Theorem mulpiord 10883

Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulpiord	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))

Proof of Theorem mulpiord

Step	Hyp	Ref	Expression
1		opelxpi 5713	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6910	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·o ‘⟨𝐴, 𝐵⟩))
3		df-ov 7415	. . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4		df-mi 10872	. . . . 5 ⊢ ·N = ( ·o ↾ (N × N))
5	4	fveq1i 6892	. . . 4 ⊢ ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2759	. . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 7415	. . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2796	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
9	1, 8	syl 17	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ⟨cop 4634   × cxp 5674   ↾ cres 5678  ‘cfv 6543  (class class class)co 7412   ·_o comu 8467  Ncnpi 10842   ·_N cmi 10844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688  df-iota 6495  df-fv 6551  df-ov 7415  df-mi 10872

This theorem is referenced by:  mulidpi  10884  mulclpi  10891  mulcompi  10894  mulasspi  10895  distrpi  10896  mulcanpi  10898  ltmpi  10902