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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrpnegap 9101 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (𝐴 ∈ ℝ+ ⊻ -𝐴 ∈ ℝ+))
 
Theorem0nrp 9102 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
¬ 0 ∈ ℝ+
 
Theoremltsubrp 9103 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴𝐵) < 𝐴)
 
Theoremltaddrp 9104 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
 
Theoremdifrp 9105 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℝ+))
 
Theoremelrpd 9106 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 ∈ ℝ+)
 
Theoremnnrpd 9107 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ+)
 
Theoremrpred 9108 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℝ)
 
Theoremrpxrd 9109 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℝ*)
 
Theoremrpcnd 9110 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℂ)
 
Theoremrpgt0d 9111 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < 𝐴)
 
Theoremrpge0d 9112 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 ≤ 𝐴)
 
Theoremrpne0d 9113 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ≠ 0)
 
Theoremrpap0d 9114 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 # 0)
 
Theoremrpregt0d 9115 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremrprege0d 9116 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrprene0d 9117 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
 
Theoremrpcnne0d 9118 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
 
Theoremrpreccld 9119 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 / 𝐴) ∈ ℝ+)
 
Theoremrprecred 9120 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 / 𝐴) ∈ ℝ)
 
Theoremrphalfcld 9121 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 / 2) ∈ ℝ+)
 
Theoremreclt1d 9122 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
 
Theoremrecgt1d 9123 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 < 𝐴 ↔ (1 / 𝐴) < 1))
 
Theoremrpaddcld 9124 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 + 𝐵) ∈ ℝ+)
 
Theoremrpmulcld 9125 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 · 𝐵) ∈ ℝ+)
 
Theoremrpdivcld 9126 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ+)
 
Theoremltrecd 9127 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
 
Theoremlerecd 9128 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
 
Theoremltrec1d 9129 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → (1 / 𝐴) < 𝐵)       (𝜑 → (1 / 𝐵) < 𝐴)
 
Theoremlerec2d 9130 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐴 ≤ (1 / 𝐵))       (𝜑𝐵 ≤ (1 / 𝐴))
 
Theoremlediv2ad 9131 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))
 
Theoremltdiv2d 9132 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 / 𝐵) < (𝐶 / 𝐴)))
 
Theoremlediv2d 9133 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
 
Theoremledivdivd 9134 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑 → (𝐴 / 𝐵) ≤ (𝐶 / 𝐷))       (𝜑 → (𝐷 / 𝐶) ≤ (𝐵 / 𝐴))
 
Theoremdivge1 9135 The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐴𝐵) → 1 ≤ (𝐵 / 𝐴))
 
Theoremdivlt1lt 9136 A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) < 1 ↔ 𝐴 < 𝐵))
 
Theoremdivle1le 9137 A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴𝐵))
 
Theoremledivge1le 9138 If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 1 ≤ 𝐶)) → (𝐴𝐵 → (𝐴 / 𝐶) ≤ 𝐵))
 
Theoremge0p1rpd 9139 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (𝐴 + 1) ∈ ℝ+)
 
Theoremrerpdivcld 9140 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremltsubrpd 9141 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵) < 𝐴)
 
Theoremltaddrpd 9142 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑𝐴 < (𝐴 + 𝐵))
 
Theoremltaddrp2d 9143 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑𝐴 < (𝐵 + 𝐴))
 
Theoremltmulgt11d 9144 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐴𝐵 < (𝐵 · 𝐴)))
 
Theoremltmulgt12d 9145 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐴𝐵 < (𝐴 · 𝐵)))
 
Theoremgt0divd 9146 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵)))
 
Theoremge0divd 9147 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵)))
 
Theoremrpgecld 9148 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐵𝐴)       (𝜑𝐴 ∈ ℝ+)
 
Theoremdivge0d 9149 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → 0 ≤ (𝐴 / 𝐵))
 
