P. 152 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1cxpd 15101	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 15102	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 15103	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 15104	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 15105	Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11178. (Contributed by AV, 23-Dec-2022.)
|- ( A e. RR+ -> ( ( sqr ` A ) ^c 2 ) = A )
Theorem	cxprecd 15106	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	rpcxpcld 15107	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 15108	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 15109	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 15110	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 15111	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 15112	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 15113	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 15114	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 15115	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 ) ) )
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 15034 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 15116	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 15117*	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 15118	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 15119	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 15120	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 15121	The logarithm of 1 to an arbitrary base B is 0. Property 1(b) of [Cohen4] p. 361. See log1 15042. (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 15122	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 15123	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 15124	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 15125	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 15126	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 15127	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 15128	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 15129	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 15130	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 15131	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 15132	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 15133	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 15134	The general logarithm function is monotone/increasing. See logleb 15051. (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 15135	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 15050. (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 15136	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 15137	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 15138	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 15139	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 15140	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 15141	Lemma for logbgcd1irrap 15143. 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 15142	Lemma for logbgcd1irrap 15143. 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 15143	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 15144	Example for logbgcd1irr 15140. The logarithm of nine to base two is not rational. Also see 2logb9irrap 15150 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 15145	The logarithm of a prime to a different prime base is not rational. For example, ( 2 logb 3 ) e. ( RR \ QQ ) (see 2logb3irr 15146). (Contributed by AV, 31-Dec-2022.)
|- ( ( X e. Prime /\ B e. Prime /\ X =/= B ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	2logb3irr 15146	Example for logbprmirr 15145. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
|- ( 2 logb 3 ) e. ( RR \ QQ )
Theorem	2logb9irrALT 15147	Alternate proof of 2logb9irr 15144: 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 15148	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 12305 resp. 2logb9irr 15144), satisfying the statement in 2irrexpq 15149. (Contributed by AV, 29-Dec-2022.)
|- ( ( sqr ` 2 ) ^c ( 2 logb 9 ) ) = 3
Theorem	2irrexpq 15149*	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 12305, 2logb9irr 15144 and sqrt2cxp2logb9e3 15148. 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 15151. (Contributed by AV, 23-Dec-2022.)
|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ
Theorem	2logb9irrap 15150	Example for logbgcd1irrap 15143. 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 15151*	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 12321, 2logb9irrap 15150 and sqrt2cxp2logb9e3 15148. 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 15152	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11630, 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  Basic number theory
11.3.1  Wilson's theorem
Theorem	wilthlem1 15153	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 12377, ( 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 ) ) ) )
11.3.2  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 15155 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 15154	Extend class notation with the Legendre symbol function.
class /L
Definition	df-lgs 15155*	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 15156	{ -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 15157	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 15158	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 )
Theorem	lgslem3 15159*	The set Z of all integers with absolute value at most 1 is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }   =>    |- ( ( A e. Z /\ B e. Z ) -> ( A x. B ) e. Z )
Theorem	lgslem4 15160*	Lemma for lgsfcl2 15163. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.)
|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }   =>    |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
Theorem	lgsval 15161*	Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) = if ( N = 0 , if ( ( A ^ 2 ) = 1 , 1 , 0 ) , ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , F ) ` ( abs ` N ) ) ) ) )
Theorem	lgsfvalg 15162*	Value of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> ( F ` M ) = 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 ) )
Theorem	lgsfcl2 15163*	The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 15156). (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   &    |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }   =>    |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )
Theorem	lgscllem 15164*	The Legendre symbol is an element of Z. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   &    |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }   =>    |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. Z )
Theorem	lgsfcl 15165*	Closure of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> ZZ )
Theorem	lgsfle1 15166*	The function F has magnitude less or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ M e. NN ) -> ( abs ` ( F ` M ) ) <_ 1 )
Theorem	lgsval2lem 15167*	Lemma for lgsval2 15173. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. Prime ) -> ( A /L N ) = if ( N = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( N - 1 ) / 2 ) ) + 1 ) mod N ) - 1 ) ) )
Theorem	lgsval4lem 15168*	Lemma for lgsval4 15177. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) )
Theorem	lgscl2 15169*	The Legendre symbol is an integer with absolute value less than or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }   =>    |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. Z )
Theorem	lgs0 15170	The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
Theorem	lgscl 15171	The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
Theorem	lgsle1 15172	The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 15171 this implies that it takes values in { -u 1 , 0 , 1 }. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( abs ` ( A /L N ) ) <_ 1 )
Theorem	lgsval2 15173	The Legendre symbol at a prime (this is the traditional domain of the Legendre symbol, except for the addition of prime 2). (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ P e. Prime ) -> ( A /L P ) = if ( P = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) ) )
Theorem	lgs2 15174	The Legendre symbol at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( A e. ZZ -> ( A /L 2 ) = if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) )
Theorem	lgsval3 15175	The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
Theorem	lgsvalmod 15176	The Legendre symbol is equivalent to a ^ ( ( p - 1 ) / 2 ), mod p. This theorem is also called "Euler's criterion", see theorem 9.2 in [ApostolNT] p. 180, or a representation of Euler's criterion using the Legendre symbol, (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
Theorem	lgsval4 15177*	Restate lgsval 15161 for nonzero N, where the function F has been abbreviated into a self-referential expression taking the value of /L on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A /L N ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , F ) ` ( abs ` N ) ) ) )
Theorem	lgsfcl3 15178*	Closure of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> ZZ )
Theorem	lgsval4a 15179*	Same as lgsval4 15177 for positive N. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. NN ) -> ( A /L N ) = ( seq 1 ( x. , F ) ` N ) )
Theorem	lgscl1 15180	The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by AV, 13-Jul-2021.)
