Theorem List for Intuitionistic Logic Explorer - 8501-8600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | divcanap1zi 8501 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | divcanap2zi 8502 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divrecapzi 8503 |
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 8504 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4zi 8505 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | rec11api 8506 |
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 8507 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) ∈ ℂ |
|
Theorem | divcanap2i 8508 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 |
|
Theorem | divcanap1i 8509 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 |
|
Theorem | divrecapi 8510 |
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 8511 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 |
|
Theorem | divcanap4i 8512 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 |
|
Theorem | divap0i 8513 |
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 8514 |
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 8515 |
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 8516 |
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 8517 |
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 8518 |
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 8519 |
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 8520 |
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 8521 |
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 8522 |
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 8523 |
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 8524 |
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 8525 |
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 8526 |
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 8527 |
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 8528 |
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 8529 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ
⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℝ) |
|
Theorem | rerecclapi 8530 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 # 0
⇒ ⊢ (1 / 𝐴) ∈ ℝ |
|
Theorem | redivclapzi 8531 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ) |
|
Theorem | redivclapi 8532 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ |
|
Theorem | div1d 8533 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 / 1) = 𝐴) |
|
Theorem | recclapd 8534 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
|
Theorem | recap0d 8535 |
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 8536 |
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 8537 |
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 8538 |
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 8539 |
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 8540 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (0 / 𝐴) = 0) |
|
Theorem | apmul1 8541 |
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 8542 |
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 8543 |
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 8544 |
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 8545 |
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 8546 |
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 8547 |
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 8548 |
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 8549 |
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 8550 |
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 8551 |
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 8552 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 8458. Generalization of diveqap1d 8551.
(Contributed by Jim Kingdon, 19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) |
|
Theorem | diveqap0ad 8553 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 8435. (Contributed by Jim Kingdon, 19-Mar-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) |
|
Theorem | divap1d 8554 |
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 8555 |
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 8556 |
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 8557 |
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 8558 |
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 8559 |
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 8560 |
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 8561 |
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 8562 |
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 8563 |
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 8564 |
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 8565 |
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 8566 |
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 8567 |
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 8568 |
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 8569 |
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 8570 |
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 8571 |
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 8572 |
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 8573 |
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 8574 |
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 8575 |
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 8576 |
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 8577 |
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 8578 |
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 8579 |
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 8580 |
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 8581 |
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 8582 |
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 8583 |
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 8584 |
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 8585 |
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 | rerecclapd 8586 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
|
Theorem | redivclapd 8587 |
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 8588 |
If two complex numbers are equal, their quotient is one. One-way
deduction form of diveqap1 8458. Converse of diveqap1d 8551. (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 8589 |
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 8590 |
Swap subtrahend and minuend inside the numerator and denominator of a
fraction. Deduction form of div2subap 8589. (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 8591 |
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 8592 |
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 8593 |
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 8594 |
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 8595 |
A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
|
Theorem | lep1 8596 |
A number is less than or equal to itself plus 1. (Contributed by NM,
5-Jan-2006.)
|
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1)) |
|
Theorem | ltm1 8597 |
A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
|
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) |
|
Theorem | lem1 8598 |
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 8599 |
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 8600 |
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) |