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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnrp 12401 | A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+) | ||
Theorem | rpgt0 12402 | A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.) |
⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | ||
Theorem | rpge0 12403 | A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.) |
⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) | ||
Theorem | rpregt0 12404 | 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)) | ||
Theorem | rprege0 12405 | A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | ||
Theorem | rpne0 12406 | A positive real is nonzero. (Contributed by NM, 18-Jul-2008.) |
⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | ||
Theorem | rprene0 12407 | A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) | ||
Theorem | rpcnne0 12408 | A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | ||
Theorem | rpcndif0 12409 | A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.) |
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ {0})) | ||
Theorem | ralrp 12410 | Quantification over positive reals. (Contributed by NM, 12-Feb-2008.) |
⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑)) | ||
Theorem | rexrp 12411 | Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.) |
⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑)) | ||
Theorem | rpaddcl 12412 | 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+) | ||
Theorem | rpmulcl 12413 | 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+) | ||
Theorem | rpmtmip 12414 | "Minus times minus is plus", see also nnmtmip 11664, 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 11079, 𝐴 and 𝐵 are complex numbers because of rpcn 12400, and (𝐴 · 𝐵) ∈ ℝ+ because of rpmulcl 12413. 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+) | ||
Theorem | rpdivcl 12415 | Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+) | ||
Theorem | rpreccl 12416 | Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.) |
⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) | ||
Theorem | rphalfcl 12417 | Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+) | ||
Theorem | rpgecl 12418 | A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ+) | ||
Theorem | rphalflt 12419 | Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴) | ||
Theorem | rerpdivcl 12420 | Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) | ||
Theorem | ge0p1rp 12421 | A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+) | ||
Theorem | rpneg 12422 | 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+)) | ||
Theorem | negelrp 12423 | Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.) |
⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+ ↔ 𝐴 < 0)) | ||
Theorem | negelrpd 12424 | The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℝ+) | ||
Theorem | 0nrp 12425 | Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.) |
⊢ ¬ 0 ∈ ℝ+ | ||
Theorem | ltsubrp 12426 | Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴) | ||
Theorem | ltaddrp 12427 | Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵)) | ||
Theorem | difrp 12428 | Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ+)) | ||
Theorem | elrpd 12429 | Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ+) | ||
Theorem | nnrpd 12430 | A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ+) | ||
Theorem | zgt1rpn0n1 12431 | 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)) | ||
Theorem | rpred 12432 | A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | rpxrd 12433 | A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
Theorem | rpcnd 12434 | A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | rpgt0d 12435 | A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 0 < 𝐴) | ||
Theorem | rpge0d 12436 | A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) | ||
Theorem | rpne0d 12437 | A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | rpregt0d 12438 | A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | ||
Theorem | rprege0d 12439 | A positive real is real and greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) | ||
Theorem | rprene0d 12440 | A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) | ||
Theorem | rpcnne0d 12441 | A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | ||
Theorem | rpreccld 12442 | Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+) | ||
Theorem | rprecred 12443 | Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) | ||
Theorem | rphalfcld 12444 | Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+) | ||
Theorem | reclt1d 12445 | The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴))) | ||
Theorem | recgt1d 12446 | The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (1 < 𝐴 ↔ (1 / 𝐴) < 1)) | ||
Theorem | rpaddcld 12447 | Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ+) | ||
Theorem | rpmulcld 12448 | Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ+) | ||
Theorem | rpdivcld 12449 | Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ+) | ||
Theorem | ltrecd 12450 | The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))) | ||
