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Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrprege0 12401 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrpne0 12402 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
(𝐴 ∈ ℝ+𝐴 ≠ 0)
 
Theoremrprene0 12403 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
 
Theoremrpcnne0 12404 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
 
Theoremrpcndif0 12405 A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
(𝐴 ∈ ℝ+𝐴 ∈ (ℂ ∖ {0}))
 
Theoremralrp 12406 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
(∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))
 
Theoremrexrp 12407 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
(∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
 
Theoremrpaddcl 12408 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 12409 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+)
 
Theoremrpmtmip 12410 "Minus times minus is plus", see also nnmtmip 11660, holds for positive reals, too (formalized to "The product of two negative reals is a positive real"). "The reason for this" in this case is that (-𝐴 · -𝐵) = (𝐴 · 𝐵) for all complex numbers 𝐴 and 𝐵 because of mul2neg 11077, 𝐴 and 𝐵 are complex numbers because of rpcn 12396, and (𝐴 · 𝐵) ∈ ℝ+ because of rpmulcl 12409. Note that the opposites -𝐴 and -𝐵 of the positive reals 𝐴 and 𝐵 are negative reals. (Contributed by AV, 23-Dec-2022.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (-𝐴 · -𝐵) ∈ ℝ+)
 
Theoremrpdivcl 12411 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
 
Theoremrpreccl 12412 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
(𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
 
Theoremrphalfcl 12413 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
 
Theoremrpgecl 12414 A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐵 ∈ ℝ+)
 
Theoremrphalflt 12415 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴)
 
Theoremrerpdivcl 12416 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 12417 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+)
 
Theoremrpneg 12418 Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20. (Contributed by NM, 7-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 ∈ ℝ+ ↔ ¬ -𝐴 ∈ ℝ+))
 
Theoremnegelrp 12419 Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
(𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+𝐴 < 0))
 
Theoremnegelrpd 12420 The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → -𝐴 ∈ ℝ+)
 
Theorem0nrp 12421 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
¬ 0 ∈ ℝ+
 
Theoremltsubrp 12422 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴𝐵) < 𝐴)
 
Theoremltaddrp 12423 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
 
Theoremdifrp 12424 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 12425 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 ∈ ℝ+)
 
Theoremnnrpd 12426 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ+)
 
Theoremzgt1rpn0n1 12427 An integer greater than 1 is a positive real number not equal to 0 or 1. Useful for working with integer logarithm bases (which is a common case, e.g. base 2, base 3 or base 10). (Contributed by Thierry Arnoux, 26-Sep-2017.) (Proof shortened by AV, 9-Jul-2022.)
(𝐵 ∈ (ℤ‘2) → (𝐵 ∈ ℝ+𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
 
Theoremrpred 12428 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℝ)
 
Theoremrpxrd 12429 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℝ*)
 
Theoremrpcnd 12430 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ∈ ℂ)
 
Theoremrpgt0d 12431 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < 𝐴)
 
Theoremrpge0d 12432 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 ≤ 𝐴)
 
Theoremrpne0d 12433 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑𝐴 ≠ 0)
 
Theoremrpregt0d 12434 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremrprege0d 12435 A positive real is real and greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrprene0d 12436 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
 
Theoremrpcnne0d 12437 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
 
Theoremrpreccld 12438 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 / 𝐴) ∈ ℝ+)
 
Theoremrprecred 12439 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (1 / 𝐴) ∈ ℝ)
 
Theoremrphalfcld 12440 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 / 2) ∈ ℝ+)
 
Theoremreclt1d 12441 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 12442 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 12443 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 12444 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 12445 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ+)
 
Theoremltrecd 12446 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 12447 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 12448 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 12449 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 12450 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 12451 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 12452 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 12453 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 12454 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 12455 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 12456 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 12457 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 12458 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 12459 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 12460 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵) < 𝐴)
 
Theoremltaddrpd 12461 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑𝐴 < (𝐴 + 𝐵))
 
Theoremltaddrp2d 12462 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑𝐴 < (𝐵 + 𝐴))
 
Theoremltmulgt11d 12463 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐴𝐵 < (𝐵 · 𝐴)))
 
Theoremltmulgt12d 12464 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐴𝐵 < (𝐴 · 𝐵)))
 
Theoremgt0divd 12465 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 12466 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 12467 A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐵𝐴)       (𝜑𝐴 ∈ ℝ+)
 
Theoremdivge0d 12468 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 12469 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 12470 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 12471 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 12472 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 12473 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 12474 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 12475 '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 12476 '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 12477 '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 12478 '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 12479 '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 12480 '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 12481 '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 12482 '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 12483 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 12484 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 12485 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 12486 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 12487 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))
 
Theoremmul2lt0rlt0 12488 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) < 0)       ((𝜑𝐵 < 0) → 0 < 𝐴)
 
Theoremmul2lt0rgt0 12489 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) < 0)       ((𝜑 ∧ 0 < 𝐵) → 𝐴 < 0)
 
Theoremmul2lt0llt0 12490 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) < 0)       ((𝜑𝐴 < 0) → 0 < 𝐵)
 
Theoremmul2lt0lgt0 12491 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) < 0)       ((𝜑 ∧ 0 < 𝐴) → 𝐵 < 0)
 
Theoremmul2lt0bi 12492 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴 · 𝐵) < 0 ↔ ((𝐴 < 0 ∧ 0 < 𝐵) ∨ (0 < 𝐴𝐵 < 0))))
 
Theoremprodge0rd 12493 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.) (Revised by AV, 9-Jul-2022.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ (𝐴 · 𝐵))       (𝜑 → 0 ≤ 𝐵)
 
Theoremprodge0ld 12494 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by AV, 9-Jul-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → 0 ≤ (𝐴 · 𝐵))       (𝜑 → 0 ≤ 𝐴)
 
Theoremltdiv23d 12495 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 12496 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 12497 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 12498 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 12499 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 12500 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))
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