Theorem mulpiord 3807

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 2455	. 2 ⊢ ((A ∈ N ∧ B ∈ N) → ⟨A, B⟩ ∈ (N × N))
2		fvres 2840	. . 3 ⊢ (⟨A, B⟩ ∈ (N × N) → (( ·o ↾ (N × N)) ‘⟨A, B⟩) = ( ·o ‘⟨A, B⟩))
3		df-opr 3003	. . . 4 ⊢ (A ·N B) = ( ·N ‘⟨A, B⟩)
4		df-mi 3796	. . . . 5 ⊢ ·N = ( ·o ↾ (N × N))
5	4	fveq1i 2833	. . . 4 ⊢ ( ·N ‘⟨A, B⟩) = (( ·o ↾ (N × N)) ‘⟨A, B⟩)
6	3, 5	eqtr 1119	. . 3 ⊢ (A ·N B) = (( ·o ↾ (N × N)) ‘⟨A, B⟩)
7		df-opr 3003	. . 3 ⊢ (A ·o B) = ( ·o ‘⟨A, B⟩)
8	2, 6, 7	3eqtr4g 1147	. 2 ⊢ (⟨A, B⟩ ∈ (N × N) → (A ·N B) = (A ·o B))
9	1, 8	syl 12	1 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) = (A ·o B))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408   ↾ cres 2412   ‘cfv 2422  (class class class)co 3001   ·_o comu 3102  Ncnpi 3766   ·_N cmi 3768

This theorem is referenced by:  mulidpi 3808  mulclpi 3815  mulcompi 3818  mulasspi 3819  distrpi 3820  mulcanpi 3821  ltmpi 3825

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-mi 3796

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