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Theorem mulpiord 9908
 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 5288 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6348 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩))
3 df-ov 6795 . . . 4 (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4 df-mi 9897 . . . . 5 ·N = ( ·𝑜 ↾ (N × N))
54fveq1i 6333 . . . 4 ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2792 . . 3 (𝐴 ·N 𝐵) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 6795 . . 3 (𝐴 ·𝑜 𝐵) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2829 . 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 382   = wceq 1630   ∈ wcel 2144  ⟨cop 4320   × cxp 5247   ↾ cres 5251  ‘cfv 6031  (class class class)co 6792   ·𝑜 comu 7710  Ncnpi 9867   ·N cmi 9869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-ov 6795  df-mi 9897 This theorem is referenced by:  mulidpi  9909  mulclpi  9916  mulcompi  9919  mulasspi  9920  distrpi  9921  mulcanpi  9923  ltmpi  9927
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