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Theorem mulpiord 6305
Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
mulpiord ((A N B N) → (A ·N B) = (A ·𝑜 B))

Proof of Theorem mulpiord
StepHypRef Expression
1 opelxpi 4322 . 2 ((A N B N) → ⟨A, B (N × N))
2 fvres 5144 . . 3 (⟨A, B (N × N) → (( ·𝑜 ↾ (N × N))‘⟨A, B⟩) = ( ·𝑜 ‘⟨A, B⟩))
3 df-ov 5461 . . . 4 (A ·N B) = ( ·N ‘⟨A, B⟩)
4 df-mi 6294 . . . . 5 ·N = ( ·𝑜 ↾ (N × N))
54fveq1i 5125 . . . 4 ( ·N ‘⟨A, B⟩) = (( ·𝑜 ↾ (N × N))‘⟨A, B⟩)
63, 5eqtri 2060 . . 3 (A ·N B) = (( ·𝑜 ↾ (N × N))‘⟨A, B⟩)
7 df-ov 5461 . . 3 (A ·𝑜 B) = ( ·𝑜 ‘⟨A, B⟩)
82, 6, 73eqtr4g 2097 . 2 (⟨A, B (N × N) → (A ·N B) = (A ·𝑜 B))
91, 8syl 14 1 ((A N B N) → (A ·N B) = (A ·𝑜 B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1243   wcel 1393  cop 3373   × cxp 4289  cres 4293  cfv 4848  (class class class)co 5458   ·𝑜 comu 5942  Ncnpi 6260   ·N cmi 6262
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-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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-br 3759  df-opab 3813  df-xp 4297  df-res 4303  df-iota 4813  df-fv 4856  df-ov 5461  df-mi 6294
This theorem is referenced by:  mulidpi  6306  mulclpi  6316  mulcompig  6319  mulasspig  6320  distrpig  6321  mulcanpig  6323  ltmpig  6327  archnqq  6405  enq0enq  6419  addcmpblnq0  6431  mulcmpblnq0  6432  mulcanenq0ec  6433  addclnq0  6439  mulclnq0  6440  nqpnq0nq  6441  nqnq0a  6442  nqnq0m  6443  nq0m0r  6444  distrnq0  6447  addassnq0lemcl  6449
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