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Theorem dmmulpi 6481
Description: Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
Assertion
Ref Expression
dmmulpi dom ·N = (N × N)

Proof of Theorem dmmulpi
StepHypRef Expression
1 dmres 4659 . . 3 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 )
2 fnom 6060 . . . . 5 ·𝑜 Fn (On × On)
3 fndm 5025 . . . . 5 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
42, 3ax-mp 7 . . . 4 dom ·𝑜 = (On × On)
54ineq2i 3162 . . 3 ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2076 . 2 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 6461 . . 3 ·N = ( ·𝑜 ↾ (N × N))
87dmeqi 4563 . 2 dom ·N = dom ( ·𝑜 ↾ (N × N))
9 df-ni 6459 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3097 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3002 . . . . . 6 N ⊆ ω
12 omsson 4362 . . . . . 6 ω ⊆ On
1311, 12sstri 2981 . . . . 5 N ⊆ On
14 anidm 382 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 138 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4472 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 7 . . 3 (N × N) ⊆ (On × On)
18 dfss 2959 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 137 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2086 1 dom ·N = (N × N)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  cdif 2941  cin 2943  wss 2944  c0 3251  {csn 3402  Oncon0 4127  ωcom 4340   × cxp 4370  dom cdm 4372  cres 4374   Fn wfn 4924   ·𝑜 comu 6029  Ncnpi 6427   ·N cmi 6429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-ni 6459  df-mi 6461
This theorem is referenced by: (None)
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