Theorem mulpiord 5025

Description: Positive integer multiplication in terms of ordinal multiplication.

Assertion

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

Proof of Theorem mulpiord

Step	Hyp	Ref	Expression
1		opelxpi 3223	. 2 ⊢ ((A ∈ N ⋀ B ∈ N) → ⟨A, B⟩ ∈ (N × N))
2		fvres 3740	. . 3 ⊢ (⟨A, B⟩ ∈ (N × N) → (( ·o ↾ (N × N)) ‘⟨A, B⟩) = ( ·o ‘⟨A, B⟩))
3		df-opr 3971	. . . 4 ⊢ (A ·N B) = ( ·N ‘⟨A, B⟩)
4		df-mi 5014	. . . . 5 ⊢ ·N = ( ·o ↾ (N × N))
5	4	fveq1i 3731	. . . 4 ⊢ ( ·N ‘⟨A, B⟩) = (( ·o ↾ (N × N)) ‘⟨A, B⟩)
6	3, 5	eqtr 1498	. . 3 ⊢ (A ·N B) = (( ·o ↾ (N × N)) ‘⟨A, B⟩)
7		df-opr 3971	. . 3 ⊢ (A ·o B) = ( ·o ‘⟨A, B⟩)
8	2, 6, 7	3eqtr4g 1534	. 2 ⊢ (⟨A, B⟩ ∈ (N × N) → (A ·N B) = (A ·o B))
9	1, 8	syl 10	1 ⊢ ((A ∈ N ⋀ B ∈ N) → (A ·N B) = (A ·o B))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ⟨cop 2415   × cxp 3174   ↾ cres 3178   ‘cfv 3188  (class class class)co 3969   ·_o comu 4137  Ncnpi 4984   ·_N cmi 4986

This theorem is referenced by:  mulidpi 5026  mulclpi 5033  mulcompi 5036  mulasspi 5037  distrpi 5038  mulcanpi 5039  ltmpi 5043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-mi 5014

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