Theorem mulpiord 10798

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 5661	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6853	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( ·o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( ·o ‘⟨𝐴, 𝐵⟩))
3		df-ov 7361	. . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘⟨𝐴, 𝐵⟩)
4		df-mi 10787	. . . . 5 ⊢ ·N = ( ·o ↾ (N × N))
5	4	fveq1i 6835	. . . 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 7361	. . 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 1541   ∈ wcel 2113  ⟨cop 4586   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ·_o comu 8395  Ncnpi 10757   ·_N cmi 10759

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-res 5636  df-iota 6448  df-fv 6500  df-ov 7361  df-mi 10787

This theorem is referenced by:  mulidpi  10799  mulclpi  10806  mulcompi  10809  mulasspi  10810  distrpi  10811  mulcanpi  10813  ltmpi  10817