Theorem dmaddpi

Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	dmaddpi	⊢ dom +N = ( N × N )

Proof

Step	Hyp	Ref	Expression
1		dmres	⊢ dom ( +o ↾ ( N × N ) ) = ( ( N × N ) ∩ dom +o )
2		fnoa	⊢ +o Fn ( On × On )
3		fndm	⊢ ( +o Fn ( On × On ) → dom +o = ( On × On ) )
4	2 3	ax-mp	⊢ dom +o = ( On × On )
5	4	ineq2i	⊢ ( ( N × N ) ∩ dom +o ) = ( ( N × N ) ∩ ( On × On ) )
6	1 5	eqtri	⊢ dom ( +o ↾ ( N × N ) ) = ( ( N × N ) ∩ ( On × On ) )
7		df-pli	⊢ +N = ( +o ↾ ( N × N ) )
8	7	dmeqi	⊢ dom +N = dom ( +o ↾ ( N × N ) )
9		df-ni	⊢ N = ( ω ∖ { ∅ } )
10		difss	⊢ ( ω ∖ { ∅ } ) ⊆ ω
11	9 10	eqsstri	⊢ N ⊆ ω
12		omsson	⊢ ω ⊆ On
13	11 12	sstri	⊢ N ⊆ On
14		anidm	⊢ ( ( N ⊆ On ∧ N ⊆ On ) ↔ N ⊆ On )
15	13 14	mpbir	⊢ ( N ⊆ On ∧ N ⊆ On )
16		xpss12	⊢ ( ( N ⊆ On ∧ N ⊆ On ) → ( N × N ) ⊆ ( On × On ) )
17	15 16	ax-mp	⊢ ( N × N ) ⊆ ( On × On )
18		dfss	⊢ ( ( N × N ) ⊆ ( On × On ) ↔ ( N × N ) = ( ( N × N ) ∩ ( On × On ) ) )
19	17 18	mpbi	⊢ ( N × N ) = ( ( N × N ) ∩ ( On × On ) )
20	6 8 19	3eqtr4i	⊢ dom +N = ( N × N )