Proof: Divide Simplify 2

Let's prove the following theorem:

if not (d = 0), then (b ⋅ d) ⋅ (c / d) = b ⋅ c

Proof:

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Given
1 not (d = 0)
Proof Table
# Claim Reason
1 (b ⋅ d) ⋅ c = (b ⋅ c) ⋅ d (b ⋅ d) ⋅ c = (b ⋅ c) ⋅ d
2 ((b ⋅ d) ⋅ c) / d = ((b ⋅ c) ⋅ d) / d if (b ⋅ d) ⋅ c = (b ⋅ c) ⋅ d, then ((b ⋅ d) ⋅ c) / d = ((b ⋅ c) ⋅ d) / d
3 d / d = 1 if not (d = 0), then d / d = 1
4 ((b ⋅ c) ⋅ d) / d = (b ⋅ c) ⋅ (d / d) ((b ⋅ c) ⋅ d) / d = (b ⋅ c) ⋅ (d / d)
5 ((b ⋅ c) ⋅ d) / d = (b ⋅ c) ⋅ 1 if d / d = 1 and ((b ⋅ c) ⋅ d) / d = (b ⋅ c) ⋅ (d / d), then ((b ⋅ c) ⋅ d) / d = (b ⋅ c) ⋅ 1
6 (b ⋅ c) ⋅ 1 = b ⋅ c (b ⋅ c) ⋅ 1 = b ⋅ c
7 ((b ⋅ c) ⋅ d) / d = b ⋅ c if (b ⋅ c) ⋅ 1 = b ⋅ c and ((b ⋅ c) ⋅ d) / d = (b ⋅ c) ⋅ 1, then ((b ⋅ c) ⋅ d) / d = b ⋅ c
8 ((b ⋅ d) ⋅ c) / d = b ⋅ c if ((b ⋅ c) ⋅ d) / d = b ⋅ c and ((b ⋅ d) ⋅ c) / d = ((b ⋅ c) ⋅ d) / d, then ((b ⋅ d) ⋅ c) / d = b ⋅ c
9 ((b ⋅ d) ⋅ c) / d = (b ⋅ d) ⋅ (c / d) ((b ⋅ d) ⋅ c) / d = (b ⋅ d) ⋅ (c / d)
10 (b ⋅ d) ⋅ (c / d) = b ⋅ c if ((b ⋅ d) ⋅ c) / d = b ⋅ c and ((b ⋅ d) ⋅ c) / d = (b ⋅ d) ⋅ (c / d), then (b ⋅ d) ⋅ (c / d) = b ⋅ c

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