Theorem lgsfcl3

Description: Closure of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015)

	Ref	Expression
Hypothesis	lgsval4.1	⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( ( 𝐴 /L 𝑛 ) ↑ ( 𝑛 pCnt 𝑁 ) ) , 1 ) )
Assertion	lgsfcl3	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → 𝐹 : ℕ ⟶ ℤ )

Proof

Step	Hyp	Ref	Expression
1		lgsval4.1	⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( ( 𝐴 /L 𝑛 ) ↑ ( 𝑛 pCnt 𝑁 ) ) , 1 ) )
2		eqid	⊢ ( 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 ) ) = ( 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 ) )
3	2	lgsfcl	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 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 ) ) : ℕ ⟶ ℤ )
4	2	lgsval4lem	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 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 ) ) = ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( ( 𝐴 /L 𝑛 ) ↑ ( 𝑛 pCnt 𝑁 ) ) , 1 ) ) )
5	4 1	eqtr4di	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 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 ) ) = 𝐹 )
6	5	feq1d	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( ( 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 ) ) : ℕ ⟶ ℤ ↔ 𝐹 : ℕ ⟶ ℤ ) )
7	3 6	mpbid	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → 𝐹 : ℕ ⟶ ℤ )