Theorem List for Intuitionistic Logic Explorer - 8601-8700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | negap0 8601 |
A number is apart from zero iff its negative is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ -𝐴 # 0)) |
|
Theorem | negap0d 8602 |
The negative of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 25-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → -𝐴 # 0) |
|
Theorem | ltleap 8603 |
Less than in terms of non-strict order and apartness. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵))) |
|
Theorem | ltap 8604 |
'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴) |
|
Theorem | gtapii 8605 |
'Greater than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 # 𝐴 |
|
Theorem | ltapii 8606 |
'Less than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 # 𝐵 |
|
Theorem | ltapi 8607 |
'Less than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴) |
|
Theorem | gtapd 8608 |
'Greater than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 # 𝐴) |
|
Theorem | ltapd 8609 |
'Less than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 # 𝐵) |
|
Theorem | leltapd 8610 |
≤ implies 'less than' is 'apart'. (Contributed by
Jim Kingdon,
13-Aug-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴)) |
|
Theorem | ap0gt0 8611 |
A nonnegative number is apart from zero if and only if it is positive.
(Contributed by Jim Kingdon, 11-Aug-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴)) |
|
Theorem | ap0gt0d 8612 |
A nonzero nonnegative number is positive. (Contributed by Jim
Kingdon, 11-Aug-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → 0 < 𝐴) |
|
Theorem | apsub1 8613 |
Subtraction respects apartness. Analogue of subcan2 8196 for apartness.
(Contributed by Jim Kingdon, 6-Jan-2022.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 − 𝐶) # (𝐵 − 𝐶))) |
|
Theorem | subap0 8614 |
Two numbers being apart is equivalent to their difference being apart from
zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) # 0 ↔ 𝐴 # 𝐵)) |
|
Theorem | subap0d 8615 |
Two numbers apart from each other have difference apart from zero.
(Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ,
15-Aug-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) # 0) |
|
Theorem | cnstab 8616 |
Equality of complex numbers is stable. Stability here means
¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 832. This theorem for real
numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim
Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB
𝐴 = 𝐵) |
|
Theorem | aprcl 8617 |
Reverse closure for apartness. (Contributed by Jim Kingdon,
19-Dec-2023.)
|
⊢ (𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
|
Theorem | apsscn 8618* |
The points apart from a given point are complex numbers. (Contributed
by Jim Kingdon, 19-Dec-2023.)
|
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ |
|
Theorem | lt0ap0 8619 |
A number which is less than zero is apart from zero. (Contributed by Jim
Kingdon, 25-Feb-2024.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 # 0) |
|
Theorem | lt0ap0d 8620 |
A real number less than zero is apart from zero. Deduction form.
(Contributed by Jim Kingdon, 24-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → 𝐴 # 0) |
|
Theorem | aptap 8621 |
Complex apartness (as defined at df-ap 8553) is a tight apartness (as
defined at df-tap 7263). (Contributed by Jim Kingdon, 16-Feb-2025.)
|
⊢ # TAp ℂ |
|
4.3.7 Reciprocals
|
|
Theorem | recextlem1 8622 |
Lemma for recexap 8624. (Contributed by Eric Schmidt, 23-May-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵))) |
|
Theorem | recexaplem2 8623 |
Lemma for recexap 8624. (Contributed by Jim Kingdon, 20-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0) |
|
Theorem | recexap 8624* |
Existence of reciprocal of nonzero complex number. (Contributed by Jim
Kingdon, 20-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1) |
|
Theorem | mulap0 8625 |
The product of two numbers apart from zero is apart from zero. Lemma
2.15 of [Geuvers], p. 6. (Contributed
by Jim Kingdon, 22-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0) |
|
Theorem | mulap0b 8626 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) |
|
Theorem | mulap0i 8627 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 23-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 # 0 & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 · 𝐵) # 0 |
|
Theorem | mulap0bd 8628 |
The product of two numbers apart from zero is apart from zero. Exercise
11.11 of [HoTT], p. (varies).
(Contributed by Jim Kingdon,
24-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) |
|
Theorem | mulap0d 8629 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 23-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) # 0) |
|
Theorem | mulap0bad 8630 |
A factor of a complex number apart from zero is apart from zero.
Partial converse of mulap0d 8629 and consequence of mulap0bd 8628.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐴 # 0) |
|
Theorem | mulap0bbd 8631 |
A factor of a complex number apart from zero is apart from zero.
Partial converse of mulap0d 8629 and consequence of mulap0bd 8628.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐵 # 0) |
|
Theorem | mulcanapd 8632 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanap2d 8633 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanapad 8634 |
Cancellation of a nonzero factor on the left in an equation. One-way
deduction form of mulcanapd 8632. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | mulcanap2ad 8635 |
Cancellation of a nonzero factor on the right in an equation. One-way
deduction form of mulcanap2d 8633. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | mulcanap 8636 |
Cancellation law for multiplication (full theorem form). (Contributed by
Jim Kingdon, 21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanap2 8637 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanapi 8638 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵) |
|
Theorem | muleqadd 8639 |
Property of numbers whose product equals their sum. Equation 5 of
[Kreyszig] p. 12. (Contributed by NM,
13-Nov-2006.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1)) |
|
Theorem | receuap 8640* |
Existential uniqueness of reciprocals. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) |
|
Theorem | mul0eqap 8641 |
If two numbers are apart from each other and their product is zero, one
of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵)
& ⊢ (𝜑 → (𝐴 · 𝐵) = 0) ⇒ ⊢ (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0)) |
|
Theorem | recapb 8642* |
A complex number has a multiplicative inverse if and only if it is apart
from zero. Theorem 11.2.4 of [HoTT], p.
