Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmmulpi GIF version

Theorem dmmulpi 6424
 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 4632 . . 3 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 )
2 fnom 6030 . . . . 5 ·𝑜 Fn (On × On)
3 fndm 4998 . . . . 5 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
42, 3ax-mp 7 . . . 4 dom ·𝑜 = (On × On)
54ineq2i 3135 . . 3 ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2060 . 2 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 6404 . . 3 ·N = ( ·𝑜 ↾ (N × N))
87dmeqi 4536 . 2 dom ·N = dom ( ·𝑜 ↾ (N × N))
9 df-ni 6402 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3070 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 2975 . . . . . 6 N ⊆ ω
12 omsson 4335 . . . . . 6 ω ⊆ On
1311, 12sstri 2954 . . . . 5 N ⊆ On
14 anidm 376 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 134 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4445 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 7 . . 3 (N × N) ⊆ (On × On)
18 dfss 2932 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 133 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2070 1 dom ·N = (N × N)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∖ cdif 2914   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  {csn 3375  Oncon0 4100  ωcom 4313   × cxp 4343  dom cdm 4345   ↾ cres 4347   Fn wfn 4897   ·𝑜 comu 5999  Ncnpi 6370   ·N cmi 6372 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-ni 6402  df-mi 6404 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator