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Theorem dmmulpi 9592
 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 5339 . . 3 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 )
2 fnom 7476 . . . . 5 ·𝑜 Fn (On × On)
3 fndm 5904 . . . . 5 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
42, 3ax-mp 5 . . . 4 dom ·𝑜 = (On × On)
54ineq2i 3773 . . 3 ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2632 . 2 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 9575 . . 3 ·N = ( ·𝑜 ↾ (N × N))
87dmeqi 5247 . 2 dom ·N = dom ( ·𝑜 ↾ (N × N))
9 df-ni 9573 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3699 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3598 . . . . . 6 N ⊆ ω
12 omsson 6961 . . . . . 6 ω ⊆ On
1311, 12sstri 3577 . . . . 5 N ⊆ On
14 anidm 674 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 220 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 5148 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3555 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 219 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2642 1 dom ·N = (N × N)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   × cxp 5036  dom cdm 5038   ↾ cres 5040  Oncon0 5640   Fn wfn 5799  ωcom 6957   ·𝑜 comu 7445  Ncnpi 9545   ·N cmi 9547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-omul 7452  df-ni 9573  df-mi 9575 This theorem is referenced by:  mulcompi  9597  mulasspi  9598  distrpi  9599  mulcanpi  9601  ltmpi  9605  ordpipq  9643
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