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. { -u 1 , 0 , 1 } )
Theorem	lgsneg 15181	The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A /L -u N ) = ( if ( A < 0 , -u 1 , 1 ) x. ( A /L N ) ) )
Theorem	lgsneg1 15182	The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. NN0 /\ N e. ZZ ) -> ( A /L -u N ) = ( A /L N ) )
Theorem	lgsmod 15183	The Legendre (Jacobi) symbol is preserved under reduction mod n when n is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ N e. NN /\ -. 2 || N ) -> ( ( A mod N ) /L N ) = ( A /L N ) )
Theorem	lgsdilem 15184	Lemma for lgsdi 15194 and lgsdir 15192: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )
Theorem	lgsdir2lem1 15185	Lemma for lgsdir2 15190. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
Theorem	lgsdir2lem2 15186	Lemma for lgsdir2 15190. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( K e. ZZ /\ 2 || ( K + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) ) )   &    |- M = ( K + 1 )   &    |- N = ( M + 1 )   &    |- N e. S   =>    |- ( N e. ZZ /\ 2 || ( N + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) ) )
Theorem	lgsdir2lem3 15187	Lemma for lgsdir2 15190. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ -. 2 || A ) -> ( A mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
Theorem	lgsdir2lem4 15188	Lemma for lgsdir2 15190. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( A mod 8 ) e. { 1 , 7 } ) -> ( ( ( A x. B ) mod 8 ) e. { 1 , 7 } <-> ( B mod 8 ) e. { 1 , 7 } ) )
Theorem	lgsdir2lem5 15189	Lemma for lgsdir2 15190. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( ( A x. B ) mod 8 ) e. { 1 , 7 } )
Theorem	lgsdir2 15190	The Legendre symbol is completely multiplicative at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A x. B ) /L 2 ) = ( ( A /L 2 ) x. ( B /L 2 ) ) )
Theorem	lgsdirprm 15191	The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.)
|- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> ( ( A x. B ) /L P ) = ( ( A /L P ) x. ( B /L P ) ) )
Theorem	lgsdir 15192	The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that A and B are odd positive integers). (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
Theorem	lgsdilem2 15193*	Lemma for lgsdi 15194. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> M =/= 0 )   &    |- ( ph -> N =/= 0 )   &    |- F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) )   =>    |- ( ph -> ( seq 1 ( x. , F ) ` ( abs ` M ) ) = ( seq 1 ( x. , F ) ` ( abs ` ( M x. N ) ) ) )
Theorem	lgsdi 15194	The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that M and N are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)
|- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
Theorem	lgsne0 15195	The Legendre symbol is nonzero (and hence equal to 1 or -u 1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( A /L N ) =/= 0 <-> ( A gcd N ) = 1 ) )
Theorem	lgsabs1 15196	The Legendre symbol is nonzero (and hence equal to 1 or -u 1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) )
Theorem	lgssq 15197	The Legendre symbol at a square is equal to 1. Together with lgsmod 15183 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.)
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = 1 )
Theorem	lgssq2 15198	The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.)
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L ( N ^ 2 ) ) = 1 )
Theorem	lgsprme0 15199	The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.)
|- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) )
Theorem	1lgs 15200	The Legendre symbol at 1. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.)
|- ( N e. ZZ -> ( 1 /L N ) = 1 )