P. 160 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15901-16000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cxprec 15901	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 15902	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 15903	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 15904	Product of exponents law for complex exponentiation. Variation on cxpmul 15903 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 15905	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 15906	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 15907	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 15908	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 15909	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 15910	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 15911	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 15912	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 15913	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 15914	Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( A e. RR+ -> ( log ` ( sqr ` A ) ) = ( ( log ` A ) / 2 ) )
Theorem	rpcxp0d 15915	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 15916	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 15917	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 15918	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 15919	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 15920	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 15921	Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11741. (Contributed by AV, 23-Dec-2022.)
|- ( A e. RR+ -> ( ( sqr ` A ) ^c 2 ) = A )
Theorem	cxprecd 15922	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 15923	Product of exponents law for complex exponentiation. Variation on cxpmul 15903 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 15924	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 15925	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 15926	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 15927	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 15928	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 15929	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 15930	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 15931	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 15932	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 15933	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 15849 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 15934	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 15935*	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 15936	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 15937	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 15938	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 15939	The logarithm of 1 to an arbitrary base B is 0. Property 1(b) of [Cohen4] p. 361. See log1 15857. (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 15940	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 15941	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 15942	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 15943	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 15944	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 15945	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 15946	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 15947	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 15948	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 15949	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 )
Theorem	nnlogbexp 15950	Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ M e. ZZ ) -> ( B logb ( B ^ M ) ) = M )
Theorem	logbrec 15951	Logarithm of a reciprocal changes sign. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = -u ( B logb A ) )
Theorem	logbleb 15952	The general logarithm function is monotone/increasing. See logleb 15866. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ /\ Y e. RR+ ) -> ( X <_ Y <-> ( B logb X ) <_ ( B logb Y ) ) )
Theorem	logblt 15953	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 15865. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ /\ Y e. RR+ ) -> ( X < Y <-> ( B logb X ) < ( B logb Y ) ) )
Theorem	rplogbcxp 15954	Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR ) -> ( B logb ( B ^c X ) ) = X )
Theorem	rpcxplogb 15955	Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B ^c ( B logb X ) ) = X )
Theorem	relogbcxpbap 15956	The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ X e. RR+ /\ Y e. RR ) -> ( ( B logb X ) = Y <-> ( B ^c Y ) = X ) )
Theorem	logbgt0b 15957	The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. RR+ /\ ( B e. RR+ /\ 1 < B ) ) -> ( 0 < ( B logb A ) <-> 1 < A ) )
Theorem	logbgcd1irr 15958	The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, ( 2 logb 9 ) e. ( RR \ QQ ). (Contributed by AV, 29-Dec-2022.)
|- ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) /\ ( X gcd B ) = 1 ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	logbgcd1irraplemexp 15959	Lemma for logbgcd1irrap 15961. Apartness of X ^ N and B ^ M. (Contributed by Jim Kingdon, 11-Jul-2024.)
|- ( ph -> X e. ( ZZ>= ` 2 ) )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> ( X gcd B ) = 1 )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( X ^ N ) # ( B ^ M ) )
Theorem	logbgcd1irraplemap 15960	Lemma for logbgcd1irrap 15961. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
|- ( ph -> X e. ( ZZ>= ` 2 ) )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> ( X gcd B ) = 1 )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( B logb X ) # ( M / N ) )
Theorem	logbgcd1irrap 15961	The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, ( 2 logb 9 ) # Q where Q is rational. (Contributed by AV, 29-Dec-2022.)
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( B logb X ) # Q )
Theorem	2logb9irr 15962	Example for logbgcd1irr 15958. The logarithm of nine to base two is not rational. Also see 2logb9irrap 15968 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.)
|- ( 2 logb 9 ) e. ( RR \ QQ )
Theorem	logbprmirr 15963	The logarithm of a prime to a different prime base is not rational. For example, ( 2 logb 3 ) e. ( RR \ QQ ) (see 2logb3irr 15964). (Contributed by AV, 31-Dec-2022.)
