Theorem addpiord

Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	addpiord	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) = ( 𝐴 +o 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		opelxpi	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( N × N ) )
2		fvres	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( N × N ) → ( ( +o ↾ ( N × N ) ) ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( +o ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
3		df-ov	⊢ ( 𝐴 +N 𝐵 ) = ( +N ‘ ⟨ 𝐴 , 𝐵 ⟩ )
4		df-pli	⊢ +N = ( +o ↾ ( N × N ) )
5	4	fveq1i	⊢ ( +N ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( ( +o ↾ ( N × N ) ) ‘ ⟨ 𝐴 , 𝐵 ⟩ )
6	3 5	eqtri	⊢ ( 𝐴 +N 𝐵 ) = ( ( +o ↾ ( N × N ) ) ‘ ⟨ 𝐴 , 𝐵 ⟩ )
7		df-ov	⊢ ( 𝐴 +o 𝐵 ) = ( +o ‘ ⟨ 𝐴 , 𝐵 ⟩ )
8	2 6 7	3eqtr4g	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( N × N ) → ( 𝐴 +N 𝐵 ) = ( 𝐴 +o 𝐵 ) )
9	1 8	syl	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) = ( 𝐴 +o 𝐵 ) )