Theorem mulpiord 10875

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 5711	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6906	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·o ‘⟨𝐴, 𝐵⟩))
3		df-ov 7406	. . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4		df-mi 10864	. . . . 5 ⊢ ·N = ( ·o ↾ (N × N))
5	4	fveq1i 6888	. . . 4 ⊢ ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2761	. . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 7406	. . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2798	. 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 397   = wceq 1542   ∈ wcel 2107  ⟨cop 4632   × cxp 5672   ↾ cres 5676  ‘cfv 6539  (class class class)co 7403   ·_o comu 8458  Ncnpi 10834   ·_N cmi 10836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-xp 5680  df-res 5686  df-iota 6491  df-fv 6547  df-ov 7406  df-mi 10864

This theorem is referenced by:  mulidpi  10876  mulclpi  10883  mulcompi  10886  mulasspi  10887  distrpi  10888  mulcanpi  10890  ltmpi  10894