Proof: Addition Theorem

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	a ⋅ 1 = a	a ⋅ 1 = a (Multiplicative Identity)
2	(a ⋅ 1) + a = a + a	if a ⋅ 1 = a, then (a ⋅ 1) + a = a + a (Additive Property of Equality)
3	a + a = (a ⋅ 1) + a	if (a ⋅ 1) + a = a + a, then a + a = (a ⋅ 1) + a (Symmetric Property of Equality)
4	a = a ⋅ 1	a = a ⋅ 1 (multiplicative_identity_2)
5	a + a = (a ⋅ 1) + (a ⋅ 1)	if a + a = (a ⋅ 1) + a and a = a ⋅ 1, then a + a = (a ⋅ 1) + (a ⋅ 1) (Term 2 Substitution)
6	(a ⋅ 1) + (a ⋅ 1) = a ⋅ (1 + 1)	(a ⋅ 1) + (a ⋅ 1) = a ⋅ (1 + 1) (Distributive Property Variation 4)
7	1 + 1 = 2	1 + 1 = 2 (Fixed Value Statement)
8	a ⋅ (1 + 1) = a ⋅ 2	if 1 + 1 = 2, then a ⋅ (1 + 1) = a ⋅ 2 (Multiplicative Property of Equality Variation 1)
9	(a ⋅ 1) + (a ⋅ 1) = a ⋅ 2	if (a ⋅ 1) + (a ⋅ 1) = a ⋅ (1 + 1) and a ⋅ (1 + 1) = a ⋅ 2, then (a ⋅ 1) + (a ⋅ 1) = a ⋅ 2 (Transitive Property of Equality)
10	a + a = a ⋅ 2	if a + a = (a ⋅ 1) + (a ⋅ 1) and (a ⋅ 1) + (a ⋅ 1) = a ⋅ 2, then a + a = a ⋅ 2 (Transitive Property of Equality)

Comments

Please log in to add comments