Proof: Divide Simplify
Let's prove the following theorem:
(b ⋅ d) ⋅ (a / b) = d ⋅ a
Proof:
| # | Claim | Reason |
|---|---|---|
| 1 | (b ⋅ d) ⋅ a = (d ⋅ a) ⋅ b | (b ⋅ d) ⋅ a = (d ⋅ a) ⋅ b |
| 2 | ((b ⋅ d) ⋅ a) / b = ((d ⋅ a) ⋅ b) / b | if (b ⋅ d) ⋅ a = (d ⋅ a) ⋅ b, then ((b ⋅ d) ⋅ a) / b = ((d ⋅ a) ⋅ b) / b |
| 3 | b / b = 1 | b / b = 1 |
| 4 | ((d ⋅ a) ⋅ b) / b = (d ⋅ a) ⋅ (b / b) | ((d ⋅ a) ⋅ b) / b = (d ⋅ a) ⋅ (b / b) |
| 5 | ((d ⋅ a) ⋅ b) / b = (d ⋅ a) ⋅ 1 | if ((d ⋅ a) ⋅ b) / b = (d ⋅ a) ⋅ (b / b) and b / b = 1, then ((d ⋅ a) ⋅ b) / b = (d ⋅ a) ⋅ 1 |
| 6 | (d ⋅ a) ⋅ 1 = d ⋅ a | (d ⋅ a) ⋅ 1 = d ⋅ a |
| 7 | ((d ⋅ a) ⋅ b) / b = d ⋅ a | if ((d ⋅ a) ⋅ b) / b = (d ⋅ a) ⋅ 1 and (d ⋅ a) ⋅ 1 = d ⋅ a, then ((d ⋅ a) ⋅ b) / b = d ⋅ a |
| 8 | ((b ⋅ d) ⋅ a) / b = d ⋅ a | if ((b ⋅ d) ⋅ a) / b = ((d ⋅ a) ⋅ b) / b and ((d ⋅ a) ⋅ b) / b = d ⋅ a, then ((b ⋅ d) ⋅ a) / b = d ⋅ a |
| 9 | ((b ⋅ d) ⋅ a) / b = (b ⋅ d) ⋅ (a / b) | ((b ⋅ d) ⋅ a) / b = (b ⋅ d) ⋅ (a / b) |
| 10 | (b ⋅ d) ⋅ (a / b) = d ⋅ a | if ((b ⋅ d) ⋅ a) / b = (b ⋅ d) ⋅ (a / b) and ((b ⋅ d) ⋅ a) / b = d ⋅ a, then (b ⋅ d) ⋅ (a / b) = d ⋅ a |
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