Theoremltmul1d 9150 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
 
Theoremltmul2d 9151 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
 
Theoremlemul1d 9152 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
 
Theoremlemul2d 9153 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
 
Theoremltdiv1d 9154 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
 
Theoremlediv1d 9155 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)))
 
Theoremltmuldivd 9156 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐶) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltmuldiv2d 9157 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐶 · 𝐴) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremlemuldivd 9158 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐶) ≤ 𝐵𝐴 ≤ (𝐵 / 𝐶)))
 
Theoremlemuldiv2d 9159 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐶 · 𝐴) ≤ 𝐵𝐴 ≤ (𝐵 / 𝐶)))
 
Theoremltdivmuld 9160 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 / 𝐶) < 𝐵𝐴 < (𝐶 · 𝐵)))
 
Theoremltdivmul2d 9161 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 / 𝐶) < 𝐵𝐴 < (𝐵 · 𝐶)))
 
Theoremledivmuld 9162 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵𝐴 ≤ (𝐶 · 𝐵)))
 
Theoremledivmul2d 9163 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵𝐴 ≤ (𝐵 · 𝐶)))
 
Theoremltmul1dd 9164 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐶))
 
Theoremltmul2dd 9165 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐶 · 𝐴) < (𝐶 · 𝐵))
 
Theoremltdiv1dd 9166 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴 / 𝐶) < (𝐵 / 𝐶))
 
Theoremlediv1dd 9167 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 / 𝐶) ≤ (𝐵 / 𝐶))
 
Theoremlediv12ad 9168 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶))
 
Theoremltdiv23d 9169 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐴 / 𝐵) < 𝐶)       (𝜑 → (𝐴 / 𝐶) < 𝐵)
 
Theoremlediv23d 9170 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐴 / 𝐵) ≤ 𝐶)       (𝜑 → (𝐴 / 𝐶) ≤ 𝐵)
 
Theoremlt2mul2divd 9171 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵)))
 
Theoremnnledivrp 9172 Division of a positive integer by a positive number is less than or equal to the integer iff the number is greater than or equal to 1. (Contributed by AV, 19-Jun-2021.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → (1 ≤ 𝐵 ↔ (𝐴 / 𝐵) ≤ 𝐴))
 
Theoremnn0ledivnn 9173 Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴)
 
Theoremaddlelt 9174 If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)
((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 + 𝐴) ≤ 𝑁𝑀 < 𝑁))
 
3.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 9175 Extend class notation to include the negative of an extended real.
class -𝑒𝐴
 
Syntaxcxad 9176 Extend class notation to include addition of extended reals.
class +𝑒
 
Syntaxcxmu 9177 Extend class notation to include multiplication of extended reals.
class ·e
 
Definitiondf-xneg 9178 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
-𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
 
Definitiondf-xadd 9179* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
+𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
 
Definitiondf-xmul 9180* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
 
Theoremltxr 9181 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
 
Theoremelxr 9182 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
 
Theoremxrnemnf 9183 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
 
Theoremxrnepnf 9184 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
 
Theoremxrltnr 9185 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
 
Theoremltpnf 9186 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 < +∞)
 
Theorem0ltpnf 9187 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 < +∞
 
Theoremmnflt 9188 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → -∞ < 𝐴)
 
Theoremmnflt0 9189 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ < 0
 
Theoremmnfltpnf 9190 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
-∞ < +∞
 
Theoremmnfltxr 9191 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
 
Theorempnfnlt 9192 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
 
Theoremnltmnf 9193 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
 
Theorempnfge 9194 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ*𝐴 ≤ +∞)
 
Theorem0lepnf 9195 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞
 
Theoremnn0pnfge0 9196 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑁 ∈ ℕ0𝑁 = +∞) → 0 ≤ 𝑁)
 
Theoremmnfle 9197 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(𝐴 ∈ ℝ* → -∞ ≤ 𝐴)
 
Theoremxrltnsym 9198 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
 
Theoremxrltnsym2 9199 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
 
Theoremxrlttr 9200 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
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