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Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrec11d 10701 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 10702 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 10703 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 10704 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 10705 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 10706 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 10707 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 10708 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 10709 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 10710 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 10711 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 10712 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 10713 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 10714 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 10715 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 10716 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 10717 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 10718 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 10719 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 10720 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 10721 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 10722 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 10723 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 10724 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 10725 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 10726 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 10727 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 10728 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 10597. Converse of diveq1d 10688. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 / 𝐵) = 1)

Theoremdiv2sub 10729 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 10730 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 10729. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐶𝐷)       (𝜑 → ((𝐴𝐵) / (𝐶𝐷)) = ((𝐵𝐴) / (𝐷𝐶)))

Theoremrereccld 10731 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremredivcld 10732 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)

Theoremsubrec 10733 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 10734 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 10735 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 10736 Move LHS left multiplication to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 / 𝐴))

Theoremmvllmuli 10737 Move LHS left multiplication to RHS. Uses divcan4i 10651. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 ≠ 0    &   (𝐴 · 𝐵) = 𝐶       𝐵 = (𝐶 / 𝐴)

5.3.7  Ordering on reals (cont.)

Theoremelimgt0 10738 Hypothesis for weak deduction theorem to eliminate 0 < 𝐴. (Contributed by NM, 15-May-1999.)
0 < if(0 < 𝐴, 𝐴, 1)

Theoremelimge0 10739 Hypothesis for weak deduction theorem to eliminate 0 ≤ 𝐴. (Contributed by NM, 30-Jul-1999.)
0 ≤ if(0 ≤ 𝐴, 𝐴, 0)

Theoremltp1 10740 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))

Theoremlep1 10741 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1))

Theoremltm1 10742 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)

Theoremlem1 10743 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴)

Theoremletrp1 10744 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 10745 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 10746 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 10747 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 10748 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)

Theoremprodge0 10749 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐵)

Theoremprodge02 10750 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐵 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐴)

Theoremltmul1a 10751 Lemma for ltmul1 10752. 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 10752 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 10753 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 10754 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 10755 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 10756 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 10757 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 10758 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 10759 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 10760 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 10761 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 10762 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐴 · 𝐵)))

Theoremltmulgt12 10763 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐵 · 𝐴)))

Theoremlemulge11 10764 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 10765 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 10766 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 10767 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 10768 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 10769 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 10770 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 10771 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 10772 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 10773 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 10774 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 10775 '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 10776 '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 10777 '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 10778 '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 10779 '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 10780 '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 10781 '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 10782 '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 10783 '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 10784 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)))

Theoremlerec 10785 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))

Theoremlt2msq1 10786 Lemma for lt2msq 10787. (Contributed by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 · 𝐴) < (𝐵 · 𝐵))

Theoremlt2msq 10787 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵)))

Theoremltdiv2 10788 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 / 𝐵) < (𝐶 / 𝐴)))

Theoremltrec1 10789 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((1 / 𝐴) < 𝐵 ↔ (1 / 𝐵) < 𝐴))

Theoremlerec2 10790 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ (1 / 𝐵) ↔ 𝐵 ≤ (1 / 𝐴)))

Theoremledivdiv 10791 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 / 𝐵) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (𝐵 / 𝐴)))

Theoremlediv2 10792 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))

Theoremltdiv23 10793 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵))

Theoremlediv23 10794 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐵) ≤ 𝐶 ↔ (𝐴 / 𝐶) ≤ 𝐵))

Theoremlediv12a 10795 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 < 𝐶𝐶𝐷))) → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶))

Theoremlediv2a 10796 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))

Theoremreclt1 10797 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))

Theoremrecgt1 10798 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐴 ↔ (1 / 𝐴) < 1))

Theoremrecgt1i 10799 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (0 < (1 / 𝐴) ∧ (1 / 𝐴) < 1))

Theoremrecp1lt1 10800 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) < 1)

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