Theorem dmaddpi 10577

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 5902	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 8300	. . . . 5 ⊢ +o Fn (On × On)
3	2	fndmi 6521	. . . 4 ⊢ dom +o = (On × On)
4	3	ineq2i 4140	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2766	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-pli 10560	. . 3 ⊢ +N = ( +o ↾ (N × N))
7	6	dmeqi 5802	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
8		df-ni 10559	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4062	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3951	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7691	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3926	. . . . 5 ⊢ N ⊆ On
13		anidm 564	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 230	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5595	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3901	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
18	16, 17	mpbi 229	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
19	5, 7, 18	3eqtr4i 2776	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 395   = wceq 1539   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558   × cxp 5578  dom cdm 5580   ↾ cres 5582  Oncon0 6251  ωcom 7687   +_o coa 8264  Ncnpi 10531   +_N cpli 10532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-oadd 8271  df-ni 10559  df-pli 10560

This theorem is referenced by:  addcompi  10581  addasspi  10582  distrpi  10585  addcanpi  10586  addnidpi  10588  ltapi  10590