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Theorem dmmulpi 9915
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 5560 . . 3 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 )
2 fnom 7743 . . . . 5 ·𝑜 Fn (On × On)
3 fndm 6130 . . . . 5 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
42, 3ax-mp 5 . . . 4 dom ·𝑜 = (On × On)
54ineq2i 3962 . . 3 ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2793 . 2 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 9898 . . 3 ·N = ( ·𝑜 ↾ (N × N))
87dmeqi 5463 . 2 dom ·N = dom ( ·𝑜 ↾ (N × N))
9 df-ni 9896 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3888 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3784 . . . . . 6 N ⊆ ω
12 omsson 7216 . . . . . 6 ω ⊆ On
1311, 12sstri 3761 . . . . 5 N ⊆ On
14 anidm 554 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 221 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 5264 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3738 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 220 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2803 1 dom ·N = (N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  cdif 3720  cin 3722  wss 3723  c0 4063  {csn 4316   × cxp 5247  dom cdm 5249  cres 5251  Oncon0 5866   Fn wfn 6026  ωcom 7212   ·𝑜 comu 7711  Ncnpi 9868   ·N cmi 9870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-omul 7718  df-ni 9896  df-mi 9898
This theorem is referenced by:  mulcompi  9920  mulasspi  9921  distrpi  9922  mulcanpi  9924  ltmpi  9928  ordpipq  9966
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