Proofs
A proof is a series of claims that lead to a conclusion. Some proofs are conditional, which means that the claims can only be made under certain conditions. Click on a statement to see the proof
if the following are true:
- (a + b) + c = d
- a = e
then (e + b) + c = d
if (a + b) + c = d, then b + c = d + (a ⋅ (-1))
if a + b = c, then ((x + a) + b) + y = (x + c) + y
if (a + b) + c = d, then a + c = d - b
if the following are true:
- a / b = c / d
- d = e
then a / b = c / e
if the following are true:
- a = b / c
- b = d
then a = d / c
if the following are true:
- a = x
- b = y
then a / b = x / y
if the following are true:
- a / b = c
- d = a
then c = d / b
if the following are true:
- c = a / b
- d = a
then c = d / b
if the following are true:
- a / b = c
- d = a
then d / b = c
if the following are true:
- a = w
- b = x
- c = y
- d = z
then (a - b) / (c - d) = (w - x) / (y - z)
if the following are true:
- a = b + c
- b > 0
then a > c
if the following are true:
- f = (a - b) / (c - d)
- a = w
- b = x
- c = y
- d = z
then f = (w - x) / (y - z)
a + (b + c) = (a + b) + c
0 + a = a
(a ⋅ (-1)) + a = 0
((-1) ⋅ a) + a = 0
a + (b ⋅ (-1)) = a - b
a - a = 0
a - 0 = a
(a + b) - c = a + (b - c)
(b + a) - a = b
a = a ⋅ 1
1 ⋅ a = a
(b + c) ⋅ a = (a ⋅ b) + (a ⋅ c)
(a + b) ⋅ c = (a ⋅ c) + (b ⋅ c)
(a ⋅ b) + (a ⋅ c) = a ⋅ (b + c)
(a ⋅ c) + (b ⋅ c) = (a + b) ⋅ c
a ⋅ (1 / b) = a / b
a ⋅ (b / c) = (a ⋅ b) / c
(a ⋅ b) / c = a ⋅ (b / c)
a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c
(a ⋅ b) ⋅ c = (a ⋅ c) ⋅ b
(a ⋅ b) ⋅ c = (b ⋅ c) ⋅ a
(a - b) ⋅ c = (a ⋅ c) - (b ⋅ c)
((a + b) + c) + d = a + ((b + c) + d)
(a + b) + c = (a + c) + b
(a + b) + c = (b + a) + c
(a + b) + c = (c + a) + b
(c + a) + b = (a + b) + c
(a + b) - c = (a - c) + b
((a + b) + c) + ((d + e) + f) = ((((a + b) + c) + d) + e) + f
((((a + b) + c) + d) + e) + f = ((((a + e) + b) + d) + f) + c
((((a + b) + c) + d) + e) + f = ((a + b) + (c + d)) + (e + f)
(0 + a) + b = a + b
((a + b) + c) + d = ((a + c) + b) + d
(a ⋅ (b ⋅ (-1))) ⋅ 2 = (a ⋅ b) ⋅ (-2)
(a ⋅ (-1)) ⋅ (a ⋅ (-1)) = a ⋅ a
(a ⋅ (1 / 2)) + (a ⋅ (1 / 2)) = a
a + a = a ⋅ 2
(a + a) + a = a ⋅ 3
((a + a) + a) + a = a ⋅ 4
((a + b) + a) + b = (a + b) ⋅ 2
(a + b) ⋅ (a + b) = ((a ⋅ a) + ((a ⋅ b) ⋅ 2)) + (b ⋅ b)
if not (b = 0), then (b ⋅ d) ⋅ (a / b) = d ⋅ a
if not (d = 0), then (b ⋅ d) ⋅ (c / d) = b ⋅ c
(0 + (a ⋅ 2)) / 2 = a
if not (2 = 0), then (a ⋅ 2) ⋅ (1 / 2) = a
if not (2 = 0), then ((b ⋅ 2) + (a ⋅ 2)) / 2 = b + a
0 ⋅ a = 0
if not (a = 0), then (1 / a) ⋅ a = 1
(a ⋅ b) ⋅ c = (c ⋅ a) ⋅ b
if not (c = 0), then (b / c) ⋅ c = b
if a ⋅ 1 = b, then a = b
if the following are true:
- a = b / c
- not (c = 0)
then c ⋅ a = b
if a + a = b, then a ⋅ 2 = b
if the following are true:
- a = b
- c = d
then a - c = b - d
if the following are true:
- a ⋅ 2 = b
- not (2 = 0)
then a = b ⋅ (1 / 2)
if the following are true:
- a + a = b
- not (2 = 0)
then a = b ⋅ (1 / 2)
if (a + b) + c = d, then (a + c) + b = d
if (a + b) + c = d, then (b + a) + c = d
if the following are true:
- x = c + d
- y = e + f
then ((a + b) + (c + d)) + (e + f) = ((a + b) + x) + y
if the following are true:
- a ⋅ b = c
- not (b = 0)
then a = c / b
if (a + b) + b = 180, then a + (b ⋅ 2) = 180
if x = (0 + (a ⋅ 2)) / 2, then x = a
if the following are true:
- x = ((b ⋅ 2) + (a ⋅ 2)) / 2
- not (2 = 0)
then x = b + a
if f = a - 0, then f = a
if f = (a + b) - b, then f = a
if the following are true:
- a < b
- c = a
then c < b
if a = b, then x - a = x - b
if the following are true:
- a = b
- f = c - a
then f = c - b
if a = b, then a - c = b - c
if a - b = c, then a = c + b
if a - b = 0, then a = b
if c = a + a, then c = a ⋅ 2
if the following are true:
- a / b = c / d
- not (b = 0)
- not (d = 0)
then d ⋅ a = b ⋅ c
((x ⋅ 4) ⋅ 2) - (2 ⋅ 2) = ((x ⋅ 4) ⋅ 2) - 4
((x ⋅ 4) ⋅ 2) - 4 = (x ⋅ 8) - 4
(c ⋅ a) - (c ⋅ b) = c ⋅ (a - b)
(a ⋅ 2) + (a ⋅ (-2)) = 0
((a ⋅ 1) / 2) + ((a ⋅ 1) / 2) = a
if not (c = 0), then (a ⋅ (1 / c)) ⋅ c = a
if the following are true:
- a = b
- b ⋅ c = d
then a ⋅ c = d
if the following are true:
- a / c = b / c
- not (c = 0)
then a = b
if the following are true:
- a / c = b / c
- not (c = 0)
then b = a
if the following are true:
- a / b = c / d
- a = w
- b = x
- d = z
then w / x = c / z
((a ⋅ b) ⋅ c) ⋅ d = ((b ⋅ d) ⋅ a) ⋅ c
if not (b = 0), then ((a / b) ⋅ c) ⋅ b = a ⋅ c
if the following are true:
- a / b = c / d
- not (b = 0)
- not (d = 0)
then a ⋅ d = b ⋅ c
if the following are true:
- a ⋅ b = c
- d = b
then a ⋅ d = c