P. 157 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15601-15700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	logblt 15601	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 15513. (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 15602	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 15603	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 15604	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 15605	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 15606	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 15607	Lemma for logbgcd1irrap 15609. 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 15608	Lemma for logbgcd1irrap 15609. 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 15609	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 15610	Example for logbgcd1irr 15606. The logarithm of nine to base two is not rational. Also see 2logb9irrap 15616 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 15611	The logarithm of a prime to a different prime base is not rational. For example, ( 2 logb 3 ) e. ( RR \ QQ ) (see 2logb3irr 15612). (Contributed by AV, 31-Dec-2022.)
|- ( ( X e. Prime /\ B e. Prime /\ X =/= B ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	2logb3irr 15612	Example for logbprmirr 15611. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
|- ( 2 logb 3 ) e. ( RR \ QQ )
Theorem	2logb9irrALT 15613	Alternate proof of 2logb9irr 15610: 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 15614	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 12652 resp. 2logb9irr 15610), satisfying the statement in 2irrexpq 15615. (Contributed by AV, 29-Dec-2022.)
|- ( ( sqr ` 2 ) ^c ( 2 logb 9 ) ) = 3
Theorem	2irrexpq 15615*	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 12652, 2logb9irr 15610 and sqrt2cxp2logb9e3 15614. 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 15617. (Contributed by AV, 23-Dec-2022.)
|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ
Theorem	2logb9irrap 15616	Example for logbgcd1irrap 15609. 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 15617*	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 12668, 2logb9irrap 15616 and sqrt2cxp2logb9e3 15614. 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 15618	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11961, 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 15619	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 12724, ( 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  Number-theoretical functions
Syntax	csgm 15620	Extend class notation with the divisor function.
class sigma
Definition	df-sgm 15621*	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 15622*	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 15623*	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 15624*	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 15625	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 15626	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 15627	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 15628*	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 15629*	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 15630*	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 8090. (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 15631*	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 15632	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 15633	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 15634	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 15635	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 ) ) )
11.3.3  Perfect Number Theorem
Theorem	mersenne 15636	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 15637	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 15638	Lemma for perfect 15640. (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 15639	Lemma for perfect 15640. (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 15640*	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 15636. 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 ) ) ) ) )
11.3.4  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 15642 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 15641	Extend class notation with the Legendre symbol function.
class /L
Definition	df-lgs 15642*	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 15643	{ -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 15644	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 15645	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 15646*	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 15647*	Lemma for lgsfcl2 15650. (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 15648*	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 15649*	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 15650*	The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 15643). (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 15651*	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 15652*	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 15653*	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 15654*	Lemma for lgsval2 15660. (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 15655*	Lemma for lgsval4 15664. (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 15656*	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 15657	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 15658	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 15659	The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 15658 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 15660	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 15661	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 15662	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 15663	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 15664*	Restate lgsval 15648 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 15665*	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 15666*	Same as lgsval4 15664 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 15667	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 15668	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 15669	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 15670	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 15671	Lemma for lgsdi 15681 and lgsdir 15679: 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 15672	Lemma for lgsdir2 15677. (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 15673	Lemma for lgsdir2 15677. (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 15674	Lemma for lgsdir2 15677. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ -. 2 || A ) -> ( A mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
Theorem	lgsdir2lem4 15675	Lemma for lgsdir2 15677. (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 15676	Lemma for lgsdir2 15677. (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 15677	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 15678	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 15679	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 15680*	Lemma for lgsdi 15681. (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 15681	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 15682	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 15683	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 15684	The Legendre symbol at a square is equal to 1. Together with lgsmod 15670 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 15685	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 15686	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 15687	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 )
Theorem	lgs1 15688	The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.)
|- ( A e. ZZ -> ( A /L 1 ) = 1 )
Theorem	lgsmodeq 15689	The Legendre (Jacobi) symbol is preserved under reduction mod n when n is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ ( N e. NN /\ -. 2 || N ) ) -> ( ( A mod N ) = ( B mod N ) -> ( A /L N ) = ( B /L N ) ) )
Theorem	lgsmulsqcoprm 15690	The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( ( A ^ 2 ) x. B ) /L N ) = ( B /L N ) )
Theorem	lgsdirnn0 15691	Variation on lgsdir 15679 valid for all A , B but only for positive N. (The exact location of the failure of this law is for A = 0, B < 0, N = -u 1 in which case ( 0 /L -u 1 ) = 1 but ( B /L -u 1 ) = -u 1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
|- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
Theorem	lgsdinn0 15692	Variation on lgsdi 15681 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
11.3.5  Gauss' Lemma Gauss' Lemma is valid for any integer not dividing the given prime number. In the following, only the special case for 2 (not dividing any odd prime) is proven, see gausslemma2d 15713. The general case is still to prove.
Theorem	gausslemma2dlem0a 15693	Auxiliary lemma 1 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   =>    |- ( ph -> P e. NN )
Theorem	gausslemma2dlem0b 15694	Auxiliary lemma 2 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> H e. NN )
Theorem	gausslemma2dlem0c 15695	Auxiliary lemma 3 for gausslemma2d 15713. (Contributed by AV, 13-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> ( ( ! ` H ) gcd P ) = 1 )
Theorem	gausslemma2dlem0d 15696	Auxiliary lemma 4 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> M e. NN0 )
Theorem	gausslemma2dlem0e 15697	Auxiliary lemma 5 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> ( M x. 2 ) < ( P / 2 ) )
Theorem	gausslemma2dlem0f 15698	Auxiliary lemma 6 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> ( M + 1 ) <_ H )
Theorem	gausslemma2dlem0g 15699	Auxiliary lemma 7 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> M <_ H )
Theorem	gausslemma2dlem0h 15700	Auxiliary lemma 8 for gausslemma2d 15713. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- N = ( H - M )   =>    |- ( ph -> N e. NN0 )