PROOF 01
Prove.
(−1) × (−1) = 1
On the integers ℤ. Two operations: + and ×.
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
−1 is the number satisfying 1 + (−1) = 0.
Proof.
The existence of such a number is guaranteed by A3.A3
∎
Lemma L1
For all a, a × 0 = 0
Proof.
(1)0 + 0 = 0A1
(2)a(0 + 0) = a × 0E
(3)a×0 + a×0 = a×0A6
(4)(a×0 + a×0) + (−(a×0)) = a×0 + (−(a×0))E
(5)a×0 + (a×0 + (−(a×0))) = 0A2, A3
(6)a×0 + 0 = 0A3
(7)a × 0 = 0A1
∎
Lemma L2
If x + a = 0 and y + a = 0, then x = y
Proof.
(1)x = x + 0A1
(2) = x + (y + a)assumption
(3) = x + (a + y)A4
(4) = (x + a) + yA2
(5) = 0 + yassumption
(6) = yA4, A1
∎
Theorem T
(−1) × (−1) = 1
Proof.
(1)1 + (−1) = 0D1
(2)(−1) + 1 = 0(1), A4
(3)(−1)((−1) + 1) = (−1) × 0(2), E
(4)(−1)(−1) + (−1)×1 = (−1) × 0(3), A6
(5)(−1) × 1 = −1A5
(6)(−1)(−1) + (−1) = (−1) × 0(4), (5)
(7)(−1)(−1) + (−1) = 0(6), L1
(8)(−1) × (−1) = 1(7), (1), L2
∎
Remark
Can A6 be removed from this proof?
Remove it and see which lines lose their justification.