Theorem dmaddpi 3812

Description: Domain of addition on positive integers.

Assertion

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

Proof of Theorem dmaddpi

Step	Hyp	Ref	Expression
1		dmres 2584	. . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2		fnoa 3117	. . . . 5 ⊢ +o Fn (On × On)
3		fndm 2723	. . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On))
4	2, 3	ax-mp 6	. . . 4 ⊢ dom +o = (On × On)
5	4	ineq2i 1642	. . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
6	1, 5	eqtr 1119	. 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7		df-pli 3795	. . 3 ⊢ +N = ( +o ↾ (N × N))
8	7	dmeqi 2532	. 2 ⊢ dom +N = dom ( +o ↾ (N × N))
9		df-ni 3794	. . . . . . 7 ⊢ N = (ω ∖ {∅})
10		difss 1596	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
11	9, 10	eqsstr 1530	. . . . . 6 ⊢ N ⊆ ω
12		omsson 2377	. . . . . 6 ⊢ ω ⊆ On
13	11, 12	sstri 1512	. . . . 5 ⊢ N ⊆ On
14		anidm 331	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
15	13, 14	mpbir 165	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
16		ssxp 2487	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
17	15, 16	ax-mp 6	. . 3 ⊢ (N × N) ⊆ (On × On)
18		dfss 1493	. . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
19	17, 18	mpbi 164	. 2 ⊢ (N × N) = ((N × N) ∩ (On × On))
20	6, 8, 19	3eqtr4 1126	1 ⊢ dom +N = (N × N)

Syntax hints:   ∧ wa 196   = wceq 1091   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {csn 1808  Oncon0 2199  ωcom 2372   × cxp 2408  dom cdm 2410   ↾ cres 2412   Fn wfn 2417   +_o coa 3101  Ncnpi 3766   +_N cpli 3767

This theorem is referenced by:  addcompi 3816  addasspi 3817  distrpi 3820  addnidpi 3822  ltapi 3824

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-res 2430  df-fun 2432  df-fn 2433  df-fv 2438  df-oprab 3004  df-oadd 3106  df-ni 3794  df-pli 3795

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