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Theorem mulpiord 9586
Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulpiord ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))

Proof of Theorem mulpiord
StepHypRef Expression
1 opelxpi 5072 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6117 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩))
3 df-ov 6552 . . . 4 (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4 df-mi 9575 . . . . 5 ·N = ( ·𝑜 ↾ (N × N))
54fveq1i 6104 . . . 4 ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2632 . . 3 (𝐴 ·N 𝐵) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 6552 . . 3 (𝐴 ·𝑜 𝐵) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2669 . 2 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
91, 8syl 17 1 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cop 4131   × cxp 5036  cres 5040  cfv 5804  (class class class)co 6549   ·𝑜 comu 7445  Ncnpi 9545   ·N cmi 9547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-ov 6552  df-mi 9575
This theorem is referenced by:  mulidpi  9587  mulclpi  9594  mulcompi  9597  mulasspi  9598  distrpi  9599  mulcanpi  9601  ltmpi  9605
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