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Metamath Proof Explorer

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