Theorem 2lgsoddprmlem4

Description: Lemma 4 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021)

	Ref	Expression
Assertion	2lgsoddprmlem4	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ) → ( 2 ∥ ( ( ( 𝑁 ↑ 2 ) − 1 ) / 8 ) ↔ ( 𝑁 mod 8 ) ∈ { 1 , 7 } ) )

Proof

Step	Hyp	Ref	Expression
1		eqidd	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ) → ( 𝑁 mod 8 ) = ( 𝑁 mod 8 ) )
2		2lgsoddprmlem2	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ ( 𝑁 mod 8 ) = ( 𝑁 mod 8 ) ) → ( 2 ∥ ( ( ( 𝑁 ↑ 2 ) − 1 ) / 8 ) ↔ 2 ∥ ( ( ( ( 𝑁 mod 8 ) ↑ 2 ) − 1 ) / 8 ) ) )
3	1 2	mpd3an3	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ) → ( 2 ∥ ( ( ( 𝑁 ↑ 2 ) − 1 ) / 8 ) ↔ 2 ∥ ( ( ( ( 𝑁 mod 8 ) ↑ 2 ) − 1 ) / 8 ) ) )
4		2lgsoddprmlem3	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ ( 𝑁 mod 8 ) = ( 𝑁 mod 8 ) ) → ( 2 ∥ ( ( ( ( 𝑁 mod 8 ) ↑ 2 ) − 1 ) / 8 ) ↔ ( 𝑁 mod 8 ) ∈ { 1 , 7 } ) )
5	1 4	mpd3an3	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ) → ( 2 ∥ ( ( ( ( 𝑁 mod 8 ) ↑ 2 ) − 1 ) / 8 ) ↔ ( 𝑁 mod 8 ) ∈ { 1 , 7 } ) )
6	3 5	bitrd	⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ) → ( 2 ∥ ( ( ( 𝑁 ↑ 2 ) − 1 ) / 8 ) ↔ ( 𝑁 mod 8 ) ∈ { 1 , 7 } ) )