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Metamath Proof Explorer

Theorem addcompi

Description: Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995) (New usage is discouraged.)

Ref Expression
Assertion addcompi ( 𝐴 +N 𝐵 ) = ( 𝐵 +N 𝐴 )

Proof

Step Hyp Ref Expression
1 pinn ⊢ ( 𝐴 ∈ N → 𝐴 ∈ ω )
2 pinn ⊢ ( 𝐵 ∈ N → 𝐵 ∈ ω )
3 nnacom ⊢ ( ( 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 addpiord ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) = ( 𝐴 +o 𝐵 ) )
6 addpiord ⊢ ( ( 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 dmaddpi ⊢ 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 )