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Metamath Proof Explorer

Theorem dvdseq

Description: If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014) (Proof shortened by AV, 7-Aug-2021)

Ref Expression
Assertion dvdseq ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀 ) ) → 𝑀 = 𝑁 )

Proof

Step Hyp Ref Expression
1 dvdsabseq ⊢ ( ( 𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀 ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) )
2 nn0re ⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ )
3 nn0ge0 ⊢ ( 𝑀 ∈ ℕ0 → 0 ≤ 𝑀 )
4 2 3 absidd ⊢ ( 𝑀 ∈ ℕ0 → ( abs ‘ 𝑀 ) = 𝑀 )
5 4 adantr ⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( abs ‘ 𝑀 ) = 𝑀 )
6 5 eqcomd ⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 𝑀 = ( abs ‘ 𝑀 ) )
7 6 adantr ⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) → 𝑀 = ( abs ‘ 𝑀 ) )
8 simpr ⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) → ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) )
9 nn0re ⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ )
10 nn0ge0 ⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 )
11 9 10 absidd ⊢ ( 𝑁 ∈ ℕ0 → ( abs ‘ 𝑁 ) = 𝑁 )
12 11 ad2antlr ⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) → ( abs ‘ 𝑁 ) = 𝑁 )
13 7 8 12 3eqtrd ⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( abs ‘ 𝑀 ) = ( abs ‘ 𝑁 ) ) → 𝑀 = 𝑁 )
14 1 13 sylan2 ⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀 ) ) → 𝑀 = 𝑁 )