Theorem dmaddpi 10876

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

Assertion

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

Proof of Theorem dmaddpi

Step	Hyp	Ref	Expression
1		dmres 6013	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8494	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6641	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4171	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2786	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10859	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5896	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10858	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4091	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3984	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7867	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3947	. . . . 5 ⊢ N ⊆ On
13		anidm 574	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 234	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5678	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3925	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 233	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2796	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 400   = wceq 1570   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  {csn 4590   × cxp 5661  dom cdm 5663   ↾ cres 5665  Oncon0 6362  ωcom 7863   +_o coa 8451  Ncnpi 10830   +_N cpli 10831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-oadd 8458  df-ni 10858  df-pli 10859

This theorem is referenced by:  addcompi  10880  addasspi  10881  distrpi  10884  addcanpi  10885  addnidpi  10887  ltapi  10889