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

Subtract to Zero 3

if a - b = 0, then a = b

Sum to Double

if c = a + a, then c = a ⋅ 2

Proportion Product

if the following are true:

a / b = c / d
not (b = 0)
not (d = 0)

then d ⋅ a = b ⋅ c

Simplify3

((x ⋅ 4) ⋅ 2) - (2 ⋅ 2) = ((x ⋅ 4) ⋅ 2) - 4

Simplify4

((x ⋅ 4) ⋅ 2) - 4 = (x ⋅ 8) - 4

Distribute Subtract2

(c ⋅ a) - (c ⋅ b) = c ⋅ (a - b)

Additive Inverse 2

(a ⋅ 2) + (a ⋅ (-2)) = 0

Half Half One

((a ⋅ 1) / 2) + ((a ⋅ 1) / 2) = a

Multiply by One

if not (c = 0), then (a ⋅ (1 / c)) ⋅ c = a

Substitute 12

if the following are true:

a = b
b ⋅ c = d

then a ⋅ c = d

Divide Numerators

if the following are true:

a / c = b / c
not (c = 0)

then a = b

Divide Numerators 2

if the following are true:

a / c = b / c
not (c = 0)

then b = a

Substitution 2

if the following are true:

a / b = c / d
a = w
b = x
d = z

then w / x = c / z

Move Terms 2 and 4 Theorem

((a ⋅ b) ⋅ c) ⋅ d = ((b ⋅ d) ⋅ a) ⋅ c

Reduction Example

if not (b = 0), then ((a / b) ⋅ c) ⋅ b = a ⋅ c

Cross Multiply Theorem

if the following are true:

a / b = c / d
not (b = 0)
not (d = 0)

then a ⋅ d = b ⋅ c

Multiply by One 2

if the following are true:

a ⋅ b = c
d = b

then a ⋅ d = c

Divide by 1

if a = 1, then b / a = b

Divide by 2

if the following are true:

not (b = 0)
not (2 = 0)
not (b ⋅ 2 = 0)

then (a ⋅ 2) / (b ⋅ 2) = a / b

Algebra

if the following are true:

a = (b + c) + d
b = x + y

then a = ((x + y) + c) + d

Algebra2

if the following are true:

a = ((b + c) + d) + e
b = x + y

then a = (((x + y) + c) + d) + e

Distribute 4

(a ⋅ 2) + (a ⋅ 2) = a ⋅ 4

Angle180 90

if 180 = a + 90, then a = 90

Example: Divide 180 by 2

if a ⋅ 2 = 180, then a = 90

One Eighty 4

if a + a = 180, then a = 90

Simplify 4

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

Simplify 5

(((-1) ⋅ 3) ⋅ x) + (4 ⋅ x) = x

Simplify If

if (3 ⋅ x) + 20 = 4 ⋅ x, then 20 = x

Reorder Terms 3

if ((a + b) + c) + e = ((a + b) + g) + h, then ((a + b) + c) + e = ((a + g) + b) + h

Reorder Terms 4

if (a + b) + c = 180, then a + c = 180 + (b ⋅ (-1))

Reorder Terms 5

if the following are true:

a = b + c
c = d

then b + d = a

Transitive With Four

if the following are true:

a = b
b = d
a = c

then d = c

Reorder Terms 6

if the following are true:

((a + b) + a) + b = 360
not (2 = 0)

then a + b = 180

Reduce Addition

if the following are true:

((a + a) + a) + a = 360
not (4 = 0)

then a = 90

Reduce Addition 2

if the following are true:

a + a = 90
not (2 = 0)

then a = 45

Three Angles

if the following are true:

60 + (a ⋅ 2) = 180
not (2 = 0)

then a = 60

Algebra 3

if the following are true:

x = 12
d = (x ⋅ 2) + 1

then d = 25

Algebra 4

if the following are true:

x = 12
d = x + 13

then d = 25

Zero Plus a 2

if the following are true:

a = 0
b = e

then a + b = e

Zero Plus a Equation

if the following are true:

y = (a + b) / 2
a = 0
b = e

then y = e / 2

Subtraction Example

if x = a - a, then x = 0

Zero Numerator Property Example

if the following are true:

x = 0 / y
not (y = 0)

then x = 0

Subtract Zero Example 3

if the following are true:

b = 0
a = 0

then b - a = 0

Algebra 5

if a + b = 180 - 90, then a + b = 90

Compute 1

if a = 90 - 67, then a = 23

Compute 2

if a = 90 - 23, then a = 67

Algebra 6

(a / b) ⋅ d = (d / b) ⋅ a

Algebra 7

if not (x = 0), then (a ⋅ x) / x = a

Algebra Divide

if the following are true:

a / b = c / d
not (a = 0)
not (d = 0)

then d / b = c / a

Algebra One

if the following are true:

a = 1
a ⋅ a = b

then 1 = b

Squares Theorem

if a = b, then a ⋅ a = b ⋅ b

Algebra Square Sum

if the following are true:

x = (a ⋅ a) + (b ⋅ b)
m = a
n = b

then x = (n ⋅ n) + (m ⋅ m)

