Distributive Property 2

Equality Example

Distributive Property 3

Distributive Property 5

Transitive Property of Equality Variation 2

Subtract Commutative

Commutative Property Example 2

Subtract Commutative 2

Multiplicative Identity 3

Proof: Simplify 4

Let's prove the following theorem:

(((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = 20

Proof:

View as a tree | View dependent proofs | Try proving it

Proof Table

#	Claim	Reason
1	(((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20	(((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 (associative)
2	(((-1) ⋅ 3) ⋅ x) + (3 ⋅ x) = (((-1) ⋅ 3) + 3) ⋅ x	(((-1) ⋅ 3) ⋅ x) + (3 ⋅ x) = (((-1) ⋅ 3) + 3) ⋅ x (distributive_property_5)
3	((-1) ⋅ 3) + 3 = 0	((-1) ⋅ 3) + 3 = 0 (subtract_commutative_2)
4	(((-1) ⋅ 3) + 3) ⋅ x = 0 ⋅ x	if ((-1) ⋅ 3) + 3 = 0, then (((-1) ⋅ 3) + 3) ⋅ x = 0 ⋅ x (Multiplicative Property of Equality)
5	0 ⋅ x = 0	0 ⋅ x = 0 (multiply_by_0)
6	(((-1) ⋅ 3) + 3) ⋅ x = 0	if (((-1) ⋅ 3) + 3) ⋅ x = 0 ⋅ x and 0 ⋅ x = 0, then (((-1) ⋅ 3) + 3) ⋅ x = 0 (Transitive Property of Equality)
7	(((-1) ⋅ 3) ⋅ x) + (3 ⋅ x) = 0	if (((-1) ⋅ 3) ⋅ x) + (3 ⋅ x) = (((-1) ⋅ 3) + 3) ⋅ x and (((-1) ⋅ 3) + 3) ⋅ x = 0, then (((-1) ⋅ 3) ⋅ x) + (3 ⋅ x) = 0 (Transitive Property of Equality)
8	((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 = 0 + 20	if (((-1) ⋅ 3) ⋅ x) + (3 ⋅ x) = 0, then ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 = 0 + 20 (Additive Property of Equality)
9	0 + 20 = 20	0 + 20 = 20 (zero_plus_a)
10	((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 = 20	if ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 = 0 + 20 and 0 + 20 = 20, then ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 = 20 (Transitive Property of Equality)
11	(((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = 20	if (((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 and ((((-1) ⋅ 3) ⋅ x) + (3 ⋅ x)) + 20 = 20, then (((-1) ⋅ 3) ⋅ x) + ((3 ⋅ x) + 20) = 20 (Transitive Property of Equality)

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