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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-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))
 
Theoremdivvalap 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))
 
Theoremdivmulap 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))
 
Theoremdivmulap2 8404 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))
 
Theoremdivmulap3 8405 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐵 · 𝐶)))
 
Theoremdivclap 8406 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremrecclap 8407 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ)
 
Theoremdivcanap2 8408 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivcanap1 8409 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdiveqap0 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))
 
Theoremdivap0b 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))
 
Theoremdivap0 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)
 
Theoremrecap0 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)
 
Theoremrecidap 8414 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremrecidap2 8415 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1)
 
Theoremdivrecap 8416 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivrecap2 8417 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
Theoremdivassap 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)))
 
Theoremdiv23ap 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))
 
Theoremdiv32ap 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)))
 
Theoremdiv13ap 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))
 
Theoremdiv12ap 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)))
 
Theoremdivmulassap 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)))
 
Theoremdivmulasscomap 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)))
 
Theoremdivdirap 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)))
 
Theoremdivcanap3 8426 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4 8427 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremdiv11ap 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))
 
Theoremdividap 8429 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1)
 
Theoremdiv0ap 8430 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0)
 
Theoremdiv1 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)
 
Theorem1div1e1 8432 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1
 
Theoremdiveqap1 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))
 
Theoremdivnegap 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))
 
Theoremmuldivdirap 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)))
 
Theoremdivsubdirap 8436 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
Theoremrecrecap 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)
 
Theoremrec11ap 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))
 
Theoremrec11rap 8439 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴))
 
Theoremdivmuldivap 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)))
 
Theoremdivdivdivap 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)))
 
Theoremdivcanap5 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))
 
Theoremdivmul13ap 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)))
 
Theoremdivmul24ap 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)))
 
Theoremdivmuleqap 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)))
 
Theoremrecdivap 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))
 
Theoremdivcanap6 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)
 
Theoremdivdiv32ap 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))
 
Theoremdivcanap7 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))
 
Theoremdmdcanap 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))
 
Theoremdivdivap1 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)))
 
Theoremdivdivap2 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))
 
Theoremrecdivap2 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)))
 
Theoremddcanap 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)
 
Theoremdivadddivap 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)))
 
Theoremdivsubdivap 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)))
 
Theoremconjmulap 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))
 
Theoremrerecclap 8458 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ)
 
Theoremredivclap 8459 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremeqneg 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 = -𝐴𝐴 = 0))
 
Theoremeqnegd 8461 A complex number equals its negative iff it is zero. Deduction form of eqneg 8460. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = -𝐴𝐴 = 0))
 
Theoremeqnegad 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = -𝐴)       (𝜑𝐴 = 0)
 
Theoremdiv2negap 8463 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivneg2ap 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))
 
Theoremrecclapzi 8465 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecap0apzi 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)
 
Theoremrecidapzi 8467 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremdiv1i 8468 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
𝐴 ∈ ℂ       (𝐴 / 1) = 𝐴
 
Theoremeqnegi 8469 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 = -𝐴𝐴 = 0)
 
Theoremrecclapi 8470 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ    &   𝐴 # 0       (1 / 𝐴) ∈ ℂ
 
Theoremrecidapi 8471 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 · (1 / 𝐴)) = 1
 
Theoremrecrecapi 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
 
Theoremdividapi 8473 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 / 𝐴) = 1
 
Theoremdiv0api 8474 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℂ    &   𝐴 # 0       (0 / 𝐴) = 0
 
Theoremdivclapzi 8475 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremdivcanap1zi 8476 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcanap2zi 8477 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivrecapzi 8478 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivcanap3zi 8479 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4zi 8480 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremrec11api 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))
 
Theoremdivclapi 8482 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℂ
 
Theoremdivcanap2i 8483 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴
 
Theoremdivcanap1i 8484 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴
 
Theoremdivrecapi 8485 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))
 
Theoremdivcanap3i 8486 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴
 
Theoremdivcanap4i 8487 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴
 
Theoremdivap0i 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
 
Theoremrec11apii 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)
 
Theoremdivassapzi 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)))
 
Theoremdivmulapzi 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))
 
Theoremdivdirapzi 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)))
 
Theoremdivdiv23apzi 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))
 
Theoremdivmulapi 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)
 
Theoremdivdiv32api 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)
 
Theoremdivassapi 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))
 
Theoremdivdirapi 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))
 
Theoremdiv23api 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)
 
Theoremdiv11api 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)
 
Theoremdivmuldivapi 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))
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