Theorem dmmulpi 10921

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 6017	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8530	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6659	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4207	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2753	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10904	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5907	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10902	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4128	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 4011	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7875	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3986	. . . . 5 ⊢ N ⊆ On
13		anidm 563	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 230	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5693	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3963	. . 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 2763	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 394   = wceq 1533   ∖ cdif 3941   ∩ cin 3943   ⊆ wss 3944  ∅c0 4322  {csn 4630   × cxp 5676  dom cdm 5678   ↾ cres 5680  Oncon0 6371  ωcom 7871   ·_o comu 8485  Ncnpi 10874   ·_N cmi 10876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-omul 8492  df-ni 10902  df-mi 10904

This theorem is referenced by:  mulcompi  10926  mulasspi  10927  distrpi  10928  mulcanpi  10930  ltmpi  10934  ordpipq  10972