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Theorem mulpiord 6876
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 4469 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 5329 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩))
3 df-ov 5655 . . . 4 (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4 df-mi 6865 . . . . 5 ·N = ( ·𝑜 ↾ (N × N))
54fveq1i 5306 . . . 4 ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2108 . . 3 (𝐴 ·N 𝐵) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 5655 . . 3 (𝐴 ·𝑜 𝐵) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2145 . 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 102   = wceq 1289  wcel 1438  cop 3449   × cxp 4436  cres 4440  cfv 5015  (class class class)co 5652   ·𝑜 comu 6179  Ncnpi 6831   ·N cmi 6833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-res 4450  df-iota 4980  df-fv 5023  df-ov 5655  df-mi 6865
This theorem is referenced by:  mulidpi  6877  mulclpi  6887  mulcompig  6890  mulasspig  6891  distrpig  6892  mulcanpig  6894  ltmpig  6898  archnqq  6976  enq0enq  6990  addcmpblnq0  7002  mulcmpblnq0  7003  mulcanenq0ec  7004  addclnq0  7010  mulclnq0  7011  nqpnq0nq  7012  nqnq0a  7013  nqnq0m  7014  nq0m0r  7015  distrnq0  7018  addassnq0lemcl  7020
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