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Theorem dmmulpi 9698
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
StepHypRef Expression
1 dmres 5407 . . 3 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 )
2 fnom 7574 . . . . 5 ·𝑜 Fn (On × On)
3 fndm 5978 . . . . 5 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
42, 3ax-mp 5 . . . 4 dom ·𝑜 = (On × On)
54ineq2i 3803 . . 3 ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2642 . 2 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 9681 . . 3 ·N = ( ·𝑜 ↾ (N × N))
87dmeqi 5314 . 2 dom ·N = dom ( ·𝑜 ↾ (N × N))
9 df-ni 9679 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3729 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3627 . . . . . 6 N ⊆ ω
12 omsson 7054 . . . . . 6 ω ⊆ On
1311, 12sstri 3604 . . . . 5 N ⊆ On
14 anidm 675 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 221 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 5215 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3582 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 220 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2652 1 dom ·N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1481  cdif 3564  cin 3566  wss 3567  c0 3907  {csn 4168   × cxp 5102  dom cdm 5104  cres 5106  Oncon0 5711   Fn wfn 5871  ωcom 7050   ·𝑜 comu 7543  Ncnpi 9651   ·N cmi 9653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-omul 7550  df-ni 9679  df-mi 9681
This theorem is referenced by:  mulcompi  9703  mulasspi  9704  distrpi  9705  mulcanpi  9707  ltmpi  9711  ordpipq  9749
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