Theorem gcd0val

Description: The value, by convention, of the gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011)

	Ref	Expression
Assertion	gcd0val	⊢ ( 0 gcd 0 ) = 0

Proof

Step	Hyp	Ref	Expression
1		0z	⊢ 0 ∈ ℤ
2		gcdval	⊢ ( ( 0 ∈ ℤ ∧ 0 ∈ ℤ ) → ( 0 gcd 0 ) = if ( ( 0 = 0 ∧ 0 = 0 ) , 0 , sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 0 ∧ 𝑛 ∥ 0 ) } , ℝ , < ) ) )
3	1 1 2	mp2an	⊢ ( 0 gcd 0 ) = if ( ( 0 = 0 ∧ 0 = 0 ) , 0 , sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 0 ∧ 𝑛 ∥ 0 ) } , ℝ , < ) )
4		eqid	⊢ 0 = 0
5		iftrue	⊢ ( ( 0 = 0 ∧ 0 = 0 ) → if ( ( 0 = 0 ∧ 0 = 0 ) , 0 , sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 0 ∧ 𝑛 ∥ 0 ) } , ℝ , < ) ) = 0 )
6	4 4 5	mp2an	⊢ if ( ( 0 = 0 ∧ 0 = 0 ) , 0 , sup ( { 𝑛 ∈ ℤ ∣ ( 𝑛 ∥ 0 ∧ 𝑛 ∥ 0 ) } , ℝ , < ) ) = 0
7	3 6	eqtri	⊢ ( 0 gcd 0 ) = 0