|- ( ( X e. Prime /\ B e. Prime /\ X =/= B ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	2logb3irr 15964	Example for logbprmirr 15963. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
|- ( 2 logb 3 ) e. ( RR \ QQ )
Theorem	2logb9irrALT 15965	Alternate proof of 2logb9irr 15962: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( 2 logb 9 ) e. ( RR \ QQ )
Theorem	sqrt2cxp2logb9e3 15966	The square root of two to the power of the logarithm of nine to base two is three. ( sqr ` 2 ) and ( 2 logb 9 ) are not rational (see sqrt2irr0 12886 resp. 2logb9irr 15962), satisfying the statement in 2irrexpq 15967. (Contributed by AV, 29-Dec-2022.)
|- ( ( sqr ` 2 ) ^c ( 2 logb 9 ) ) = 3
Theorem	2irrexpq 15967*	There exist real numbers a and b which are not rational such that ( a ^ b ) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named non-rational numbers ( sqr ` 2 ) and ( 2 logb 9 ), see sqrt2irr0 12886, 2logb9irr 15962 and sqrt2cxp2logb9e3 15966. Therefore, this proof is acceptable/usable in intuitionistic logic. For a theorem which is the same but proves that a and b are irrational (in the sense of being apart from any rational number), see 2irrexpqap 15969. (Contributed by AV, 23-Dec-2022.)
|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ
Theorem	2logb9irrap 15968	Example for logbgcd1irrap 15961. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.)
|- ( Q e. QQ -> ( 2 logb 9 ) # Q )
Theorem	2irrexpqap 15969*	There exist real numbers a and b which are irrational (in the sense of being apart from any rational number) such that ( a ^ b ) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers ( sqr ` 2 ) and ( 2 logb 9 ), see sqrt2irrap 12902, 2logb9irrap 15968 and sqrt2cxp2logb9e3 15966. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
|- E. a e. RR E. b e. RR ( A. p e. QQ a # p /\ A. q e. QQ b # q /\ ( a ^c b ) e. QQ )
11.2.5  Quartic binomial expansion
Theorem	binom4 15970	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12195, but it turns out to be just as much work to put it into this form after clearing all the sums and calculating binomial coefficients.) (Contributed by Mario Carneiro, 6-May-2015.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 4 ) = ( ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) )
11.3  Pell equations
11.3.1  Pell equations 1: A nontrivial solution always exists
Theorem	pellexlem1 15971	Lemma for pellex . Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.)
|- ( ( ( D e. NN /\ A e. NN /\ B e. NN ) /\ -. ( sqr ` D ) e. QQ ) -> ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) =/= 0 )
Theorem	pellexlem2 15972	Lemma for pellex . Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)
|- ( ( ( D e. NN /\ A e. NN /\ B e. NN ) /\ ( abs ` ( ( A / B ) - ( sqr ` D ) ) ) < ( B ^ -u 2 ) ) -> ( abs ` ( ( A ^ 2 ) - ( D x. ( B ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) )
Theorem	pellexlem3 15973*	Lemma for pellex . To each good rational approximation of ( sqr ` D ), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
|- ( ( D e. NN /\ -. ( sqr ` D ) e. QQ ) -> { x e. QQ | ( 0 < x /\ ( abs ` ( x - ( sqr ` D ) ) ) < ( (denom ` x ) ^ -u 2 ) ) } ~<_ { <. y , z >. | ( ( y e. NN /\ z e. NN ) /\ ( ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) =/= 0 /\ ( abs ` ( ( y ^ 2 ) - ( D x. ( z ^ 2 ) ) ) ) < ( 1 + ( 2 x. ( sqr ` D ) ) ) ) ) } )
11.4  Basic number theory
11.4.1  Wilson's theorem
Theorem	wilthlem1 15974	The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ZZ / P ZZ are 1 and -u 1 == P - 1. (Note that from prmdiveq 12958, ( N ^ ( P - 2 ) ) mod P is the modular inverse of N in ZZ / P ZZ. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )
11.4.2  Number-theoretical functions
Syntax	csgm 15975	Extend class notation with the divisor function.
