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Theorem dmmulpi 10931
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 6030 . . 3 dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2 fnom 8547 . . . . 5 ·o Fn (On × On)
32fndmi 6672 . . . 4 dom ·o = (On × On)
43ineq2i 4217 . . 3 ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
51, 4eqtri 2765 . 2 dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6 df-mi 10914 . . 3 ·N = ( ·o ↾ (N × N))
76dmeqi 5915 . 2 dom ·N = dom ( ·o ↾ (N × N))
8 df-ni 10912 . . . . . . 7 N = (ω ∖ {∅})
9 difss 4136 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
108, 9eqsstri 4030 . . . . . 6 N ⊆ ω
11 omsson 7891 . . . . . 6 ω ⊆ On
1210, 11sstri 3993 . . . . 5 N ⊆ On
13 anidm 564 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1412, 13mpbir 231 . . . 4 (N ⊆ On ∧ N ⊆ On)
15 xpss12 5700 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1614, 15ax-mp 5 . . 3 (N × N) ⊆ (On × On)
17 dfss 3970 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1816, 17mpbi 230 . 2 (N × N) = ((N × N) ∩ (On × On))
195, 7, 183eqtr4i 2775 1 dom ·N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  cdif 3948  cin 3950  wss 3951  c0 4333  {csn 4626   × cxp 5683  dom cdm 5685  cres 5687  Oncon0 6384  ωcom 7887   ·o comu 8504  Ncnpi 10884   ·N cmi 10886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-omul 8511  df-ni 10912  df-mi 10914
This theorem is referenced by:  mulcompi  10936  mulasspi  10937  distrpi  10938  mulcanpi  10940  ltmpi  10944  ordpipq  10982
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