P. 162 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16101-16200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sgmcl 16101	Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)
Theorem	sgmnncl 16102	Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)
Theorem	dvdsppwf1o 16103*	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 ∈ (0...𝐴) ↦ (𝑃↑𝑛))    ⇒   ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐴)})
Theorem	mpodvdsmulf1o 16104*	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 · 𝑁. (Contributed by GG, 18-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)    &   ⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}    &   ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   ⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    ⇒   ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
Theorem	fsumdvdsmul 16105*	Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘 ∥ 𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 8302. (Revised by GG, 18-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)    &   ⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}    &   ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   ⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) = 𝐷)    &   ⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)
Theorem	sgmppw 16106*	The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝐴 σ (𝑃↑𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃↑𝑐𝐴)↑𝑘))
Theorem	0sgmppw 16107	A prime power 𝑃↑𝐾 has 𝐾 + 1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ0) → (0 σ (𝑃↑𝐾)) = (𝐾 + 1))
Theorem	1sgmprm 16108	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 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))
Theorem	1sgm2ppw 16109	The sum of the divisors of 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 17-May-2016.)
⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))
Theorem	sgmmul 16110	The divisor function for fixed parameter 𝐴 is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)))
11.4.3  Perfect Number Theorem
Theorem	mersenne 16111	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 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℙ)
Theorem	perfect1 16112	Euclid's contribution to the Euclid-Euler theorem. A number of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((2↑𝑃) · ((2↑𝑃) − 1)))
Theorem	perfectlem1 16113	Lemma for perfect 16115. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐵)    &   ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))    ⇒   ⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
Theorem	perfectlem2 16114	Lemma for perfect 16115. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by Wolf Lammen, 17-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐵)    &   ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))    ⇒   ⊢ (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1)))
Theorem	perfect 16115*	The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime), and therefore 𝑝 is also prime, see mersenne 16111. This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ ((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((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 -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 16117 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 16116	Extend class notation with the Legendre symbol function.
class /L
Definition	df-lgs 16117*	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 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, (if(𝑚 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛)))))
Theorem	zabsle1 16118	{-1, 0, 1} is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.)
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1))
Theorem	lgslem1 16119	When 𝑎 is coprime to the prime 𝑝, 𝑎↑((𝑝 − 1) / 2) is equivalent mod 𝑝 to 1 or -1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})
Theorem	lgslem2 16120	The set 𝑍 of all integers with absolute value at most 1 contains {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}    ⇒   ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍)
Theorem	lgslem3 16121*	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 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}    ⇒   ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐴 · 𝐵) ∈ 𝑍)
Theorem	lgslem4 16122*	Lemma for lgsfcl2 16125. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.)
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
Theorem	lgsval 16123*	Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))
Theorem	lgsfvalg 16124*	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 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐹‘𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))
Theorem	lgsfcl2 16125*	The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 16118). (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    &   ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍)
Theorem	lgscllem 16126*	The Legendre symbol is an element of 𝑍. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    &   ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍)
Theorem	lgsfcl 16127*	Closure of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ)
Theorem	lgsfle1 16128*	The function 𝐹 has magnitude less or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑀 ∈ ℕ) → (abs‘(𝐹‘𝑀)) ≤ 1)
Theorem	lgsval2lem 16129*	Lemma for lgsval2 16135. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℙ) → (𝐴 /L 𝑁) = if(𝑁 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑁 − 1) / 2)) + 1) mod 𝑁) − 1)))
Theorem	lgsval4lem 16130*	Lemma for lgsval4 16139. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))
Theorem	lgscl2 16131*	The Legendre symbol is an integer with absolute value less than or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍)
