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

Proof of Theorem mulpiord

Step	Hyp	Ref	Expression
1		opelxpi 5288	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨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))
5	4	fveq1i 6333	. . . 4 ⊢ ( ·N ‘⟨𝐴, 𝐵⟩) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2792	. . 3 ⊢ (𝐴 ·N 𝐵) = (( ·𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 6795	. . 3 ⊢ (𝐴 ·𝑜 𝐵) = ( ·𝑜 ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2829	. 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 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