Theorem | lerecd 12451 | The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) | ||
Theorem | ltrec1d 12452 | Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → (1 / 𝐴) < 𝐵) ⇒ ⊢ (𝜑 → (1 / 𝐵) < 𝐴) | ||
Theorem | lerec2d 12453 | Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 ≤ (1 / 𝐵)) ⇒ ⊢ (𝜑 → 𝐵 ≤ (1 / 𝐴)) | ||
Theorem | lediv2ad 12454 | Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐶) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)) | ||
Theorem | ltdiv2d 12455 | Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 / 𝐵) < (𝐶 / 𝐴))) | ||
Theorem | lediv2d 12456 | Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))) | ||
Theorem | ledivdivd 12457 | Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 / 𝐵) ≤ (𝐶 / 𝐷)) ⇒ ⊢ (𝜑 → (𝐷 / 𝐶) ≤ (𝐵 / 𝐴)) | ||
Theorem | divge1 12458 | 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 ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴)) | ||
Theorem | divlt1lt 12459 | 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)) | ||
Theorem | divle1le 12460 | 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 ↔ 𝐴 ≤ 𝐵)) | ||
Theorem | ledivge1le 12461 | 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)) → (𝐴 ≤ 𝐵 → (𝐴 / 𝐶) ≤ 𝐵)) | ||
Theorem | ge0p1rpd 12462 | A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) | ||
Theorem | rerpdivcld 12463 | Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) | ||
Theorem | ltsubrpd 12464 | Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐴) | ||
Theorem | ltaddrpd 12465 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) | ||
Theorem | ltaddrp2d 12466 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴)) | ||
Theorem | ltmulgt11d 12467 | Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴))) | ||
Theorem | ltmulgt12d 12468 | Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐴 · 𝐵))) | ||
Theorem | gt0divd 12469 | Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) | ||
Theorem | ge0divd 12470 | Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) | ||
Theorem | rpgecld 12471 | A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ+) | ||
Theorem | divge0d 12472 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵)) | ||
Theorem | ltmul1d 12473 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) | ||
Theorem | ltmul2d 12474 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) | ||
Theorem | lemul1d 12475 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) | ||
Theorem | lemul2d 12476 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) | ||
Theorem | ltdiv1d 12477 | Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) | ||
Theorem | lediv1d 12478 | Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶))) | ||
Theorem | ltmuldivd 12479 | 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶))) | ||
Theorem | ltmuldiv2d 12480 | 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶))) | ||
Theorem | lemuldivd 12481 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) | ||
Theorem | lemuldiv2d 12482 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) | ||
Theorem | ltdivmuld 12483 | 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐶 · 𝐵))) | ||
Theorem | ltdivmul2d 12484 | 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 · 𝐶))) | ||
Theorem | ledivmuld 12485 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵))) | ||
Theorem | ledivmul2d 12486 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶))) | ||
Theorem | ltmul1dd 12487 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐶)) | ||
Theorem | ltmul2dd 12488 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐶 · 𝐴) < (𝐶 · 𝐵)) | ||
Theorem | ltdiv1dd 12489 | Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) < (𝐵 / 𝐶)) | ||
Theorem | lediv1dd 12490 | Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)) | ||
Theorem | lediv12ad 12491 | Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶)) | ||
Theorem | mul2lt0rlt0 12492 | If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐴 · 𝐵) < 0) ⇒ ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) | ||
Theorem | mul2lt0rgt0 12493 | If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐴 · 𝐵) < 0) ⇒ ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐴 < 0) | ||
Theorem | mul2lt0llt0 12494 | If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐴 · 𝐵) < 0) ⇒ ⊢ ((𝜑 ∧ 𝐴 < 0) → 0 < 𝐵) | ||
Theorem | mul2lt0lgt0 12495 | If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐴 · 𝐵) < 0) ⇒ ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 < 0) | ||
Theorem | mul2lt0bi 12496 | If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 ↔ ((𝐴 < 0 ∧ 0 < 𝐵) ∨ (0 < 𝐴 ∧ 𝐵 < 0)))) | ||
Theorem | prodge0rd 12497 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) ⇒ ⊢ (𝜑 → 0 ≤ 𝐵) | ||
Theorem | prodge0ld 12498 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) | ||
Theorem | ltdiv23d 12499 | Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 / 𝐵) < 𝐶) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) < 𝐵) | ||
Theorem | lediv23d 12500 | Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 / 𝐵) ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) ≤ 𝐵) |
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