Theorem List for Intuitionistic Logic Explorer - 8701-8800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | recidap2d 8701 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
|
Theorem | recrecapd 8702 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / (1 / 𝐴)) = 𝐴) |
|
Theorem | dividapd 8703 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐴) = 1) |
|
Theorem | div0apd 8704 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (0 / 𝐴) = 0) |
|
Theorem | apmul1 8705 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶))) |
|
Theorem | apmul2 8706 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐶 · 𝐴) # (𝐶 · 𝐵))) |
|
Theorem | divclapd 8707 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) |
|
Theorem | divcanap1d 8708 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | divcanap2d 8709 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divrecapd 8710 |
Relationship between division and reciprocal. Theorem I.9 of
[Apostol] p. 18. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
|
Theorem | divrecap2d 8711 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
|
Theorem | divcanap3d 8712 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4d 8713 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | diveqap0d 8714 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) |
|
Theorem | diveqap1d 8715 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 1) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | diveqap1ad 8716 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 8622. Generalization of diveqap1d 8715.
(Contributed by Jim Kingdon, 19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) |
|
Theorem | diveqap0ad 8717 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 8599. (Contributed by Jim Kingdon, 19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) |
|
Theorem | divap1d 8718 |
If two complex numbers are apart, their quotient is apart from one.
(Contributed by Jim Kingdon, 20-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) # 1) |
|
Theorem | divap0bd 8719 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0)) |
|
Theorem | divnegapd 8720 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
|
Theorem | divneg2apd 8721 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵)) |
|
Theorem | div2negapd 8722 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) |
|
Theorem | divap0d 8723 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) # 0) |
|
Theorem | recdivapd 8724 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) |
|
Theorem | recdivap2d 8725 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵))) |
|
Theorem | divcanap6d 8726 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1) |
|
Theorem | ddcanapd 8727 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵) |
|
Theorem | rec11apd 8728 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (1 / 𝐴) = (1 / 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | divmulapd 8729 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴)) |
|
Theorem | apdivmuld 8730 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 26-Dec-2022.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ (𝐵 · 𝐶) # 𝐴)) |
|
Theorem | div32apd 8731 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵))) |
|
Theorem | div13apd 8732 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴)) |
|
Theorem | divdiv32apd 8733 |
Swap denominators in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)) |
|
Theorem | divcanap5d 8734 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵)) |
|
Theorem | divcanap5rd 8735 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵)) |
|
Theorem | divcanap7d 8736 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵)) |
|
Theorem | dmdcanapd 8737 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐵 / 𝐶) · (𝐴 / 𝐵)) = (𝐴 / 𝐶)) |
|
Theorem | dmdcanap2d 8738 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐶)) = (𝐴 / 𝐶)) |
|
Theorem | divdivap1d 8739 |
Division into a fraction. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) |
|
Theorem | divdivap2d 8740 |
Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵)) |
|
Theorem | divmulap2d 8741 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵))) |
|
Theorem | divmulap3d 8742 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) |
|
Theorem | divassapd 8743 |
An associative law for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
|
Theorem | div12apd 8744 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶))) |
|
Theorem | div23apd 8745 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) |
|
Theorem | divdirapd 8746 |
Distribution of division over addition. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) |
|
Theorem | divsubdirapd 8747 |
Distribution of division over subtraction. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
|
Theorem | div11apd 8748 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | divmuldivapd 8749 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
30-Jul-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐷 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))) |
|
Theorem | divmuleqapd 8750 |
Cross-multiply in an equality of ratios. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐷 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐶 · 𝐵))) |
|
Theorem | rerecclapd 8751 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
|
Theorem | redivclapd 8752 |
Closure law for division of reals. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
|
Theorem | diveqap1bd 8753 |
If two complex numbers are equal, their quotient is one. One-way
deduction form of diveqap1 8622. Converse of diveqap1d 8715. (Contributed
by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
|
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = 1) |
|
Theorem | div2subap 8754 |
Swap the order of subtraction in a division. (Contributed by Scott
Fenton, 24-Jun-2013.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 # 𝐷)) → ((𝐴 − 𝐵) / (𝐶 − 𝐷)) = ((𝐵 − 𝐴) / (𝐷 − 𝐶))) |
|
Theorem | div2subapd 8755 |
Swap subtrahend and minuend inside the numerator and denominator of a
fraction. Deduction form of div2subap 8754. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 𝐷) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) / (𝐶 − 𝐷)) = ((𝐵 − 𝐴) / (𝐷 − 𝐶))) |
|
Theorem | subrecap 8756 |
Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) − (1 / 𝐵)) = ((𝐵 − 𝐴) / (𝐴 · 𝐵))) |
|
Theorem | subrecapi 8757 |
Subtraction of reciprocals. (Contributed by Scott Fenton,
9-Jan-2017.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 # 0 & ⊢ 𝐵 # 0
⇒ ⊢ ((1 / 𝐴) − (1 / 𝐵)) = ((𝐵 − 𝐴) / (𝐴 · 𝐵)) |
|
Theorem | subrecapd 8758 |
Subtraction of reciprocals. (Contributed by Scott Fenton,
9-Jan-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴) − (1 / 𝐵)) = ((𝐵 − 𝐴) / (𝐴 · 𝐵))) |
|
Theorem | mvllmulapd 8759 |
Move LHS left multiplication to RHS. (Contributed by Jim Kingdon,
10-Jun-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → (𝐴 · 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = (𝐶 / 𝐴)) |
|
4.3.9 Ordering on reals (cont.)
|
|
Theorem | ltp1 8760 |
A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
|
Theorem | lep1 8761 |
A number is less than or equal to itself plus 1. (Contributed by NM,
5-Jan-2006.)
|
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1)) |
|
Theorem | ltm1 8762 |
A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
|
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) |
|
Theorem | lem1 8763 |
A number minus 1 is less than or equal to itself. (Contributed by Mario
Carneiro, 2-Oct-2015.)
|
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴) |
|
Theorem | letrp1 8764 |
A transitive property of 'less than or equal' and plus 1. (Contributed by
NM, 5-Aug-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ (𝐵 + 1)) |
|
Theorem | p1le 8765 |
A transitive property of plus 1 and 'less than or equal'. (Contributed by
NM, 16-Aug-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
|
Theorem | recgt0 8766 |
The reciprocal of a positive number is positive. Exercise 4 of [Apostol]
p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro,
27-May-2016.)
|
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) |
|
Theorem | prodgt0gt0 8767 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 8768 for the same theorem with 0 < 𝐴
replaced by the weaker condition 0 ≤ 𝐴. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵) |
|
Theorem | prodgt0 8768 |
Infer that a multiplicand is positive from a nonnegative multiplier and
positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario
Carneiro, 27-May-2016.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵) |
|
Theorem | prodgt02 8769 |
Infer that a multiplier is positive from a nonnegative multiplicand and
positive product. (Contributed by NM, 24-Apr-2005.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐵 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐴) |
|
Theorem | prodge0 8770 |
Infer that a multiplicand is nonnegative from a positive multiplier and
nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario
Carneiro, 27-May-2016.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐵) |
|
Theorem | prodge02 8771 |
Infer that a multiplier is nonnegative from a positive multiplicand and
nonnegative product. (Contributed by NM, 2-Jul-2005.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐵 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐴) |
|
Theorem | ltmul2 8772 |
Multiplication of both sides of 'less than' by a positive number. Theorem
I.19 of [Apostol] p. 20. (Contributed by
NM, 13-Feb-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
|
Theorem | lemul2 8773 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by NM, 16-Mar-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
|
Theorem | lemul1a 8774 |
Multiplication of both sides of 'less than or equal to' by a nonnegative
number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies).
(Contributed by NM, 21-Feb-2005.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) |
|
Theorem | lemul2a 8775 |
Multiplication of both sides of 'less than or equal to' by a nonnegative
number. (Contributed by Paul Chapman, 7-Sep-2007.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)) |
|
Theorem | ltmul12a 8776 |
Comparison of product of two positive numbers. (Contributed by NM,
30-Dec-2005.)
|
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 < 𝐷))) → (𝐴 · 𝐶) < (𝐵 · 𝐷)) |
|
Theorem | lemul12b 8777 |
Comparison of product of two nonnegative numbers. (Contributed by NM,
22-Feb-2008.)
|
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 ≤ 𝐷))) → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))) |
|
Theorem | lemul12a 8778 |
Comparison of product of two nonnegative numbers. (Contributed by NM,
22-Feb-2008.)
|
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))) |
|
Theorem | mulgt1 8779 |
The product of two numbers greater than 1 is greater than 1. (Contributed
by NM, 13-Feb-2005.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (1 < 𝐴 ∧ 1 < 𝐵)) → 1 < (𝐴 · 𝐵)) |
|
Theorem | ltmulgt11 8780 |
Multiplication by a number greater than 1. (Contributed by NM,
24-Dec-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵))) |
|
Theorem | ltmulgt12 8781 |
Multiplication by a number greater than 1. (Contributed by NM,
24-Dec-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐵 · 𝐴))) |
|
Theorem | lemulge11 8782 |
Multiplication by a number greater than or equal to 1. (Contributed by
NM, 17-Dec-2005.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐴 · 𝐵)) |
|
Theorem | lemulge12 8783 |
Multiplication by a number greater than or equal to 1. (Contributed by
Paul Chapman, 21-Mar-2011.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐵 · 𝐴)) |
|
Theorem | ltdiv1 8784 |
Division of both sides of 'less than' by a positive number. (Contributed
by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
|
Theorem | lediv1 8785 |
Division of both sides of a less than or equal to relation by a positive
number. (Contributed by NM, 18-Nov-2004.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶))) |
|
Theorem | gt0div 8786 |
Division of a positive number by a positive number. (Contributed by NM,
28-Sep-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) |
|
Theorem | ge0div 8787 |
Division of a nonnegative number by a positive number. (Contributed by
NM, 28-Sep-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) |
|
Theorem | divgt0 8788 |
The ratio of two positive numbers is positive. (Contributed by NM,
12-Oct-1999.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 / 𝐵)) |
|
Theorem | divge0 8789 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by NM, 27-Sep-1999.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) |
|
Theorem | ltmuldiv 8790 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro,
27-May-2016.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶))) |
|
Theorem | ltmuldiv2 8791 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 18-Nov-2004.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶))) |
|
Theorem | ltdivmul 8792 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 18-Nov-2004.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐶 · 𝐵))) |
|
Theorem | ledivmul 8793 |
'Less than or equal to' relationship between division and multiplication.
(Contributed by NM, 9-Dec-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵))) |
|
Theorem | ltdivmul2 8794 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 24-Feb-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 · 𝐶))) |
|
Theorem | lt2mul2div 8795 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 8-Jan-2006.)
|
⊢ (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) |
|
Theorem | ledivmul2 8796 |
'Less than or equal to' relationship between division and multiplication.
(Contributed by NM, 9-Dec-2005.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶))) |
|
Theorem | lemuldiv 8797 |
'Less than or equal' relationship between division and multiplication.
(Contributed by NM, 10-Mar-2006.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) |
|
Theorem | lemuldiv2 8798 |
'Less than or equal' relationship between division and multiplication.
(Contributed by NM, 10-Mar-2006.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) |
|
Theorem | ltrec 8799 |
The reciprocal of both sides of 'less than'. (Contributed by NM,
26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))) |
|
Theorem | lerec 8800 |
The reciprocal of both sides of 'less than or equal to'. (Contributed by
NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) |