(varies), generalized from
real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)) |
|
4.3.8 Division
|
|
Syntax | cdiv 8643 |
Extend class notation to include division.
|
class / |
|
Definition | df-div 8644* |
Define division. Theorem divmulap 8646 relates it to multiplication, and
divclap 8649 and redivclap 8702 prove its closure laws. (Contributed by NM,
2-Feb-1995.) Use divvalap 8645 instead. (Revised by Mario Carneiro,
1-Apr-2014.) (New usage is discouraged.)
|
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦
(℩𝑧 ∈
ℂ (𝑦 · 𝑧) = 𝑥)) |
|
Theorem | divvalap 8645* |
Value of division: the (unique) element 𝑥 such that
(𝐵
· 𝑥) = 𝐴. This is meaningful
only when 𝐵 is apart from
zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
|
Theorem | divmulap 8646 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴)) |
|
Theorem | divmulap2 8647 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵))) |
|
Theorem | divmulap3 8648 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) |
|
Theorem | divclap 8649 |
Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) |
|
Theorem | recclap 8650 |
Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ) |
|
Theorem | divcanap2 8651 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divcanap1 8652 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | diveqap0 8653 |
A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) |
|
Theorem | divap0b 8654 |
The ratio of numbers apart from zero is apart from zero. (Contributed by
Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0)) |
|
Theorem | divap0 8655 |
The ratio of numbers apart from zero is apart from zero. (Contributed by
Jim Kingdon, 22-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / 𝐵) # 0) |
|
Theorem | recap0 8656 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) # 0) |
|
Theorem | recidap 8657 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1) |
|
Theorem | recidap2 8658 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1) |
|
Theorem | divrecap 8659 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
|
Theorem | divrecap2 8660 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
|
Theorem | divassap 8661 |
An associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
|
Theorem | div23ap 8662 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) |
|
Theorem | div32ap 8663 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵))) |
|
Theorem | div13ap 8664 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴)) |
|
Theorem | div12ap 8665 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶))) |
|
Theorem | divmulassap 8666 |
An associative law for division and multiplication. (Contributed by Jim
Kingdon, 24-Jan-2022.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷))) |
|
Theorem | divmulasscomap 8667 |
An associative/commutative law for division and multiplication.
(Contributed by Jim Kingdon, 24-Jan-2022.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷))) |
|
Theorem | divdirap 8668 |
Distribution of division over addition. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) |
|
Theorem | divcanap3 8669 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4 8670 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | div11ap 8671 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | dividap 8672 |
A number divided by itself is one. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1) |
|
Theorem | div0ap 8673 |
Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0) |
|
Theorem | div1 8674 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof
shortened by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) |
|
Theorem | 1div1e1 8675 |
1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
7-Dec-2018.)
|
⊢ (1 / 1) = 1 |
|
Theorem | diveqap1 8676 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) |
|
Theorem | divnegap 8677 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
|
Theorem | muldivdirap 8678 |
Distribution of division over addition with a multiplication.
(Contributed by Jim Kingdon, 11-Nov-2021.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶))) |
|
Theorem | divsubdirap 8679 |
Distribution of division over subtraction. (Contributed by NM,
4-Mar-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
|
Theorem | recrecap 8680 |
A number is equal to the reciprocal of its reciprocal. (Contributed by
Jim Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴) |
|
Theorem | rec11ap 8681 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | rec11rap 8682 |
Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴)) |
|
Theorem | divmuldivap 8683 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷))) |
|
Theorem | divdivdivap 8684 |
Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by
Jim Kingdon, 25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))) |
|
Theorem | divcanap5 8685 |
Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵)) |
|
Theorem | divmul13ap 8686 |
Swap the denominators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐵 / 𝐶) · (𝐴 / 𝐷))) |
|
Theorem | divmul24ap 8687 |
Swap the numerators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 / 𝐷) · (𝐵 / 𝐶))) |
|
Theorem | divmuleqap 8688 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) = (𝐵 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐵 · 𝐶))) |
|
Theorem | recdivap 8689 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) |
|
Theorem | divcanap6 8690 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1) |
|
Theorem | divdiv32ap 8691 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)) |
|
Theorem | divcanap7 8692 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵)) |
|
Theorem | dmdcanap 8693 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · (𝐶 / 𝐴)) = (𝐶 / 𝐵)) |
|
Theorem | divdivap1 8694 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) |
|
Theorem | divdivap2 8695 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵)) |
|
Theorem | recdivap2 8696 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵))) |
|
Theorem | ddcanap 8697 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵) |
|
Theorem | divadddivap 8698 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) + (𝐵 / 𝐷)) = (((𝐴 · 𝐷) + (𝐵 · 𝐶)) / (𝐶 · 𝐷))) |
|
Theorem | divsubdivap 8699 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) − (𝐵 / 𝐷)) = (((𝐴 · 𝐷) − (𝐵 · 𝐶)) / (𝐶 · 𝐷))) |
|
Theorem | conjmulap 8700 |
Two numbers whose reciprocals sum to 1 are called "conjugates" and
satisfy
this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
⊢ (((𝑃 ∈ ℂ ∧ 𝑃 # 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 # 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1)) |