This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nnmsuc

Description: Multiplication with successor. Theorem 4J(A2) of Enderton p. 80. (Contributed by NM, 20-Sep-1995) (Revised by Mario Carneiro, 14-Nov-2014)

		Ref	Expression
	Assertion	nnmsuc	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·_o suc 𝐵 ) = ( ( 𝐴 ·_o 𝐵 ) +_o 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		nnon	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On )
2		onmsuc	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ ω ) → ( 𝐴 ·_o suc 𝐵 ) = ( ( 𝐴 ·_o 𝐵 ) +_o 𝐴 ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·_o suc 𝐵 ) = ( ( 𝐴 ·_o 𝐵 ) +_o 𝐴 ) )