 Home Intuitionistic Logic ExplorerTheorem List (p. 94 of 129) < 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 - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrpssre 9301 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
+ ⊆ ℝ

Theoremrpgt0 9302 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
(𝐴 ∈ ℝ+ → 0 < 𝐴)

Theoremrpge0 9303 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
(𝐴 ∈ ℝ+ → 0 ≤ 𝐴)

Theoremrpregt0 9304 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Theoremrprege0 9305 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))

Theoremrpne0 9306 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
(𝐴 ∈ ℝ+𝐴 ≠ 0)

Theoremrpap0 9307 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(𝐴 ∈ ℝ+𝐴 # 0)

Theoremrprene0 9308 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))

Theoremrpreap0 9309 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 # 0))

Theoremrpcnne0 9310 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))

Theoremrpcnap0 9311 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 # 0))

Theoremralrp 9312 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
(∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))

Theoremrexrp 9313 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
(∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))

Theoremrpaddcl 9314 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 + 𝐵) ∈ ℝ+)

Theoremrpmulcl 9315 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+)

Theoremrpdivcl 9316 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)

Theoremrpreccl 9317 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
(𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)

Theoremrphalfcl 9318 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)

Theoremrpgecl 9319 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐵 ∈ ℝ+)

Theoremrphalflt 9320 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴)

Theoremrerpdivcl 9321 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ)

Theoremge0p1rp 9322 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+)

Theoremrpnegap 9323 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 9324 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
¬ 0 ∈ ℝ+

Theoremltsubrp 9325 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴𝐵) < 𝐴)

Theoremltaddrp 9326 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))

Theoremdifrp 9327 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 9328 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 ∈ ℝ+)

Theoremnnrpd 9329 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ+)

Theoremrpred 9330 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℝ)

Theoremrpxrd 9331 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℝ*)

Theoremrpcnd 9332 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℂ)

Theoremrpgt0d 9333 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < 𝐴)

Theoremrpge0d 9334 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 ≤ 𝐴)

Theoremrpne0d 9335 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ≠ 0)

Theoremrpap0d 9336 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 # 0)

Theoremrpregt0d 9337 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Theoremrprege0d 9338 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 9339 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))

Theoremrpcnne0d 9340 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))

Theoremrpreccld 9341 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 / 𝐴) ∈ ℝ+)

Theoremrprecred 9342 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremrphalfcld 9343 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 / 2) ∈ ℝ+)

Theoremreclt1d 9344 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 9345 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 9346 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 9347 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 9348 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ+)

Theoremltrecd 9349 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 9350 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 9351 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 9352 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 9353 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 9354 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 9355 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 9356 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 9357 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 9358 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 9359 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 9360 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 9361 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 9362 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 9363 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵) < 𝐴)

Theoremltaddrpd 9364 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑𝐴 < (𝐴 + 𝐵))

Theoremltaddrp2d 9365 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑𝐴 < (𝐵 + 𝐴))

Theoremltmulgt11d 9366 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐴𝐵 < (𝐵 · 𝐴)))

Theoremltmulgt12d 9367 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐴𝐵 < (𝐴 · 𝐵)))

Theoremgt0divd 9368 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 9369 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 9370 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 9371 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 9372 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 9373 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 9374 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 9375 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 9376 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 9377 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 9378 '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 9379 '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 9380 '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 9381 '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 9382 '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 9383 '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 9384 '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 9385 '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 9386 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 9387 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 9388 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 9389 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 9390 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 9391 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 9392 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 9393 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 9394 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 9395 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 9396 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 9397 Extend class notation to include the negative of an extended real.
class -𝑒𝐴