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Theorem mulpiord 6301
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 4319 . 2 ((A N B N) → ⟨A, B (N × N))
2 fvres 5141 . . 3 (⟨A, B (N × N) → (( ·𝑜 ↾ (N × N))‘⟨A, B⟩) = ( ·𝑜 ‘⟨A, B⟩))
3 df-ov 5458 . . . 4 (A ·N B) = ( ·N ‘⟨A, B⟩)
4 df-mi 6290 . . . . 5 ·N = ( ·𝑜 ↾ (N × N))
54fveq1i 5122 . . . 4 ( ·N ‘⟨A, B⟩) = (( ·𝑜 ↾ (N × N))‘⟨A, B⟩)
63, 5eqtri 2057 . . 3 (A ·N B) = (( ·𝑜 ↾ (N × N))‘⟨A, B⟩)
7 df-ov 5458 . . 3 (A ·𝑜 B) = ( ·𝑜 ‘⟨A, B⟩)
82, 6, 73eqtr4g 2094 . 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 1242   wcel 1390  cop 3370   × cxp 4286  cres 4290  cfv 4845  (class class class)co 5455   ·𝑜 comu 5938  Ncnpi 6256   ·N cmi 6258
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-res 4300  df-iota 4810  df-fv 4853  df-ov 5458  df-mi 6290
This theorem is referenced by:  mulidpi  6302  mulclpi  6312  mulcompig  6315  mulasspig  6316  distrpig  6317  mulcanpig  6319  ltmpig  6323  archnqq  6400  enq0enq  6413  addcmpblnq0  6425  mulcmpblnq0  6426  mulcanenq0ec  6427  addclnq0  6433  mulclnq0  6434  nqpnq0nq  6435  nqnq0a  6436  nqnq0m  6437  nq0m0r  6438  distrnq0  6441  addassnq0lemcl  6443
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