Theorem dmmulpi 10875

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

Assertion

Ref	Expression
dmmulpi	⊢ dom ·N = (N × N)

Proof of Theorem dmmulpi

Step	Hyp	Ref	Expression
1		dmres 6012	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8493	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6640	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4178	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2792	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10858	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5895	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10856	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4098	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3991	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7865	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3954	. . . . 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 5677	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3932	. . 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 2802	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 400   = wceq 1567   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594   × cxp 5660  dom cdm 5662   ↾ cres 5664  Oncon0 6361  ωcom 7861   ·_o comu 8450  Ncnpi 10828   ·_N cmi 10830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-omul 8457  df-ni 10856  df-mi 10858

This theorem is referenced by:  mulcompi  10880  mulasspi  10881  distrpi  10882  mulcanpi  10884  ltmpi  10888  ordpipq  10926