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

Theorem mulpiord 7347
Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
mulpiord ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))

Proof of Theorem mulpiord
StepHypRef Expression
1 opelxpi 4676 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 5558 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·o ‘⟨𝐴, 𝐵⟩))
3 df-ov 5900 . . . 4 (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4 df-mi 7336 . . . . 5 ·N = ( ·o ↾ (N × N))
54fveq1i 5535 . . . 4 ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2210 . . 3 (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 5900 . . 3 (𝐴 ·o 𝐵) = ( ·o ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2247 . 2 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
91, 8syl 14 1 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  cop 3610   × cxp 4642  cres 4646  cfv 5235  (class class class)co 5897   ·o comu 6440  Ncnpi 7302   ·N cmi 7304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-res 4656  df-iota 5196  df-fv 5243  df-ov 5900  df-mi 7336
This theorem is referenced by:  mulidpi  7348  mulclpi  7358  mulcompig  7361  mulasspig  7362  distrpig  7363  mulcanpig  7365  ltmpig  7369  archnqq  7447  enq0enq  7461  addcmpblnq0  7473  mulcmpblnq0  7474  mulcanenq0ec  7475  addclnq0  7481  mulclnq0  7482  nqpnq0nq  7483  nqnq0a  7484  nqnq0m  7485  nq0m0r  7486  distrnq0  7489  addassnq0lemcl  7491
  Copyright terms: Public domain W3C validator