class sigma
Definition	df-sgm 15976*	Define the sum of positive divisors function ( x sigma n ), which is the sum of the xth powers of the positive integer divisors of n, see definition in [ApostolNT] p. 38. For x = 0, ( x sigma n ) counts the number of divisors of n, i.e. ( 0 sigma n ) is the divisor function, see remark in [ApostolNT] p. 38. (Contributed by Mario Carneiro, 22-Sep-2014.)
|- sigma = ( x e. CC , n e. NN |-> sum_ k e. { p e. NN | p || n } ( k ^c x ) )
Theorem	sgmval 15977*	The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
|- ( ( A e. CC /\ B e. NN ) -> ( A sigma B ) = sum_ k e. { p e. NN | p || B } ( k ^c A ) )
Theorem	sgmval2 15978*	The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A sigma B ) = sum_ k e. { p e. NN | p || B } ( k ^ A ) )
Theorem	0sgm 15979*	The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)
|- ( A e. NN -> ( 0 sigma A ) = (♯ ` { p e. NN | p || A } ) )
Theorem	sgmf 15980	The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
|- sigma : ( CC X. NN ) --> CC
Theorem	sgmcl 15981	Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
|- ( ( A e. CC /\ B e. NN ) -> ( A sigma B ) e. CC )
Theorem	sgmnncl 15982	Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
|- ( ( A e. NN0 /\ B e. NN ) -> ( A sigma B ) e. NN )
Theorem	dvdsppwf1o 15983*	A bijection between the divisors of a prime power and the integers less than or equal to the exponent. (Contributed by Mario Carneiro, 5-May-2016.)
|- F = ( n e. ( 0 ... A ) |-> ( P ^ n ) )   =>    |- ( ( P e. Prime /\ A e. NN0 ) -> F : ( 0 ... A ) -1-1-onto-> { x e. NN | x || ( P ^ A ) } )
Theorem	mpodvdsmulf1o 15984*	If M and N are two coprime integers, multiplication forms a bijection from the set of pairs <. j , k >. where j || M and k || N, to the set of divisors of M x. N. (Contributed by GG, 18-Apr-2025.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. NN )   &    |- ( ph -> ( M gcd N ) = 1 )   &    |- X = { x e. NN | x || M }   &    |- Y = { x e. NN | x || N }   &    |- Z = { x e. NN | x || ( M x. N ) }   =>    |- ( ph -> ( ( x e. CC , y e. CC |-> ( x x. y ) ) |` ( X X. Y ) ) : ( X X. Y ) -1-1-onto-> Z )
Theorem	fsumdvdsmul 15985*	Product of two divisor sums. (This is also the main part of the proof that " sum_ k || N F ( k ) is a multiplicative function if F is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 8266. (Revised by GG, 18-Apr-2025.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. NN )   &    |- ( ph -> ( M gcd N ) = 1 )   &    |- X = { x e. NN | x || M }   &    |- Y = { x e. NN | x || N }   &    |- Z = { x e. NN | x || ( M x. N ) }   &    |- ( ( ph /\ j e. X ) -> A e. CC )   &    |- ( ( ph /\ k e. Y ) -> B e. CC )   &    |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> ( A x. B ) = D )   &    |- ( i = ( j x. k ) -> C = D )   =>    |- ( ph -> ( sum_ j e. X A x. sum_ k e. Y B ) = sum_ i e. Z C )
Theorem	sgmppw 15986*	The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)
|- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( A sigma ( P ^ N ) ) = sum_ k e. ( 0 ... N ) ( ( P ^c A ) ^ k ) )
Theorem	0sgmppw 15987	A prime power P ^ K has K + 1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)
|- ( ( P e. Prime /\ K e. NN0 ) -> ( 0 sigma ( P ^ K ) ) = ( K + 1 ) )
Theorem	1sgmprm 15988	The sum of divisors for a prime is P + 1 because the only divisors are 1 and P. (Contributed by Mario Carneiro, 17-May-2016.)
|- ( P e. Prime -> ( 1 sigma P ) = ( P + 1 ) )
Theorem	1sgm2ppw 15989	The sum of the divisors of 2 ^ ( N - 1 ). (Contributed by Mario Carneiro, 17-May-2016.)
