Theorem dmmulpi 10981

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 6004	. . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2		fnom 8522	. . . . 5 ⊢ ·o Fn (On × On)
3	2	fndmi 6646	. . . 4 ⊢ dom ·o = (On × On)
4	3	ineq2i 4113	. . 3 ⊢ ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
5	1, 4	eqtri 2783	. 2 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6		df-mi 10964	. . 3 ⊢ ·N = ( ·o ↾ (N × N))
7	6	dmeqi 5886	. 2 ⊢ dom ·N = dom ( ·o ↾ (N × N))
8		df-ni 10962	. . . . . . 7 ⊢ N = (ω ∖ {∅})
9		difss 4071	. . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω
10	8, 9	eqsstri 3955	. . . . . 6 ⊢ N ⊆ ω
11		omsson 7883	. . . . . 6 ⊢ ω ⊆ On
12	10, 11	sstri 3918	. . . . 5 ⊢ N ⊆ On
13		anidm 575	. . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
14	12, 13	mpbir 234	. . . 4 ⊢ (N ⊆ On ∧ N ⊆ On)
15		xpss12 5665	. . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
16	14, 15	ax-mp 5	. . 3 ⊢ (N × N) ⊆ (On × On)
17		dfss 4094	. . 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 2793	1 ⊢ dom ·N = (N × N)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088  ∅c0 4277  {csn 4584   × cxp 5648  dom cdm 5650   ↾ cres 5652  Oncon0 6364  ωcom 7879   ·_o comu 8479  Ncnpi 10934   ·_N cmi 10936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-omul 8486  df-ni 10962  df-mi 10964

This theorem is used by:  mulcompi  10986  mulasspi  10987  distrpi  10988  mulcanpi  10990  ltmpi  10994  ordpipq  11032