Theorem euclemma

Description: Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. Theorem 1.9 in ApostolNT p. 17. (Contributed by Paul Chapman, 17-Nov-2012)

	Ref	Expression
Assertion	euclemma	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ↔ ( 𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		coprm	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ) → ( ¬ 𝑃 ∥ 𝑀 ↔ ( 𝑃 gcd 𝑀 ) = 1 ) )
2	1	3adant3	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ¬ 𝑃 ∥ 𝑀 ↔ ( 𝑃 gcd 𝑀 ) = 1 ) )
3	2	anbi2d	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ∧ ¬ 𝑃 ∥ 𝑀 ) ↔ ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ∧ ( 𝑃 gcd 𝑀 ) = 1 ) ) )
4		prmz	⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℤ )
5		coprmdvds	⊢ ( ( 𝑃 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ∧ ( 𝑃 gcd 𝑀 ) = 1 ) → 𝑃 ∥ 𝑁 ) )
6	4 5	syl3an1	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ∧ ( 𝑃 gcd 𝑀 ) = 1 ) → 𝑃 ∥ 𝑁 ) )
7	3 6	sylbid	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ∧ ¬ 𝑃 ∥ 𝑀 ) → 𝑃 ∥ 𝑁 ) )
8	7	expd	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑃 ∥ ( 𝑀 · 𝑁 ) → ( ¬ 𝑃 ∥ 𝑀 → 𝑃 ∥ 𝑁 ) ) )
9		df-or	⊢ ( ( 𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁 ) ↔ ( ¬ 𝑃 ∥ 𝑀 → 𝑃 ∥ 𝑁 ) )
10	8 9	syl6ibr	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑃 ∥ ( 𝑀 · 𝑁 ) → ( 𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁 ) ) )
11		ordvdsmul	⊢ ( ( 𝑃 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁 ) → 𝑃 ∥ ( 𝑀 · 𝑁 ) ) )
12	4 11	syl3an1	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁 ) → 𝑃 ∥ ( 𝑀 · 𝑁 ) ) )
13	10 12	impbid	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑃 ∥ ( 𝑀 · 𝑁 ) ↔ ( 𝑃 ∥ 𝑀 ∨ 𝑃 ∥ 𝑁 ) ) )