Theorem List for Intuitionistic Logic Explorer - 8601-8700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | divdirap 8601 |
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 8602 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4 8603 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | div11ap 8604 |
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 8605 |
A number divided by itself is one. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1) |
|
Theorem | div0ap 8606 |
Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0) |
|
Theorem | div1 8607 |
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 8608 |
1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
7-Dec-2018.)
|
⊢ (1 / 1) = 1 |
|
Theorem | diveqap1 8609 |
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 8610 |
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 8611 |
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 8612 |
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 8613 |
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 8614 |
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 8615 |
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 8616 |
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 8617 |
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 8618 |
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 8619 |
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 8620 |
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 8621 |
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 8622 |
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 8623 |
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 8624 |
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 8625 |
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 8626 |
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 8627 |
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 8628 |
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 8629 |
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 8630 |
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 8631 |
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 8632 |
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 8633 |
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)) |
|
Theorem | rerecclap 8634 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) |
|
Theorem | redivclap 8635 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ) |
|
Theorem | eqneg 8636 |
A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.)
(Revised by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
|
Theorem | eqnegd 8637 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8636. (Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
|
Theorem | eqnegad 8638 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8636. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) |
|
Theorem | div2negap 8639 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) |
|
Theorem | divneg2ap 8640 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵)) |
|
Theorem | recclapzi 8641 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℂ) |
|
Theorem | recap0apzi 8642 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) # 0) |
|
Theorem | recidapzi 8643 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1) |
|
Theorem | div1i 8644 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 / 1) = 𝐴 |
|
Theorem | eqnegi 8645 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) |
|
Theorem | recclapi 8646 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (1 / 𝐴) ∈ ℂ |
|
Theorem | recidapi 8647 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 |
|
Theorem | recrecapi 8648 |
A number is equal to the reciprocal of its reciprocal. Theorem I.10
of [Apostol] p. 18. (Contributed by
NM, 9-Feb-1995.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (1 / (1 / 𝐴)) = 𝐴 |
|
Theorem | dividapi 8649 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (𝐴 / 𝐴) = 1 |
|
Theorem | div0api 8650 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (0 / 𝐴) = 0 |
|
Theorem | divclapzi 8651 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ) |
|
Theorem | divcanap1zi 8652 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | divcanap2zi 8653 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divrecapzi 8654 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
|
Theorem | divcanap3zi 8655 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4zi 8656 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | rec11api 8657 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ ((𝐴 # 0 ∧ 𝐵 # 0) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | divclapi 8658 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) ∈ ℂ |
|
Theorem | divcanap2i 8659 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 |
|
Theorem | divcanap1i 8660 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 |
|
Theorem | divrecapi 8661 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)) |
|
Theorem | divcanap3i 8662 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 |
|
Theorem | divcanap4i 8663 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 |
|
Theorem | divap0i 8664 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 # 0 & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) # 0 |
|
Theorem | rec11apii 8665 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 # 0 & ⊢ 𝐵 # 0
⇒ ⊢ ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵) |
|
Theorem | divassapzi 8666 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ (𝐶 # 0 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
|
Theorem | divmulapzi 8667 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴)) |
|
Theorem | divdirapzi 8668 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ (𝐶 # 0 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) |
|
Theorem | divdiv23apzi 8669 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐵 # 0 ∧ 𝐶 # 0) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)) |
|
Theorem | divmulapi 8670 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴) |
|
Theorem | divdiv32api 8671 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵) |
|
Theorem | divassapi 8672 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)) |
|
Theorem | divdirapi 8673 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)) |
|
Theorem | div23api 8674 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵) |
|
Theorem | div11api 8675 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵) |
|
Theorem | divmuldivapi 8676 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0
⇒ ⊢ ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)) |
|
Theorem | divmul13api 8677 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0
⇒ ⊢ ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷)) |
|
Theorem | divadddivapi 8678 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0
⇒ ⊢ ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷)) |
|
Theorem | divdivdivapi 8679 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0 & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)) |
|
Theorem | rerecclapzi 8680 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ
⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℝ) |
|
Theorem | rerecclapi 8681 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 # 0
⇒ ⊢ (1 / 𝐴) ∈ ℝ |
|
Theorem | redivclapzi 8682 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ) |
|
Theorem | redivclapi 8683 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ |
|
Theorem | div1d 8684 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 / 1) = 𝐴) |
|
Theorem | recclapd 8685 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
|
Theorem | recap0d 8686 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) # 0) |
|
Theorem | recidapd 8687 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) |
|
Theorem | recidap2d 8688 |
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 8689 |
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 8690 |
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 8691 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (0 / 𝐴) = 0) |
|
Theorem | apmul1 8692 |
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 8693 |
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 8694 |
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 8695 |
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 8696 |
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 8697 |
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 8698 |
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 8699 |
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 8700 |
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) |