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Theorem mulpiord 10358
 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 5565 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6682 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·o ‘⟨𝐴, 𝐵⟩))
3 df-ov 7159 . . . 4 (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4 df-mi 10347 . . . . 5 ·N = ( ·o ↾ (N × N))
54fveq1i 6664 . . . 4 ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2781 . . 3 (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 7159 . . 3 (𝐴 ·o 𝐵) = ( ·o ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2818 . 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 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5526   ↾ cres 5530  ‘cfv 6340  (class class class)co 7156   ·o comu 8116  Ncnpi 10317   ·N cmi 10319 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-res 5540  df-iota 6299  df-fv 6348  df-ov 7159  df-mi 10347 This theorem is referenced by:  mulidpi  10359  mulclpi  10366  mulcompi  10369  mulasspi  10370  distrpi  10371  mulcanpi  10373  ltmpi  10377
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