PROOF 115

Prove.

In a proportion, the product of the extremes equals the product of the means

On the rational numbers ℚ.

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 b ≠ 0 and d ≠ 0, a : b = c : d means a/b = c/d.

Proof.

A proportion is the statement that two quotients are equal. The colon is notation, not a new operation.convention

∎

Definition D2

a/b means the x satisfying b × x = a.

Proof.

a/b has a value only when such an x exists and is unique.convention

∎

Lemma L1

If b ≠ 0 and bx = by, then x = y.

Proof.

(1)Since b ≠ 0, there is a b⁻¹.A4

(2)b⁻¹(bx) = b⁻¹(by)assumption, (1), E

(3)(b⁻¹b)x = (b⁻¹b)y(2), A1

(4)b⁻¹b = bb⁻¹ = 1A2, A4

(5)1×x = x × 1 = x, 1×y = yA2, A3

(6)x = y(3), (4), (5)

∎

Lemma L2

If b ≠ 0, then a/b has a value.

Proof.

(1)Since b ≠ 0, there is a b⁻¹. Let x = a·b⁻¹.A4

(2)bx = b(a·b⁻¹) = (ba)b⁻¹ = (ab)b⁻¹ = a(bb⁻¹)(1), A1, A2

(3)a(bb⁻¹) = a × 1 = aA4, A3

(4)Hence there exists an x with bx = a.(2), (3)

(5)Such an x is unique.L1

(6)That is, a/b has a value.(4), (5), D2

∎

Theorem T1

For b ≠ 0 and d ≠ 0, if a : b = c : d, then ad = bc.

Proof.

(1)Suppose a : b = c : d.assumption

(2)a/b = c/d, so call this value t.(1), D1

(3)bt = a(2), D2

(4)dt = c(2), D2

(5)ad = (bt)d(3), E

(6)(bt)d = b(td) = b(dt)A1, A2

(7)b(dt) = bc(4), E

(8)ad = bc(5), (6), (7)

∎

Theorem T2

For b ≠ 0 and d ≠ 0, if ad = bc, then a : b = c : d.

Proof.

(1)Suppose ad = bc.assumption

(2)a/b and c/d have values, so let x = a/b, y = c/d.L2

(3)bx = a and dy = c.(2), D2

(4)(bx)d = ad = bc = b(dy)(1), (3), E

(5)(bx)d = b(xd)A1

(6)b(xd) = b(dy) and b ≠ 0, so xd = dy(4), (5), L1

(7)xd = dx, so dx = dy(6), A2

(8)Since d ≠ 0, x = y(7), L1

(9)That is, a/b = c/d, and a : b = c : d.(2), (8), D1

∎

Corollary C

For b ≠ 0 and d ≠ 0, a : b = c : d if and only if ad = bc. The former statement uses division; the latter uses only multiplication.

Proof.

(1)If a : b = c : d, then ad = bc.T1

(2)If ad = bc, then a : b = c : d.T2

(3)Hence the two statements are equivalent.(1), (2)

(4)Unfolding a : b = c : d gives a/b = c/d, which contains division, while ad = bc contains multiplication only.D1, D2

∎

Remark

Can A4 be removed from this proof?

The two directions of the proof use different things. Check which line writes down b⁻¹.