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Metamath Proof Explorer

Theorem nnesuc

Description: Exponentiation with a successor exponent. Definition 8.30 of TakeutiZaring p. 67. (Contributed by Mario Carneiro, 14-Nov-2014)

Ref Expression
Assertion nnesuc ( ( 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 onesuc ⊢ ( ( 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 ) )