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

Theorem dmaddpi 7101
Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
Assertion
Ref Expression
dmaddpi dom +N = (N × N)

Proof of Theorem dmaddpi
StepHypRef Expression
1 dmres 4810 . . 3 dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o )
2 fnoa 6311 . . . . 5 +o Fn (On × On)
3 fndm 5192 . . . . 5 ( +o Fn (On × On) → dom +o = (On × On))
42, 3ax-mp 5 . . . 4 dom +o = (On × On)
54ineq2i 3244 . . 3 ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On))
61, 5eqtri 2138 . 2 dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-pli 7081 . . 3 +N = ( +o ↾ (N × N))
87dmeqi 4710 . 2 dom +N = dom ( +o ↾ (N × N))
9 df-ni 7080 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3172 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3099 . . . . . 6 N ⊆ ω
12 omsson 4496 . . . . . 6 ω ⊆ On
1311, 12sstri 3076 . . . . 5 N ⊆ On
14 anidm 393 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 145 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4616 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3055 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 144 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2148 1 dom +N = (N × N)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  cdif 3038  cin 3040  wss 3041  c0 3333  {csn 3497  Oncon0 4255  ωcom 4474   × cxp 4507  dom cdm 4509  cres 4511   Fn wfn 5088   +o coa 6278  Ncnpi 7048   +N cpli 7049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-ni 7080  df-pli 7081
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator