduction form of logrpap0 15419. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( log ` A ) # 0 )
Theorem	rplogcl 15421	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+ )
Theorem	logge0 15422	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 ) )
Theorem	logdivlti 15423	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 ) )
Theorem	relogcld 15424	Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( log ` A ) e. RR )
Theorem	reeflogd 15425	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 )
Theorem	relogmuld 15426	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 ) ) )
Theorem	relogdivd 15427	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 ) ) )
Theorem	logled 15428	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 ) ) )
Theorem	relogefd 15429	Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( log ` ( exp ` A ) ) = A )
Theorem	rplogcld 15430	Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 < A )   =>    |- ( ph -> ( log ` A ) e. RR+ )
Theorem	logge0d 15431	The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 <_ A )   =>    |- ( ph -> 0 <_ ( log ` A ) )
Theorem	logge0b 15432	The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( 0 <_ ( log ` A ) <-> 1 <_ A ) )
Theorem	loggt0b 15433	The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( 0 < ( log ` A ) <-> 1 < A ) )
Theorem	logle1b 15434	The logarithm of a number is less than or equal to 1 iff the number is less than or equal to Euler's constant. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( ( log ` A ) <_ 1 <-> A <_ _e ) )
Theorem	loglt1b 15435	The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( ( log ` A ) < 1 <-> A < _e ) )
Theorem	rpcxpef 15436	Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
Theorem	cxpexprp 15437	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
Theorem	cxpexpnn 15438	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
Theorem	logcxp 15439	Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR ) -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
Theorem	rpcxp0 15440	Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( A e. RR+ -> ( A ^c 0 ) = 1 )
Theorem	rpcxp1 15441	Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. RR+ -> ( A ^c 1 ) = A )
Theorem	1cxp 15442	Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. CC -> ( 1 ^c A ) = 1 )
Theorem	ecxp 15443	Write the exponential function as an exponent to the power _e. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. CC -> ( _e ^c A ) = ( exp ` A ) )
Theorem	rpcncxpcl 15444	Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
Theorem	rpcxpcl 15445	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
Theorem	cxpap0 15446	Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) # 0 )
Theorem	rpcxpadd 15447	Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )
Theorem	rpcxpp1 15448	Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B + 1 ) ) = ( ( A ^c B ) x. A ) )
Theorem	rpcxpneg 15449	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c -u B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpcxpsub 15450	Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B - C ) ) = ( ( A ^c B ) / ( A ^c C ) ) )
Theorem	rpmulcxp 15451	Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) )
Theorem	cxprec 15452	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpdivcxp 15453	Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A / B ) ^c C ) = ( ( A ^c C ) / ( B ^c C ) ) )
Theorem	cxpmul 15454	Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) )
Theorem	rpcxpmul2 15455	Product of exponents law for complex exponentiation. Variation on cxpmul 15454 with more general conditions on A and B when C is a nonnegative integer. (Contributed by Mario Carneiro, 9-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )
Theorem	rpcxproot 15456	The complex power function allows us to write n-th roots via the idiom A ^c ( 1 / N ). (Contributed by Mario Carneiro, 6-May-2015.)
|- ( ( A e. RR+ /\ N e. NN ) -> ( ( A ^c ( 1 / N ) ) ^ N ) = A )
Theorem	abscxp 15457	Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( abs ` ( A ^c B ) ) = ( A ^c ( Re ` B ) ) )
Theorem	cxplt 15458	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )
Theorem	cxple 15459	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B <_ C <-> ( A ^c B ) <_ ( A ^c C ) ) )
Theorem	rpcxple2 15460	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
Theorem	rpcxplt2 15461	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( A ^c C ) < ( B ^c C ) ) )
Theorem	cxplt3 15462	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )
Theorem	cxple3 15463	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )
Theorem	rpcxpsqrt 15464	The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other n-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.)
|- ( A e. RR+ -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
Theorem	logsqrt 15465	Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( A e. RR+ -> ( log ` ( sqr ` A ) ) = ( ( log ` A ) / 2 ) )
Theorem	rpcxp0d 15466	Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( A ^c 0 ) = 1 )
Theorem	rpcxp1d 15467	Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( A ^c 1 ) = A )
Theorem	1cxpd 15468	Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 ^c A ) = 1 )
Theorem	rpcncxpcld 15469	Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A ^c B ) e. CC )
Theorem	cxpltd 15470	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 < A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )
Theorem	cxpled 15471	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 < A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B <_ C <-> ( A ^c B ) <_ ( A ^c C ) ) )
Theorem	rpcxpsqrtth 15472	Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11412. (Contributed by AV, 23-Dec-2022.)
