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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdiveq0d 11501 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 / 𝐵) = 0)       (𝜑𝐴 = 0)
 
Theoremdiveq1d 11502 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 / 𝐵) = 1)       (𝜑𝐴 = 𝐵)
 
Theoremdiveq1ad 11503 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 11409. Generalization of diveq1d 11502. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
Theoremdiveq0ad 11504 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 11386. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
Theoremdivne1d 11505 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 11502. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 / 𝐵) ≠ 1)
 
Theoremdivne0bd 11506 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 / 𝐵) ≠ 0))
 
Theoremdivnegd 11507 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
Theoremdivneg2d 11508 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
Theoremdiv2negd 11509 Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivne0d 11510 The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) ≠ 0)
 
Theoremrecdivd 11511 The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
Theoremrecdiv2d 11512 Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
Theoremdivcan6d 11513 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
Theoremddcand 11514 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
Theoremrec11d 11515 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (1 / 𝐴) = (1 / 𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremdivmuld 11516 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
Theoremdiv32d 11517 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
Theoremdiv13d 11518 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
Theoremdivdiv32d 11519 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivcan5d 11520 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
 
Theoremdivcan5rd 11521 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵))
 
Theoremdivcan7d 11522 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
 
Theoremdmdcand 11523 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐵 / 𝐶) · (𝐴 / 𝐵)) = (𝐴 / 𝐶))
 
Theoremdmdcan2d 11524 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐶)) = (𝐴 / 𝐶))
 
Theoremdivdiv1d 11525 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
 
Theoremdivdiv2d 11526 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
 
Theoremdivmul2d 11527 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))
 
Theoremdivmul3d 11528 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐵 · 𝐶)))
 
Theoremdivassd 11529 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
Theoremdiv12d 11530 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
 
Theoremdiv23d 11531 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
Theoremdivdird 11532 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
Theoremdivsubdird 11533 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
Theoremdiv11d 11534 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremdivmuldivd 11535 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))
 
Theoremdivmul13d 11536 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷)))
 
Theoremdivmul24d 11537 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 / 𝐷) · (𝐶 / 𝐵)))
 
Theoremdivadddivd 11538 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷)))
 
Theoremdivsubdivd 11539 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) − (𝐶 / 𝐷)) = (((𝐴 · 𝐷) − (𝐶 · 𝐵)) / (𝐵 · 𝐷)))
 
Theoremdivmuleqd 11540 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐶 · 𝐵)))
 
Theoremdivdivdivd 11541 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)))
 
Theoremdiveq1bd 11542 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 11409. Converse of diveq1d 11502. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 / 𝐵) = 1)
 
Theoremdiv2sub 11543 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶𝐷)) → ((𝐴𝐵) / (𝐶𝐷)) = ((𝐵𝐴) / (𝐷𝐶)))
 
Theoremdiv2subd 11544 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 11543. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐶𝐷)       (𝜑 → ((𝐴𝐵) / (𝐶𝐷)) = ((𝐵𝐴) / (𝐷𝐶)))
 
Theoremrereccld 11545 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / 𝐴) ∈ ℝ)
 
Theoremredivcld 11546 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremsubrec 11547 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) − (1 / 𝐵)) = ((𝐵𝐴) / (𝐴 · 𝐵)))
 
Theoremsubreci 11548 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 ≠ 0    &   𝐵 ≠ 0       ((1 / 𝐴) − (1 / 𝐵)) = ((𝐵𝐴) / (𝐴 · 𝐵))
 
Theoremsubrecd 11549 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((1 / 𝐴) − (1 / 𝐵)) = ((𝐵𝐴) / (𝐴 · 𝐵)))
 
Theoremmvllmuld 11550 Move the left term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 / 𝐴))
 
Theoremmvllmuli 11551 Move the left term in a product on the LHS to the RHS, inference form. Uses divcan4i 11465. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 ≠ 0    &   (𝐴 · 𝐵) = 𝐶       𝐵 = (𝐶 / 𝐴)
 
Theoremldiv 11552 Left-division. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) = 𝐶𝐴 = (𝐶 / 𝐵)))
 
Theoremrdiv 11553 Right-division. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) = 𝐶𝐵 = (𝐶 / 𝐴)))
 