Theorem	lgs0 16132	The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (𝐴 ∈ ℤ → (𝐴 /L 0) = if((𝐴↑2) = 1, 1, 0))
Theorem	lgscl 16133	The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ)
Theorem	lgsle1 16134	The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 16133 this implies that it takes values in {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1)
Theorem	lgsval2 16135	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 ∈ ℤ ∧ 𝑃 ∈ ℙ) → (𝐴 /L 𝑃) = if(𝑃 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)))
Theorem	lgs2 16136	The Legendre symbol at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (𝐴 ∈ ℤ → (𝐴 /L 2) = if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)))
Theorem	lgsval3 16137	The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝐴 /L 𝑃) = ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
Theorem	lgsvalmod 16138	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 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃))
Theorem	lgsval4 16139*	Restate lgsval 16123 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 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))))
Theorem	lgsfcl3 16140*	Closure of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ)
Theorem	lgsval4a 16141*	Same as lgsval4 16139 for positive 𝑁. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /L 𝑁) = (seq1( · , 𝐹)‘𝑁))
Theorem	lgscl1 16142	The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by AV, 13-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ {-1, 0, 1})
Theorem	lgsneg 16143	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 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /L 𝑁)))
Theorem	lgsneg1 16144	The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁))
Theorem	lgsmod 16145	The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /L 𝑁) = (𝐴 /L 𝑁))
Theorem	lgsdilem 16146	Lemma for lgsdi 16156 and lgsdir 16154: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → if((𝑁 < 0 ∧ (𝐴 · 𝐵) < 0), -1, 1) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐵 < 0), -1, 1)))
Theorem	lgsdir2lem1 16147	Lemma for lgsdir2 16152. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (((1 mod 8) = 1 ∧ (-1 mod 8) = 7) ∧ ((3 mod 8) = 3 ∧ (-3 mod 8) = 5))
Theorem	lgsdir2lem2 16148	Lemma for lgsdir2 16152. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆)))    &   ⊢ 𝑀 = (𝐾 + 1)    &   ⊢ 𝑁 = (𝑀 + 1)    &   ⊢ 𝑁 ∈ 𝑆    ⇒   ⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆)))
Theorem	lgsdir2lem3 16149	Lemma for lgsdir2 16152. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))
Theorem	lgsdir2lem4 16150	Lemma for lgsdir2 16152. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐵 mod 8) ∈ {1, 7}))
Theorem	lgsdir2lem5 16151	Lemma for lgsdir2 16152. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})
Theorem	lgsdir2 16152	The Legendre symbol is completely multiplicative at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐵) /L 2) = ((𝐴 /L 2) · (𝐵 /L 2)))
Theorem	lgsdirprm 16153	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 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 · 𝐵) /L 𝑃) = ((𝐴 /L 𝑃) · (𝐵 /L 𝑃)))
Theorem	lgsdir 16154	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 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 · 𝐵) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐵 /L 𝑁)))
Theorem	lgsdilem2 16155*	Lemma for lgsdi 16156. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ≠ 0)    &   ⊢ (𝜑 → 𝑁 ≠ 0)    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑀)), 1))    ⇒   ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) = (seq1( · , 𝐹)‘(abs‘(𝑀 · 𝑁))))
Theorem	lgsdi 16156	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 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /L (𝑀 · 𝑁)) = ((𝐴 /L 𝑀) · (𝐴 /L 𝑁)))
Theorem	lgsne0 16157	The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 /L 𝑁) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1))
Theorem	lgsabs1 16158	The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1))
Theorem	lgssq 16159	The Legendre symbol at a square is equal to 1. Together with lgsmod 16145 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 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = 1)
Theorem	lgssq2 16160	The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 /L (𝑁↑2)) = 1)
Theorem	lgsprme0 16161	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 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 /L 𝑃) = 0 ↔ (𝐴 mod 𝑃) = 0))
Theorem	1lgs 16162	The Legendre symbol at 1. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ (𝑁 ∈ ℤ → (1 /L 𝑁) = 1)
Theorem	lgs1 16163	The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ (𝐴 ∈ ℤ → (𝐴 /L 1) = 1)
Theorem	lgsmodeq 16164	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 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁)))
Theorem	lgsmulsqcoprm 16165	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 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → (((𝐴↑2) · 𝐵) /L 𝑁) = (𝐵 /L 𝑁))
Theorem	lgsdirnn0 16166	Variation on lgsdir 16154 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 = -1 in which case (0 /L -1) = 1 but (𝐵 /L -1) = -1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 · 𝐵) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐵 /L 𝑁)))
Theorem	lgsdinn0 16167	Variation on lgsdi 16156 valid for all 𝑀, 𝑁 but only for positive 𝐴. (The exact location of the failure of this law is for 𝐴 = -1, 𝑀 = 0, and some 𝑁 in which case (-1 /L 0) = 1 but (-1 /L 𝑁) = -1 when -1 is not a quadratic residue mod 𝑁.) (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L (𝑀 · 𝑁)) = ((𝐴 /L 𝑀) · (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 16188. The general case is still to prove.
