P. 161 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16001-16100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2irrexpq 16001*	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 12920, 2logb9irr 15996 and sqrt2cxp2logb9e3 16000. 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 16003. (Contributed by AV, 23-Dec-2022.)
|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ
Theorem	2logb9irrap 16002	Example for logbgcd1irrap 15995. 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 16003*	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 12936, 2logb9irrap 16002 and sqrt2cxp2logb9e3 16000. 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 16004	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12229, 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 16005	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 16006	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 16007*	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 16008	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 12992, ( 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 16009	Extend class notation with the divisor function.
class sigma
Definition	df-sgm 16010*	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 16011*	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 16012*	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 16013*	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 16014	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 16015	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 16016	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 16017*	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 16018*	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 16019*	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 8292. (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 16020*	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 16021	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 16022	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 16023	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 16024	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 16025	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 16026	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 16027	Lemma for perfect 16029. (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 16028	Lemma for perfect 16029. (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 16029*	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 16025. 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 16031 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 16030	Extend class notation with the Legendre symbol function.
class /L
Definition	df-lgs 16031*	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 16032	{ -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 16033	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 16034	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 16035*	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 16036*	Lemma for lgsfcl2 16039. (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 16037*	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 16038*	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 16039*	The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 16032). (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 16040*	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 16041*	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 16042*	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 16043*	Lemma for lgsval2 16049. (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 16044*	Lemma for lgsval4 16053. (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 16045*	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 16046	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 16047	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 16048	The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 16047 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 16049	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 16050	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 16051	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 16052	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 16053*	Restate lgsval 16037 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 16054*	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 16055*	Same as lgsval4 16053 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 16056	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 16057	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 16058	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 16059	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 16060	Lemma for lgsdi 16070 and lgsdir 16068: 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 16061	Lemma for lgsdir2 16066. (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 16062	Lemma for lgsdir2 16066. (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 16063	Lemma for lgsdir2 16066. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( A e. ZZ /\ -. 2 || A ) -> ( A mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
Theorem	lgsdir2lem4 16064	Lemma for lgsdir2 16066. (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 16065	Lemma for lgsdir2 16066. (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 16066	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 16067	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 16068	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 16069*	Lemma for lgsdi 16070. (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 16070	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 16071	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 16072	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 16073	The Legendre symbol at a square is equal to 1. Together with lgsmod 16059 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 16074	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 16075	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 16076	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 16077	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 16078	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 16079	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 16080	Variation on lgsdir 16068 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 16081	Variation on lgsdi 16070 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 ) ) )
11.4.5  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 16102. The general case is still to prove.
Theorem	gausslemma2dlem0a 16082	Auxiliary lemma 1 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   =>    |- ( ph -> P e. NN )
Theorem	gausslemma2dlem0b 16083	Auxiliary lemma 2 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> H e. NN )
Theorem	gausslemma2dlem0c 16084	Auxiliary lemma 3 for gausslemma2d 16102. (Contributed by AV, 13-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> ( ( ! ` H ) gcd P ) = 1 )
Theorem	gausslemma2dlem0d 16085	Auxiliary lemma 4 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> M e. NN0 )
Theorem	gausslemma2dlem0e 16086	Auxiliary lemma 5 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> ( M x. 2 ) < ( P / 2 ) )
Theorem	gausslemma2dlem0f 16087	Auxiliary lemma 6 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> ( M + 1 ) <_ H )
Theorem	gausslemma2dlem0g 16088	Auxiliary lemma 7 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- H = ( ( P - 1 ) / 2 )   =>    |- ( ph -> M <_ H )
Theorem	gausslemma2dlem0h 16089	Auxiliary lemma 8 for gausslemma2d 16102. (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 )
Theorem	gausslemma2dlem0i 16090	Auxiliary lemma 9 for gausslemma2d 16102. (Contributed by AV, 14-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- N = ( H - M )   =>    |- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
Theorem	gausslemma2dlem1a 16091*	Lemma for gausslemma2dlem1 16094. (Contributed by AV, 1-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   =>    |- ( ph -> ran R = ( 1 ... H ) )
Theorem	gausslemma2dlem1cl 16092	Lemma for gausslemma2dlem1 16094. Closure of the body of the definition of R. (Contributed by Jim Kingdon, 10-Aug-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- ( ph -> A e. ( 1 ... H ) )   =>    |- ( ph -> if ( ( A x. 2 ) < ( P / 2 ) , ( A x. 2 ) , ( P - ( A x. 2 ) ) ) e. ZZ )
Theorem	gausslemma2dlem1f1o 16093*	Lemma for gausslemma2dlem1 16094. (Contributed by Jim Kingdon, 9-Aug-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   =>    |- ( ph -> R : ( 1 ... H ) -1-1-onto-> ( 1 ... H ) )
Theorem	gausslemma2dlem1 16094*	Lemma 1 for gausslemma2d 16102. (Contributed by AV, 5-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   =>    |- ( ph -> ( ! ` H ) = prod_ k e. ( 1 ... H ) ( R ` k ) )
Theorem	gausslemma2dlem2 16095*	Lemma 2 for gausslemma2d 16102. (Contributed by AV, 4-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> A. k e. ( 1 ... M ) ( R ` k ) = ( k x. 2 ) )
Theorem	gausslemma2dlem3 16096*	Lemma 3 for gausslemma2d 16102. (Contributed by AV, 4-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) )
Theorem	gausslemma2dlem4 16097*	Lemma 4 for gausslemma2d 16102. (Contributed by AV, 16-Jun-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> ( ! ` H ) = ( prod_ k e. ( 1 ... M ) ( R ` k ) x. prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) ) )
Theorem	gausslemma2dlem5a 16098*	Lemma for gausslemma2dlem5 16099. (Contributed by AV, 8-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- M = ( |_ ` ( P / 4 ) )   =>    |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( prod_ k e. ( ( M + 1 ) ... H ) ( -u 1 x. ( k x. 2 ) ) mod P ) )
Theorem	gausslemma2dlem5 16099*	Lemma 5 for gausslemma2d 16102. (Contributed by AV, 9-Jul-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- N = ( H - M )   =>    |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( ( ( -u 1 ^ N ) x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) mod P ) )
Theorem	gausslemma2dlem6 16100*	Lemma 6 for gausslemma2d 16102. (Contributed by AV, 16-Jun-2021.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- M = ( |_ ` ( P / 4 ) )   &    |- N = ( H - M )   =>    |- ( ph -> ( ( ! ` H ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) )