Theorem dmmulpi 10470

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 5858	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8214	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6460	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4110	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2759	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10453	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5758	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10451	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4032	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3921	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7626	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3896	. . . . 5 ⊢ N ⊆ On
13		anidm 568	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 234	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5551	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3871	. . 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 2769	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 399   = wceq 1543   ∖ cdif 3850   ∩ cin 3852   ⊆ wss 3853  ∅c0 4223  {csn 4527   × cxp 5534  dom cdm 5536   ↾ cres 5538  Oncon0 6191  ωcom 7622   ·_o comu 8178  Ncnpi 10423   ·_N cmi 10425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-omul 8185  df-ni 10451  df-mi 10453

This theorem is referenced by:  mulcompi  10475  mulasspi  10476  distrpi  10477  mulcanpi  10479  ltmpi  10483  ordpipq  10521