Proof: Cross Multiply Theorem

Let's prove the following theorem:

if the following are true:

a / b = c / d
not (b = 0)
not (d = 0)

then a ⋅ d = b ⋅ c

For clarity, here is the condition in fraction notation:

$\frac{a}{b} = \frac{c}{d}$

This theorem says that the product of the left numerator and the right denominator is equal to the product of the left denominator and the right numerator.

In the proof, we first multiply both sides by b. Then we show that the left side reduces to a.

Next, we multiply both sides by d. At that point, the left side is:

a ⋅ d

Then we show that the right side reduces to

b ⋅ c

Proof:

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Given

1	a / b = c / d
2	not (b = 0)
3	not (d = 0)

Proof Table

#	Claim	Reason
1	(a / b) ⋅ b = (c / d) ⋅ b	if a / b = c / d, then (a / b) ⋅ b = (c / d) ⋅ b (Multiplicative Property of Equality)
2	(a / b) ⋅ b = a	if not (b = 0), then (a / b) ⋅ b = a (Multiply By 1 Theorem)
3	a = (c / d) ⋅ b	if (a / b) ⋅ b = a and (a / b) ⋅ b = (c / d) ⋅ b, then a = (c / d) ⋅ b (Transitive Property of Equality Variation 2)
4	a ⋅ d = ((c / d) ⋅ b) ⋅ d	if a = (c / d) ⋅ b, then a ⋅ d = ((c / d) ⋅ b) ⋅ d (Multiplicative Property of Equality)
5	((c / d) ⋅ b) ⋅ d = c ⋅ b	if not (d = 0), then ((c / d) ⋅ b) ⋅ d = c ⋅ b (Reduction Example)
6	a ⋅ d = c ⋅ b	if a ⋅ d = ((c / d) ⋅ b) ⋅ d and ((c / d) ⋅ b) ⋅ d = c ⋅ b, then a ⋅ d = c ⋅ b (Transitive Property of Equality)
7	c ⋅ b = b ⋅ c	c ⋅ b = b ⋅ c (Commutative Property of Multiplication)
8	a ⋅ d = b ⋅ c	if a ⋅ d = c ⋅ b and c ⋅ b = b ⋅ c, then a ⋅ d = b ⋅ c (Transitive Property of Equality)

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