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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lesub0 11501 | Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ (𝐵 − 𝐴)) ↔ 𝐴 = 0)) | ||
Theorem | mulge0 11502 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵)) | ||
Theorem | mullt0 11503 | The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵)) | ||
Theorem | msqgt0 11504 | A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 6-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴 · 𝐴)) | ||
Theorem | msqge0 11505 | A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴)) | ||
Theorem | 0lt1 11506 | 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) |
⊢ 0 < 1 | ||
Theorem | 0le1 11507 | 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ 0 ≤ 1 | ||
Theorem | relin01 11508 | An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.) |
⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ∨ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1) ∨ 1 ≤ 𝐴)) | ||
Theorem | ltordlem 11509* | Lemma for ltord1 11510. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁)) | ||
Theorem | ltord1 11510* | Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ 𝑀 < 𝑁)) | ||
Theorem | leord1 11511* | Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑀 ≤ 𝑁)) | ||
Theorem | eqord1 11512* | A strictly increasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁)) | ||
Theorem | ltord2 11513* | Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ 𝑁 < 𝑀)) | ||
Theorem | leord2 11514* | Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑁 ≤ 𝑀)) | ||
Theorem | eqord2 11515* | A strictly decreasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) & ⊢ 𝑆 ⊆ ℝ & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁)) | ||
Theorem | wloglei 11516* | Form of wlogle 11517 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → 𝑆 ⊆ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜃) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒) | ||
Theorem | wlogle 11517* | If the predicate 𝜒(𝑥, 𝑦) is symmetric under interchange of 𝑥, 𝑦, then "without loss of generality" we can assume that 𝑥 ≤ 𝑦. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.) |
⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → 𝑆 ⊆ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒) ⇒ ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒) | ||
Theorem | leidi 11518 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ≤ 𝐴 | ||
Theorem | gt0ne0i 11519 | Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 < 𝐴 → 𝐴 ≠ 0) | ||
Theorem | gt0ne0ii 11520 | Positive implies nonzero. (Contributed by NM, 15-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 𝐴 ≠ 0 | ||
Theorem | msqgt0i 11521 | A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 ≠ 0 → 0 < (𝐴 · 𝐴)) | ||
Theorem | msqge0i 11522 | A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 0 ≤ (𝐴 · 𝐴) | ||
Theorem | addgt0i 11523 | Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵)) | ||
Theorem | addge0i 11524 | Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 + 𝐵)) | ||
Theorem | addgegt0i 11525 | Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵)) | ||
Theorem | addgt0ii 11526 | Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 < 𝐴 & ⊢ 0 < 𝐵 ⇒ ⊢ 0 < (𝐴 + 𝐵) | ||
Theorem | add20i 11527 | Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | ||
Theorem | ltnegi 11528 | Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 ↔ -𝐵 < -𝐴) | ||
Theorem | lenegi 11529 | Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴) | ||
Theorem | ltnegcon2i 11530 | Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < -𝐵 ↔ 𝐵 < -𝐴) | ||
Theorem | mulge0i 11531 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 · 𝐵)) | ||
Theorem | lesub0i 11532 | Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ (𝐵 − 𝐴)) ↔ 𝐴 = 0) | ||
Theorem | ltaddposi 11533 | Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)) | ||
Theorem | posdifi 11534 | Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)) | ||
Theorem | ltnegcon1i 11535 | Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴) | ||
Theorem | lenegcon1i 11536 | Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴) | ||
