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PROOF 15

Prove.

a⁰ = 1

On the positive reals.

What you may use

A1aᵐ⁺ⁿ = aᵐaⁿ (m, n natural)
law of exponents for natural exponents
A2aᵐ⁺ⁿ = aᵐaⁿ for integer exponents too
demand that the law extend
A3if a > 0 then a·a⁻¹ = 1
existence of multiplicative inverses
A4(xy)z = x(yz)
associativity of multiplication
A5x × 1 = x
multiplicative identity
A6n + 0 = 0 + n = n
additive identity for exponents
A7x × 0 = 0
multiplication by 0
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For a natural number n, aⁿ is a multiplied n times.

Proof.

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