HomeHome Intuitionistic Logic Explorer
Theorem List (p. 89 of 142)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlt2msq1 8801 Lemma for lt2msq 8802. (Contributed by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 · 𝐴) < (𝐵 · 𝐵))
 
Theoremlt2msq 8802 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 8803 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 8804 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 8805 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 8806 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 8807 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 8808 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 8809 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 8810 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 8811 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 8812 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 8813 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 8814 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 8815 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) < 1)
 
Theoremrecreclt 8816 Given a positive number 𝐴, construct a new positive number less than both 𝐴 and 1. (Contributed by NM, 28-Dec-2005.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((1 / (1 + (1 / 𝐴))) < 1 ∧ (1 / (1 + (1 / 𝐴))) < 𝐴))
 
Theoremle2msq 8817 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
 
Theoremmsq11 8818 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremledivp1 8819 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ 𝐴)
 
Theoremsqueeze0 8820* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥𝐴 < 𝑥)) → 𝐴 = 0)
 
Theoremltp1i 8821 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
𝐴 ∈ ℝ       𝐴 < (𝐴 + 1)
 
Theoremrecgt0i 8822 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ       (0 < 𝐴 → 0 < (1 / 𝐴))
 
Theoremrecgt0ii 8823 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       0 < (1 / 𝐴)
 
Theoremprodgt0i 8824 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵)) → 0 < 𝐵)
 
Theoremprodge0i 8825 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵)) → 0 ≤ 𝐵)
 
Theoremdivgt0i 8826 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 / 𝐵))
 
Theoremdivge0i 8827 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 ≤ (𝐴 / 𝐵))
 
Theoremltreci 8828 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
 
Theoremlereci 8829 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
 
Theoremlt2msqi 8830 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵)))
 
Theoremle2msqi 8831 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
 
Theoremmsq11i 8832 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremdivgt0i2i 8833 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐵       (0 < 𝐴 → 0 < (𝐴 / 𝐵))
 
Theoremltrecii 8834 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))
 
Theoremdivgt0ii 8835 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       0 < (𝐴 / 𝐵)
 
Theoremltmul1i 8836 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
 
Theoremltdiv1i 8837 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
 
Theoremltmuldivi 8838 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → ((𝐴 · 𝐶) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltmul2i 8839 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
 
Theoremlemul1i 8840 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
 
Theoremlemul2i 8841 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
 
Theoremltdiv23i 8842 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((0 < 𝐵 ∧ 0 < 𝐶) → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵))
 
Theoremltdiv23ii 8843 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   0 < 𝐵    &   0 < 𝐶       ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)
 
Theoremltmul1ii 8844 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   0 < 𝐶       (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))
 
Theoremltdiv1ii 8845 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   0 < 𝐶       (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))
 
Theoremltp1d 8846 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 < (𝐴 + 1))
 
Theoremlep1d 8847 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (𝐴 + 1))
 
Theoremltm1d 8848 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 − 1) < 𝐴)
 
Theoremlem1d 8849 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 − 1) ≤ 𝐴)
 
Theoremrecgt0d 8850 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑 → 0 < (1 / 𝐴))
 
Theoremdivgt0d 8851 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 / 𝐵))
 
Theoremmulgt1d 8852 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑 → 1 < 𝐵)       (𝜑 → 1 < (𝐴 · 𝐵))
 
Theoremlemulge11d 8853 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 1 ≤ 𝐵)       (𝜑𝐴 ≤ (𝐴 · 𝐵))
 
Theoremlemulge12d 8854 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 1 ≤ 𝐵)       (𝜑𝐴 ≤ (𝐵 · 𝐴))
 
Theoremlemul1ad 8855 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))
 
Theoremlemul2ad 8856 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))
 
Theoremltmul12ad 8857 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐶 < 𝐷)       (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐷))
 
Theoremlemul12ad 8858 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))
 
Theoremlemul12bd 8859 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐷)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))
 
Theoremmulle0r 8860 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 0 ∧ 0 ≤ 𝐵)) → (𝐴 · 𝐵) ≤ 0)
 
4.3.10  Suprema
 
Theoremlbreu 8861* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → ∃!𝑥𝑆𝑦𝑆 𝑥𝑦)
 
Theoremlbcl 8862* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ∈ 𝑆)
 
Theoremlble 8863* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ≤ 𝐴)
 
Theoremlbinf 8864* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) = (𝑥𝑆𝑦𝑆 𝑥𝑦))
 
Theoremlbinfcl 8865* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) ∈ 𝑆)
 
Theoremlbinfle 8866* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)
 
Theoremsuprubex 8867* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵𝐴)       (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))
 
Theoremsuprlubex 8868* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧𝐴 𝐵 < 𝑧))
 
Theoremsuprnubex 8869* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))
 
Theoremsuprleubex 8870* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))
 
Theoremnegiso 8871 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)       (𝐹 Isom < , < (ℝ, ℝ) ∧ 𝐹 = 𝐹)
 
Theoremdfinfre 8872* The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
(𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
 
Theoremsup3exmid 8873* If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
((𝑢 ⊆ ℝ ∧ ∃𝑤 𝑤𝑢 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑢 𝑦𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝑢 𝑦 < 𝑧)))       DECID 𝜑
 
4.3.11  Imaginary and complex number properties
 
Theoremcrap0 8874 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 # 0 ∨ 𝐵 # 0) ↔ (𝐴 + (i · 𝐵)) # 0))
 
Theoremcreur 8875* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → ∃!𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremcreui 8876* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
 
Theoremcju 8877* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ))
 
4.4  Integer sets
 
4.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 8878 Extend class notation to include the class of positive integers.
class
 
Definitiondf-inn 8879* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8880 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
 
Theoremdfnn2 8880* Definition of the set of positive integers. Another name for df-inn 8879. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
 
Theorempeano5nni 8881* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
 
Theoremnnssre 8882 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
ℕ ⊆ ℝ
 
Theoremnnsscn 8883 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℕ ⊆ ℂ
 
Theoremnnex 8884 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℕ ∈ V
 
Theoremnnre 8885 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
 
Theoremnncn 8886 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
 
Theoremnnrei 8887 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ∈ ℝ
 
Theoremnncni 8888 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ∈ ℂ
 
Theorem1nn 8889 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
1 ∈ ℕ
 
Theorempeano2nn 8890 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
 
Theoremnnred 8891 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ)
 
Theoremnncnd 8892 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℂ)
 
Theorempeano2nnd 8893 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (𝐴 + 1) ∈ ℕ)
 
4.4.2  Principle of mathematical induction
 
Theoremnnind 8894* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8898 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
(𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ → (𝜒𝜃))       (𝐴 ∈ ℕ → 𝜏)
 
TheoremnnindALT 8895* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 8894 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝑦 ∈ ℕ → (𝜒𝜃))    &   𝜓    &   (𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))       (𝐴 ∈ ℕ → 𝜏)
 
Theoremnn1m1nn 8896 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
 
Theoremnn1suc 8897* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜑𝜃))    &   𝜓    &   (𝑦 ∈ ℕ → 𝜒)       (𝐴 ∈ ℕ → 𝜃)
 
Theoremnnaddcl 8898 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
 
Theoremnnmulcl 8899 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
 
Theoremnnmulcli 8900 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 · 𝐵) ∈ ℕ
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14113
  Copyright terms: Public domain < Previous  Next >