Theorem List for Intuitionistic Logic Explorer - 8401-8500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Definition | df-div 8401* |
Define division. Theorem divmulap 8403 relates it to multiplication, and
divclap 8406 and redivclap 8459 prove its closure laws. (Contributed by NM,
2-Feb-1995.) Use divvalap 8402 instead. (Revised by Mario Carneiro,
1-Apr-2014.) (New usage is discouraged.)
|
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦
(℩𝑧 ∈
ℂ (𝑦 · 𝑧) = 𝑥)) |
|
Theorem | divvalap 8402* |
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 8403 |
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 8404 |
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 8405 |
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 8406 |
Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) |
|
Theorem | recclap 8407 |
Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ) |
|
Theorem | divcanap2 8408 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divcanap1 8409 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | diveqap0 8410 |
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 8411 |
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 8412 |
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 8413 |
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 8414 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1) |
|
Theorem | recidap2 8415 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1) |
|
Theorem | divrecap 8416 |
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 8417 |
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 8418 |
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 8419 |
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 8420 |
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 8421 |
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 8422 |
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 8423 |
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 8424 |
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 8425 |
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 8426 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4 8427 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | div11ap 8428 |
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 8429 |
A number divided by itself is one. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1) |
|
Theorem | div0ap 8430 |
Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0) |
|
Theorem | div1 8431 |
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 8432 |
1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
7-Dec-2018.)
|
⊢ (1 / 1) = 1 |
|
Theorem | diveqap1 8433 |
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 8434 |
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 8435 |
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 8436 |
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 8437 |
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 8438 |
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 8439 |
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 8440 |
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 8441 |
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 8442 |
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 8443 |
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 8444 |
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 8445 |
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 8446 |
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 8447 |
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 8448 |
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 8449 |
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 8450 |
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 8451 |
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 8452 |
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 8453 |
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 8454 |
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 8455 |
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 8456 |
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 8457 |
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 8458 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) |
|
Theorem | redivclap 8459 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ) |
|
Theorem | eqneg 8460 |
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 8461 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8460. (Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
|
Theorem | eqnegad 8462 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8460. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) |
|
Theorem | div2negap 8463 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) |
|
Theorem | divneg2ap 8464 |
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 8465 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℂ) |
|
Theorem | recap0apzi 8466 |
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 8467 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1) |
|
Theorem | div1i 8468 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 / 1) = 𝐴 |
|
Theorem | eqnegi 8469 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) |
|
Theorem | recclapi 8470 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (1 / 𝐴) ∈ ℂ |
|
Theorem | recidapi 8471 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 |
|
Theorem | recrecapi 8472 |
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 8473 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (𝐴 / 𝐴) = 1 |
|
Theorem | div0api 8474 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0
⇒ ⊢ (0 / 𝐴) = 0 |
|
Theorem | divclapzi 8475 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ) |
|
Theorem | divcanap1zi 8476 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | divcanap2zi 8477 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divrecapzi 8478 |
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 8479 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4zi 8480 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | rec11api 8481 |
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 8482 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 / 𝐵) ∈ ℂ |
|
Theorem | divcanap2i 8483 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 |
|
Theorem | divcanap1i 8484 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 |
|
Theorem | divrecapi 8485 |
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 8486 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 |
|
Theorem | divcanap4i 8487 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0
⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 |
|
Theorem | divap0i 8488 |
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 8489 |
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 8490 |
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 8491 |
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 8492 |
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 8493 |
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 8494 |
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 8495 |
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 8496 |
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 8497 |
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 8498 |
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 8499 |
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 8500 |
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)) |