|- ( A e. RR+ -> ( ( sqr ` A ) ^c 2 ) = A )
Theorem	cxprecd 15473	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpcxpmul2d 15474	Product of exponents law for complex exponentiation. Variation on cxpmul 15454 with more general conditions on A and B when C is a nonnegative integer. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. NN0 )   =>    |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )
Theorem	rpcxpcld 15475	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A ^c B ) e. RR+ )
Theorem	logcxpd 15476	Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
Theorem	cxplt3d 15477	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )
Theorem	cxple3d 15478	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )
Theorem	cxpmuld 15479	Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) )
Theorem	cxpcom 15480	Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. RR+ /\ B e. RR /\ C e. RR ) -> ( ( A ^c B ) ^c C ) = ( ( A ^c C ) ^c B ) )
Theorem	apcxp2 15481	Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( A ^c B ) # ( A ^c C ) ) )
Theorem	rpabscxpbnd 15482	Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   &    |- ( ph -> 0 < ( Re ` B ) )   &    |- ( ph -> M e. RR )   &    |- ( ph -> ( abs ` A ) <_ M )   =>    |- ( ph -> ( abs ` ( A ^c B ) ) <_ ( ( M ^c ( Re ` B ) ) x. ( exp ` ( ( abs ` B ) x. pi ) ) ) )
Theorem	ltexp2 15483	Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
|- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( M < N <-> ( A ^ M ) < ( A ^ N ) ) )
Theorem	ltexp2d 15484	Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> 1 < A )   =>    |- ( ph -> ( M < N <-> ( A ^ M ) < ( A ^ N ) ) )
11.2.4  Logarithms to an arbitrary base Define "log using an arbitrary base" function and then prove some of its properties. As with df-relog 15400 this is for real logarithms rather than complex logarithms. Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log". There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations): ( B logb X ) where B is the base and X is the argument of the logarithm function. An alternative would be to support the notational form ( ( logb ` B ) ` X ); that looks a little more like traditional notation.
Syntax	clogb 15485	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 15486*	Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as ( B logb X ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition will only be useful where x is a positive real apart from one and where y is a positive real, so the choice of ( CC \ { 0 , 1 } ) and ( CC \ { 0 } ) is somewhat arbitrary (we adopt the definition used in set.mm). (Contributed by David A. Wheeler, 21-Jan-2017.)
|- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
Theorem	rplogbval 15487	Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
Theorem	rplogbcl 15488	General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) e. RR )
Theorem	rplogbid1 15489	General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.)
|- ( ( A e. RR+ /\ A # 1 ) -> ( A logb A ) = 1 )
Theorem	rplogb1 15490	The logarithm of 1 to an arbitrary base B is 0. Property 1(b) of [Cohen4] p. 361. See log1 15408. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. RR+ /\ B # 1 ) -> ( B logb 1 ) = 0 )
Theorem	rpelogb 15491	The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using _e as the base in logb is the same as log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
|- ( A e. RR+ -> ( _e logb A ) = ( log ` A ) )
Theorem	rplogbchbase 15492	Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( A logb X ) = ( ( B logb X ) / ( B logb A ) ) )
Theorem	relogbval 15493	Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
Theorem	relogbzcl 15494	Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) e. RR )
Theorem	rplogbreexp 15495	Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) )
Theorem	rplogbzexp 15496	Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ N e. ZZ ) -> ( B logb ( C ^ N ) ) = ( N x. ( B logb C ) ) )
Theorem	rprelogbmul 15497	The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. C ) ) = ( ( B logb A ) + ( B logb C ) ) )
Theorem	rprelogbmulexp 15498	The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( B logb ( A x. ( C ^c E ) ) ) = ( ( B logb A ) + ( E x. ( B logb C ) ) ) )
Theorem	rprelogbdiv 15499	The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( ( B logb A ) - ( B logb C ) ) )
Theorem	relogbexpap 15500	Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
|- ( ( B e. RR+ /\ B # 1 /\ M e. ZZ ) -> ( B logb ( B ^ M ) ) = M )