Proof: Multiply By 1 Theorem

Let's prove the following theorem:

Using the Inverse Product theorem, we claim that:

(1 / c) ⋅ c = 1

Thus

b ⋅ ((1 / c) ⋅ c) = b ⋅ 1

Proof:

Proof Table

#	Claim	Reason
1	b / c = b ⋅ (1 / c)	b / c = b ⋅ (1 / c) (Division is Multiplication by Inverse)
2	(b / c) ⋅ c = (b ⋅ (1 / c)) ⋅ c	if b / c = b ⋅ (1 / c), then (b / c) ⋅ c = (b ⋅ (1 / c)) ⋅ c (Substitution)
3	(1 / c) ⋅ c = 1	if not (c = 0), then (1 / c) ⋅ c = 1 (Inverse Product Theorem)
4	(b ⋅ (1 / c)) ⋅ c = b ⋅ ((1 / c) ⋅ c)	(b ⋅ (1 / c)) ⋅ c = b ⋅ ((1 / c) ⋅ c) (Associative Property of Multiplication)
5	b ⋅ ((1 / c) ⋅ c) = b ⋅ 1	if (1 / c) ⋅ c = 1, then b ⋅ ((1 / c) ⋅ c) = b ⋅ 1 (Multiplicative Property of Equality Variation 1)
6	b ⋅ 1 = b	b ⋅ 1 = b (Multiplicative Identity)
7	(b / c) ⋅ c = b	if b ⋅ 1 = b and b ⋅ ((1 / c) ⋅ c) = b ⋅ 1 and (b ⋅ (1 / c)) ⋅ c = b ⋅ ((1 / c) ⋅ c) and (b / c) ⋅ c = (b ⋅ (1 / c)) ⋅ c, then (b / c) ⋅ c = b (Transitive Property Application 4)

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