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Theorem mulpiord 7312
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 4657 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 5537 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·o ‘⟨𝐴, 𝐵⟩))
3 df-ov 5874 . . . 4 (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4 df-mi 7301 . . . . 5 ·N = ( ·o ↾ (N × N))
54fveq1i 5514 . . . 4 ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2198 . . 3 (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 5874 . . 3 (𝐴 ·o 𝐵) = ( ·o ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2235 . 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 1353  wcel 2148  cop 3595   × cxp 4623  cres 4627  cfv 5214  (class class class)co 5871   ·o comu 6411  Ncnpi 7267   ·N cmi 7269
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-res 4637  df-iota 5176  df-fv 5222  df-ov 5874  df-mi 7301
This theorem is referenced by:  mulidpi  7313  mulclpi  7323  mulcompig  7326  mulasspig  7327  distrpig  7328  mulcanpig  7330  ltmpig  7334  archnqq  7412  enq0enq  7426  addcmpblnq0  7438  mulcmpblnq0  7439  mulcanenq0ec  7440  addclnq0  7446  mulclnq0  7447  nqpnq0nq  7448  nqnq0a  7449  nqnq0m  7450  nq0m0r  7451  distrnq0  7454  addassnq0lemcl  7456
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