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Metamath Proof Explorer

Theorem mulcompi

Description: Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995) (New usage is discouraged.)

Ref Expression
Assertion mulcompi ( 𝐴 ·N 𝐵 ) = ( 𝐵 ·N 𝐴 )

Proof

Step Hyp Ref Expression
1 pinn ⊢ ( 𝐴 ∈ N → 𝐴 ∈ ω )
2 pinn ⊢ ( 𝐵 ∈ N → 𝐵 ∈ ω )
3 nnmcom ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o 𝐵 ) = ( 𝐵 ·o 𝐴 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·o 𝐵 ) = ( 𝐵 ·o 𝐴 ) )
5 mulpiord ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·N 𝐵 ) = ( 𝐴 ·o 𝐵 ) )
6 mulpiord ⊢ ( ( 𝐵 ∈ N ∧ 𝐴 ∈ N ) → ( 𝐵 ·N 𝐴 ) = ( 𝐵 ·o 𝐴 ) )
7 6 ancoms ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐵 ·N 𝐴 ) = ( 𝐵 ·o 𝐴 ) )
8 4 5 7 3eqtr4d ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·N 𝐵 ) = ( 𝐵 ·N 𝐴 ) )
9 dmmulpi ⊢ dom ·N = ( N × N )
10 9 ndmovcom ⊢ ( ¬ ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·N 𝐵 ) = ( 𝐵 ·N 𝐴 ) )
11 8 10 pm2.61i ⊢ ( 𝐴 ·N 𝐵 ) = ( 𝐵 ·N 𝐴 )