Theorem dmmulpi 10871

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 6011	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8490	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6639	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4170	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2786	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10854	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5894	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10852	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4090	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3983	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7862	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3946	. . . . 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 5676	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 3924	. . 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 2796	1 ⊢ dom ·N = (N × N)

Syntax hints:   ∧ wa 400   = wceq 1570   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4589   × cxp 5659  dom cdm 5661   ↾ cres 5663  Oncon0 6360  ωcom 7858   ·_o comu 8447  Ncnpi 10824   ·_N cmi 10826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-omul 8454  df-ni 10852  df-mi 10854

This theorem is referenced by:  mulcompi  10876  mulasspi  10877  distrpi  10878  mulcanpi  10880  ltmpi  10884  ordpipq  10922