Algebra Substitution

if the following are true:

a ⋅ a = (b ⋅ b) + (c ⋅ c)
a = x
b = y

then x ⋅ x = (y ⋅ y) + (c ⋅ c)

Reorder Terms 7

((a ⋅ b) ⋅ c) ⋅ d = ((a ⋅ c) ⋅ b) ⋅ d

Algebra 8

(s / 2) ⋅ (s / 2) = (s ⋅ s) / 4

Algebra 9

if a = b + c, then a - b = c

Algebra 10 Help

a + (b ⋅ ((-1) / c)) = a - (b / c)

Algebra 10

(s ⋅ s) - ((s ⋅ s) / 4) = (3 / 4) ⋅ (s ⋅ s)

Algebra 11

if the following are true:

a = b / c
d = b
c = e

then a = d / e

Algebra 11a

if the following are true:

a = b / c
b = d
c = e

then a = d / e

Square Product Theorem

if a = b ⋅ c, then a ⋅ a = ((b ⋅ c) ⋅ b) ⋅ c

Square Product Example

(s ⋅ s) ⋅ (1 / 4) = (s / 2) ⋅ (s / 2)

Square Root Example

((s ⋅ s) ⋅ (1 / 4))^{(1 / 2)} = s / 2

Reduction Property 2

(3 / 4) ⋅ (s ⋅ s) = 3 ⋅ ((s ⋅ s) ⋅ (1 / 4))

Square Root Example 2

((3 / 4) ⋅ (s ⋅ s))^{(1 / 2)} = (3^{(1 / 2)}) ⋅ (s / 2)

Square Root 2

square root of (x ⋅ x) = x

Algebra 17

if not (s = 0), then s ⋅ (1 / s) = 1

Algebra 17b

if not (s = 0), then (s / 2) / s = 1 / 2

Algebra 16

if not (s = 0), then (a ⋅ (s / 2)) / s = a / 2

Divide Zero 2

if b > 0, then (c - c) / ((b + a) - a) = 0

Algebra 19

if the following are true:

a = b + c
x = y + z
a = x
b = z

then c = y

Substitute Two Numbers

if the following are true:

a = b + c
b = x
c = y

then a = x + y

Zero Plus a

0 + a = a

Add a Number to Both Sides

if a + 90 = 180, then a = 90

Add Number to Both Sides 5

if 9 + x = 14, then x = 5

Substitute Two Variables Theorem

if the following are true:

a = x
b = y

then a ⋅ b = x ⋅ y

Reorder Terms 8

(a ⋅ b) ⋅ (c ⋅ d) = ((a ⋅ c) ⋅ b) ⋅ d

Inverse Example

((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s / 2) ⋅ (s / 2)

Add Equal Terms

if the following are true:

a = x
b = y

then x + y = a + b

Adding Fractions

(a / c) + (b / c) = (a + b) / c

Product is One

if the following are true:

not (a = 0)
not (b = 0)

then (a ⋅ (1 / a)) ⋅ (b ⋅ (1 / b)) = 1

Product is One 2

if the following are true:

not (a = 0)
not (b = 0)

then ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = 1

Product is One 3

if the following are true:

not (a = 0)
not (b = 0)

then (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1

Remove One

if not (a = 0), then ((1 / a) ⋅ a) ⋅ x = x

Remove One 2

if not (a = 0), then (1 / a) ⋅ (a ⋅ x) = x

Removing the Common Term

if the following are true:

a ⋅ x = a ⋅ y
not (a = 0)

then x = y

Multiplying Denominators

if the following are true:

not (a = 0)
not (b = 0)
not (a ⋅ b = 0)

then (1 / a) ⋅ (1 / b) = 1 / (a ⋅ b)

Simplify Product

(a ⋅ (1 / b)) ⋅ (c ⋅ (1 / d)) = (a / b) ⋅ (c / d)

Simplify Product 2

(a ⋅ c) ⋅ ((1 / b) ⋅ (1 / d)) = (a / b) ⋅ (c / d)

Fraction Multiplication

if the following are true:

not (b = 0)
not (d = 0)
not (b ⋅ d = 0)

then (a / b) ⋅ (c / d) = (a ⋅ c) / (b ⋅ d)

Multiply by Inverse 2

(1 / b) ⋅ a = a / b

Divide by Term

if the following are true:

a = b ⋅ c
not (b = 0)

then c = a / b

Negative Exponent

if not (x^m = 0), then x^{((-1) ⋅ m)} = 1 / (x^m)

Add 1 to Exponent

x^{(m + 1)} = (x^m) ⋅ x

Power of 2

x² = x ⋅ x

Power of 2 2

x ⋅ x = x²

Square Root of a Square

(x ⋅ x)^{(1 / 2)} = x

Multiplication Substitution

if the following are true:

a = x
b = y
c = x ⋅ y

then c = a ⋅ b

Power Symmetry

(x^m) ⋅ (xⁿ) = x^{(m + n)}

Addition Substitution

if the following are true:

a = x
b = y
c = z
z = x + y

then c = a + b

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