Theoremmdiv 11554 A division law. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 = (𝐶 / 𝐵) ↔ 𝐵 = (𝐶 / 𝐴)))
 
Theoremlineq 11555 Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (((𝐴 · 𝑋) + 𝐵) = 𝑌𝑋 = ((𝑌𝐵) / 𝐴)))
 
5.3.7  Ordering on reals (cont.)
 
Theoremelimgt0 11556 Hypothesis for weak deduction theorem to eliminate 0 < 𝐴. (Contributed by NM, 15-May-1999.)
0 < if(0 < 𝐴, 𝐴, 1)
 
Theoremelimge0 11557 Hypothesis for weak deduction theorem to eliminate 0 ≤ 𝐴. (Contributed by NM, 30-Jul-1999.)
0 ≤ if(0 ≤ 𝐴, 𝐴, 0)
 
Theoremltp1 11558 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
Theoremlep1 11559 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1))
 
Theoremltm1 11560 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
 
Theoremlem1 11561 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴)
 
Theoremletrp1 11562 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐴 ≤ (𝐵 + 1))
 
Theoremp1le 11563 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴𝐵)
 
Theoremrecgt0 11564 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴))
 
Theoremprodgt0 11565 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
 
Theoremprodgt02 11566 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐵 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐴)
 
Theoremltmul1a 11567 Lemma for ltmul1 11568. Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) ∧ 𝐴 < 𝐵) → (𝐴 · 𝐶) < (𝐵 · 𝐶))
 
Theoremltmul1 11568 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
 
Theoremltmul2 11569 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
 
Theoremlemul1 11570 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
 
Theoremlemul2 11571 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
 
Theoremlemul1a 11572 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))
 
Theoremlemul2a 11573 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))
 
Theoremltmul12a 11574 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 ≤ 𝐶𝐶 < 𝐷))) → (𝐴 · 𝐶) < (𝐵 · 𝐷))
 
Theoremlemul12b 11575 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 ≤ 𝐷))) → ((𝐴𝐵𝐶𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)))
 
Theoremlemul12a 11576 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐵𝐶𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)))
 
Theoremmulgt1 11577 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (1 < 𝐴 ∧ 1 < 𝐵)) → 1 < (𝐴 · 𝐵))
 
Theoremltmulgt11 11578 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐴 · 𝐵)))
 
Theoremltmulgt12 11579 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐵 · 𝐴)))
 
Theoremlemulge11 11580 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐴 · 𝐵))
 
Theoremlemulge12 11581 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐵 · 𝐴))
 
Theoremltdiv1 11582 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
 
Theoremlediv1 11583 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)))
 
Theoremgt0div 11584 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵)))
 
Theoremge0div 11585 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵)))
 
Theoremdivgt0 11586 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 / 𝐵))
 
Theoremdivge0 11587 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵))
 
Theoremmulge0b 11588 A condition for multiplication to be nonnegative. (Contributed by Scott Fenton, 25-Jun-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 · 𝐵) ↔ ((𝐴 ≤ 0 ∧ 𝐵 ≤ 0) ∨ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵))))
 
Theoremmulle0b 11589 A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴𝐵 ≤ 0))))
 
Theoremmulsuble0b 11590 A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (((𝐴𝐵) · (𝐶𝐵)) ≤ 0 ↔ ((𝐴𝐵𝐵𝐶) ∨ (𝐶𝐵𝐵𝐴))))
 
Theoremltmuldiv 11591 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltmuldiv2 11592 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltdivmul 11593 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵𝐴 < (𝐶 · 𝐵)))
 
Theoremledivmul 11594 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵𝐴 ≤ (𝐶 · 𝐵)))
 
Theoremltdivmul2 11595 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵𝐴 < (𝐵 · 𝐶)))
 
Theoremlt2mul2div 11596 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
(((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵)))
 
Theoremledivmul2 11597 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵𝐴 ≤ (𝐵 · 𝐶)))
 
Theoremlemuldiv 11598 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ 𝐵𝐴 ≤ (𝐵 / 𝐶)))
 
Theoremlemuldiv2 11599 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) ≤ 𝐵𝐴 ≤ (𝐵 / 𝐶)))
 
Theoremltrec 11600 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-45962
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