PROOF 16
Prove.
a⁻ⁿ = 1/aⁿ
On the positive reals.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
For a natural number n, aⁿ is a multiplied n times.
Proof.
(1)a¹ = a, and for natural n, aⁿ⁺¹ = aⁿ × a.convention
(2)This definition has meaning only for natural exponents. a⁰ and a⁻ⁿ have no meaning yet.convention
∎
Definition D2
1/x denotes the y satisfying x × y = 1.
Proof.
A reciprocal is not a new kind of number, but a name for the number that multiplies to 1.convention
∎
Lemma L1
If a > 0, then a⁰ = 1
Proof.
(1)0 + 1 = 1A8
(2)a⁰⁺¹ = a⁰ × a¹A1, A2
(3)a⁰⁺¹ = a¹ = a(1), D1, E
(4)a⁰ × a¹ = a⁰ × aD1, E
(5)a⁰ × a = a(2), (3), (4)
(6)1 × a = aA4, A5
(7)a⁰ × a = 1 × a, and a > 0.(5), (6)
(8)a⁰ = 1(7), A6
∎
Lemma L2
If x × y = 1 and x × z = 1, then y = z
Proof.
(1)y = y × 1A5
(2) = y(xz)assumption
(3) = (yx)zA3
(4) = (xy)zA4
(5) = 1 × zassumption
(6) = z × 1A4
(7) = zA5
∎
Lemma L3
a⁻ⁿ × aⁿ = 1
Proof.
(1)(−n) + n = 0A7
(2)a⁽⁻ⁿ⁾⁺ⁿ = a⁻ⁿ × aⁿA1, A2
(3)a⁽⁻ⁿ⁾⁺ⁿ = a⁰(1), E
(4)a⁰ = 1L1
(5)a⁻ⁿ × aⁿ = 1(2), (3), (4)
∎
Theorem T
If a > 0, then a⁻ⁿ = 1/aⁿ
Proof.
(1)a⁻ⁿ × aⁿ = 1L3
(2)aⁿ × a⁻ⁿ = 1(1), A4
(3)Therefore a⁻ⁿ satisfies the condition of D2.(2), D2
(4)The y satisfying aⁿ × y = 1 is unique.L2
(5)a⁻ⁿ = 1/aⁿ(3), (4), D2
∎
Remark
Can A3 be removed from this proof?
Showing a⁻ⁿ × aⁿ = 1 and calling a⁻ⁿ the reciprocal are different statements.