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Theorem dmmulpi 10302
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 5840 . . 3 dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2 fnom 8117 . . . . 5 ·o Fn (On × On)
32fndmi 6426 . . . 4 dom ·o = (On × On)
43ineq2i 4136 . . 3 ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
51, 4eqtri 2821 . 2 dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
6 df-mi 10285 . . 3 ·N = ( ·o ↾ (N × N))
76dmeqi 5737 . 2 dom ·N = dom ( ·o ↾ (N × N))
8 df-ni 10283 . . . . . . 7 N = (ω ∖ {∅})
9 difss 4059 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
108, 9eqsstri 3949 . . . . . 6 N ⊆ ω
11 omsson 7564 . . . . . 6 ω ⊆ On
1210, 11sstri 3924 . . . . 5 N ⊆ On
13 anidm 568 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1412, 13mpbir 234 . . . 4 (N ⊆ On ∧ N ⊆ On)
15 xpss12 5534 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1614, 15ax-mp 5 . . 3 (N × N) ⊆ (On × On)
17 dfss 3899 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1816, 17mpbi 233 . 2 (N × N) = ((N × N) ∩ (On × On))
195, 7, 183eqtr4i 2831 1 dom ·N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  cdif 3878  cin 3880  wss 3881  c0 4243  {csn 4525   × cxp 5517  dom cdm 5519  cres 5521  Oncon0 6159  ωcom 7560   ·o comu 8083  Ncnpi 10255   ·N cmi 10257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-omul 8090  df-ni 10283  df-mi 10285
This theorem is referenced by:  mulcompi  10307  mulasspi  10308  distrpi  10309  mulcanpi  10311  ltmpi  10315  ordpipq  10353
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