Commutative Property Variation 1
Substitution 2
Subtract Both Sides
Substitute 7

Proof: Substitute 7

Let's prove the following theorem:

if the following are true:
  • a + (b ⋅ (-1)) = c
  • b = d

then a + (d ⋅ (-1)) = c

Proof:

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

Given
1 a + (b ⋅ (-1)) = c
2 b = d
Proof Table
# Claim Reason
1 c = a + (b ⋅ (-1)) if a + (b ⋅ (-1)) = c, then c = a + (b ⋅ (-1))
2 c + b = (a + (b ⋅ (-1))) + b if c = a + (b ⋅ (-1)), then c + b = (a + (b ⋅ (-1))) + b
3 (b ⋅ (-1)) + b = b + (b ⋅ (-1)) (b ⋅ (-1)) + b = b + (b ⋅ (-1))
4 b + (b ⋅ (-1)) = 0 b + (b ⋅ (-1)) = 0
5 (a + (b ⋅ (-1))) + b = a + ((b ⋅ (-1)) + b) (a + (b ⋅ (-1))) + b = a + ((b ⋅ (-1)) + b)
6 c + b = a + ((b ⋅ (-1)) + b) if c + b = (a + (b ⋅ (-1))) + b and (a + (b ⋅ (-1))) + b = a + ((b ⋅ (-1)) + b), then c + b = a + ((b ⋅ (-1)) + b)
7 c + b = a + (b + (b ⋅ (-1))) if c + b = a + ((b ⋅ (-1)) + b) and (b ⋅ (-1)) + b = b + (b ⋅ (-1)), then c + b = a + (b + (b ⋅ (-1)))
8 c + b = a + 0 if c + b = a + (b + (b ⋅ (-1))) and b + (b ⋅ (-1)) = 0, then c + b = a + 0
9 a + 0 = a a + 0 = a
10 c + b = a if c + b = a + 0 and a + 0 = a, then c + b = a
11 a = c + b if c + b = a, then a = c + b
12 a = c + d if a = c + b and b = d, then a = c + d
13 a + (d ⋅ (-1)) = c if a = c + d, then a + (d ⋅ (-1)) = c
Previous Lesson

Comments

Please log in to add comments