|- ( N e. NN -> ( 1 sigma ( 2 ^ ( N - 1 ) ) ) = ( ( 2 ^ N ) - 1 ) )
Theorem	sgmmul 15990	The divisor function for fixed parameter A is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)
|- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( A sigma ( M x. N ) ) = ( ( A sigma M ) x. ( A sigma N ) ) )
11.4.3  Perfect Number Theorem
Theorem	mersenne 15991	A Mersenne prime is a prime number of the form 2 ^ P - 1. This theorem shows that the P in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.)
|- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. Prime )
Theorem	perfect1 15992	Euclid's contribution to the Euclid-Euler theorem. A number of the form 2 ^ ( p - 1 ) x. ( 2 ^ p - 1 ) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)
|- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) )
Theorem	perfectlem1 15993	Lemma for perfect 15995. (Contributed by Mario Carneiro, 7-Jun-2016.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> -. 2 || B )   &    |- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )   =>    |- ( ph -> ( ( 2 ^ ( A + 1 ) ) e. NN /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN /\ ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) )
Theorem	perfectlem2 15994	Lemma for perfect 15995. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by Wolf Lammen, 17-Sep-2020.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> -. 2 || B )   &    |- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )   =>    |- ( ph -> ( B e. Prime /\ B = ( ( 2 ^ ( A + 1 ) ) - 1 ) ) )
Theorem	perfect 15995*	The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer N is a perfect number (that is, its divisor sum is 2 N) if and only if it is of the form 2 ^ ( p - 1 ) x. ( 2 ^ p - 1 ), where 2 ^ p - 1 is prime (a Mersenne prime), and therefore p is also prime, see mersenne 15991. This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.)
|- ( ( N e. NN /\ 2 || N ) -> ( ( 1 sigma N ) = ( 2 x. N ) <-> E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) )
11.4.4  Quadratic residues and the Legendre symbol If the congruence ( ( x ^ 2 ) mod p ) = ( n mod p ) has a solution we say that n is a quadratic residue mod p. If the congruence has no solution we say that n is a quadratic nonresidue mod p, see definition in [ApostolNT] p. 178. The Legendre symbol ( n /L p ) is defined in a way that its value is 1 if n is a quadratic residue mod p and -u 1 if n is a quadratic nonresidue mod p (and 0 if p divides n). Originally, the Legendre symbol ( N /L P ) was defined for odd primes P only (and arbitrary integers N) by Adrien-Marie Legendre in 1798, see definition in [ApostolNT] p. 179. It was generalized to be defined for any positive odd integer by Carl Gustav Jacob Jacobi in 1837 (therefore called "Jacobi symbol" since then), see definition in [ApostolNT] p. 188. Finally, it was generalized to be defined for any integer by Leopold Kronecker in 1885 (therefore called "Kronecker symbol" since then). The definition df-lgs 15997 for the "Legendre symbol" /L is actually the definition of the "Kronecker symbol". Since only one definition (and one class symbol) are provided in set.mm, the names "Legendre symbol", "Jacobi symbol" and "Kronecker symbol" are used synonymously for /L, but mostly it is called "Legendre symbol", even if it is used in the context of a "Jacobi symbol" or "Kronecker symbol".
Syntax	clgs 15996	Extend class notation with the Legendre symbol function.
class /L
Definition	df-lgs 15997*	Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers, and also the Jacobi symbol, which restricts the Kronecker symbol to positive odd integers). See definition in [ApostolNT] p. 179 resp. definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- /L = ( a e. ZZ , n e. ZZ |-> if ( n = 0 , if ( ( a ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( n < 0 /\ a < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || a , 0 , if ( ( a mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( a ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) ) ` ( abs ` n ) ) ) ) )
Theorem	zabsle1 15998	{ -u 1 , 0 , 1 } is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.)
|- ( Z e. ZZ -> ( Z e. { -u 1 , 0 , 1 } <-> ( abs ` Z ) <_ 1 ) )
Theorem	lgslem1 15999	When a is coprime to the prime p, a ^ ( ( p - 1 ) / 2 ) is equivalent mod p to 1 or -u 1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
Theorem	lgslem2 16000	The set Z of all integers with absolute value at most 1 contains { -u 1 , 0 , 1 }. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }   =>    |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )