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Metamath Proof Explorer

Theorem modprm1div

Description: A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018) (Proof shortened by AV, 30-May-2023)

		Ref	Expression
	Assertion	modprm1div	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ) → ( ( 𝐴 mod 𝑃 ) = 1 ↔ 𝑃 ∥ ( 𝐴 − 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		prmuz2	⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ( ℤ_≥ ‘ 2 ) )
2		modm1div	⊢ ( ( 𝑃 ∈ ( ℤ_≥ ‘ 2 ) ∧ 𝐴 ∈ ℤ ) → ( ( 𝐴 mod 𝑃 ) = 1 ↔ 𝑃 ∥ ( 𝐴 − 1 ) ) )
3	1 2	sylan	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ) → ( ( 𝐴 mod 𝑃 ) = 1 ↔ 𝑃 ∥ ( 𝐴 − 1 ) ) )