Theorem	gausslemma2dlem0a 16168	Auxiliary lemma 1 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    ⇒   ⊢ (𝜑 → 𝑃 ∈ ℕ)
Theorem	gausslemma2dlem0b 16169	Auxiliary lemma 2 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    ⇒   ⊢ (𝜑 → 𝐻 ∈ ℕ)
Theorem	gausslemma2dlem0c 16170	Auxiliary lemma 3 for gausslemma2d 16188. (Contributed by AV, 13-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    ⇒   ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1)
Theorem	gausslemma2dlem0d 16171	Auxiliary lemma 4 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    ⇒   ⊢ (𝜑 → 𝑀 ∈ ℕ0)
Theorem	gausslemma2dlem0e 16172	Auxiliary lemma 5 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    ⇒   ⊢ (𝜑 → (𝑀 · 2) < (𝑃 / 2))
Theorem	gausslemma2dlem0f 16173	Auxiliary lemma 6 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    ⇒   ⊢ (𝜑 → (𝑀 + 1) ≤ 𝐻)
Theorem	gausslemma2dlem0g 16174	Auxiliary lemma 7 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    ⇒   ⊢ (𝜑 → 𝑀 ≤ 𝐻)
Theorem	gausslemma2dlem0h 16175	Auxiliary lemma 8 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = (𝐻 − 𝑀)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ0)
Theorem	gausslemma2dlem0i 16176	Auxiliary lemma 9 for gausslemma2d 16188. (Contributed by AV, 14-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = (𝐻 − 𝑀)    ⇒   ⊢ (𝜑 → (((2 /L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /L 𝑃) = (-1↑𝑁)))
Theorem	gausslemma2dlem1a 16177*	Lemma for gausslemma2dlem1 16180. (Contributed by AV, 1-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    ⇒   ⊢ (𝜑 → ran 𝑅 = (1...𝐻))
Theorem	gausslemma2dlem1cl 16178	Lemma for gausslemma2dlem1 16180. Closure of the body of the definition of 𝑅. (Contributed by Jim Kingdon, 10-Aug-2025.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ (𝜑 → 𝐴 ∈ (1...𝐻))    ⇒   ⊢ (𝜑 → if((𝐴 · 2) < (𝑃 / 2), (𝐴 · 2), (𝑃 − (𝐴 · 2))) ∈ ℤ)
Theorem	gausslemma2dlem1f1o 16179*	Lemma for gausslemma2dlem1 16180. (Contributed by Jim Kingdon, 9-Aug-2025.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    ⇒   ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))
Theorem	gausslemma2dlem1 16180*	Lemma 1 for gausslemma2d 16188. (Contributed by AV, 5-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    ⇒   ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
Theorem	gausslemma2dlem2 16181*	Lemma 2 for gausslemma2d 16188. (Contributed by AV, 4-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = (𝑘 · 2))
Theorem	gausslemma2dlem3 16182*	Lemma 3 for gausslemma2d 16188. (Contributed by AV, 4-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = (𝑃 − (𝑘 · 2)))
Theorem	gausslemma2dlem4 16183*	Lemma 4 for gausslemma2d 16188. (Contributed by AV, 16-Jun-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    ⇒   ⊢ (𝜑 → (!‘𝐻) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
Theorem	gausslemma2dlem5a 16184*	Lemma for gausslemma2dlem5 16185. (Contributed by AV, 8-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    ⇒   ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(-1 · (𝑘 · 2)) mod 𝑃))
Theorem	gausslemma2dlem5 16185*	Lemma 5 for gausslemma2d 16188. (Contributed by AV, 9-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝑁 = (𝐻 − 𝑀)    ⇒   ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (((-1↑𝑁) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑘 · 2)) mod 𝑃))
Theorem	gausslemma2dlem6 16186*	Lemma 6 for gausslemma2d 16188. (Contributed by AV, 16-Jun-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝑁 = (𝐻 − 𝑀)    ⇒   ⊢ (𝜑 → ((!‘𝐻) mod 𝑃) = ((((-1↑𝑁) · (2↑𝐻)) · (!‘𝐻)) mod 𝑃))
Theorem	gausslemma2dlem7 16187*	Lemma 7 for gausslemma2d 16188. (Contributed by AV, 13-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝑁 = (𝐻 − 𝑀)    ⇒   ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)
Theorem	gausslemma2d 16188*	Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S = {2, 4, 6, ..., p - 1}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ 𝐻 = ((𝑃 − 1) / 2)    &   ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))    &   ⊢ 𝑀 = (⌊‘(𝑃 / 4))    &   ⊢ 𝑁 = (𝐻 − 𝑀)    ⇒   ⊢ (𝜑 → (2 /L 𝑃) = (-1↑𝑁))
11.4.6  Quadratic reciprocity
Theorem	lgseisenlem1 16189*	Lemma for lgseisen 16193. If 𝑅(𝑢) = (𝑄 · 𝑢) mod 𝑃 and 𝑀(𝑢) = (-1↑𝑅(𝑢)) · 𝑅(𝑢), then for any even 1 ≤ 𝑢 ≤ 𝑃 − 1, 𝑀(𝑢) is also an even integer 1 ≤ 𝑀(𝑢) ≤ 𝑃 − 1. To simplify these statements, we divide all the even numbers by 2, so that it becomes the statement that 𝑀(𝑥 / 2) = (-1↑𝑅(𝑥 / 2)) · 𝑅(𝑥 / 2) / 2 is an integer between 1 and (𝑃 − 1) / 2. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)    &   ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))    ⇒   ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))⟶(1...((𝑃 − 1) / 2)))
Theorem	lgseisenlem2 16190*	Lemma for lgseisen 16193. The function 𝑀 is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)    &   ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))    &   ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)    ⇒   ⊢ (𝜑 → 𝑀:(1...((𝑃 − 1) / 2))–1-1-onto→(1...((𝑃 − 1) / 2)))
Theorem	lgseisenlem3 16191*	Lemma for lgseisen 16193. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-2019.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)    &   ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))    &   ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑌)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘((-1↑𝑅) · 𝑄)))) = (1r‘𝑌))
Theorem	lgseisenlem4 16192*	Lemma for lgseisen 16193. (Contributed by Mario Carneiro, 18-Jun-2015.) (Proof shortened by AV, 15-Jun-2019.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑅 = ((𝑄 · (2 · 𝑥)) mod 𝑃)    &   ⊢ 𝑀 = (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↦ ((((-1↑𝑅) · 𝑅) mod 𝑃) / 2))    &   ⊢ 𝑆 = ((𝑄 · (2 · 𝑦)) mod 𝑃)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑌)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    ⇒   ⊢ (𝜑 → ((𝑄↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))) mod 𝑃))
Theorem	lgseisen 16193*	Eisenstein's lemma, an expression for (𝑃 /L 𝑄) when 𝑃, 𝑄 are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → (𝑄 /L 𝑃) = (-1↑Σ𝑥 ∈ (1...((𝑃 − 1) / 2))(⌊‘((𝑄 / 𝑃) · (2 · 𝑥)))))
Theorem	lgsquadlemsfi 16194*	Lemma for lgsquad 16199. 𝑆 is finite. (Contributed by Jim Kingdon, 16-Sep-2025.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑀 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = ((𝑄 − 1) / 2)    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   ⊢ (𝜑 → 𝑆 ∈ Fin)
Theorem	lgsquadlemofi 16195*	Lemma for lgsquad 16199. There are finitely many members of 𝑆 with odd first part. (Contributed by Jim Kingdon, 16-Sep-2025.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑀 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = ((𝑄 − 1) / 2)    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   ⊢ (𝜑 → {𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1st ‘𝑧)} ∈ Fin)
Theorem	lgsquadlem1 16196*	Lemma for lgsquad 16199. Count the members of 𝑆 with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑀 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = ((𝑄 − 1) / 2)    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   ⊢ (𝜑 → (-1↑Σ𝑢 ∈ (((⌊‘(𝑀 / 2)) + 1)...𝑀)(⌊‘((𝑄 / 𝑃) · (2 · 𝑢)))) = (-1↑(♯‘{𝑧 ∈ 𝑆 ∣ ¬ 2 ∥ (1st ‘𝑧)})))
Theorem	lgsquadlem2 16197*	Lemma for lgsquad 16199. Count the members of 𝑆 with even coordinates, and combine with lgsquadlem1 16196 to get the total count of lattice points in 𝑆 (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑀 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = ((𝑄 − 1) / 2)    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   ⊢ (𝜑 → (𝑄 /L 𝑃) = (-1↑(♯‘𝑆)))
Theorem	lgsquadlem3 16198*	Lemma for lgsquad 16199. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ 𝑀 = ((𝑃 − 1) / 2)    &   ⊢ 𝑁 = ((𝑄 − 1) / 2)    &   ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}    ⇒   ⊢ (𝜑 → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(𝑀 · 𝑁)))
Theorem	lgsquad 16199	The Law of Quadratic Reciprocity, see also theorem 9.8 in [ApostolNT] p. 185. If 𝑃 and 𝑄 are distinct odd primes, then the product of the Legendre symbols (𝑃 /L 𝑄) and (𝑄 /L 𝑃) is the parity of ((𝑃 − 1) / 2) · ((𝑄 − 1) / 2). This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. This is Metamath 100 proof #7. (Contributed by Mario Carneiro, 19-Jun-2015.)
⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(((𝑃 − 1) / 2) · ((𝑄 − 1) / 2))))
Theorem	lgsquad2lem1 16200	Lemma for lgsquad2 16202. (Contributed by Mario Carneiro, 19-Jun-2015.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 𝑀)    &   ⊢ (𝜑 → ((𝐴 /L 𝑁) · (𝑁 /L 𝐴)) = (-1↑(((𝐴 − 1) / 2) · ((𝑁 − 1) / 2))))    &   ⊢ (𝜑 → ((𝐵 /L 𝑁) · (𝑁 /L 𝐵)) = (-1↑(((𝐵 − 1) / 2) · ((𝑁 − 1) / 2))))    ⇒   ⊢ (𝜑 → ((𝑀 /L 𝑁) · (𝑁 /L 𝑀)) = (-1↑(((𝑀 − 1) / 2) · ((𝑁 − 1) / 2))))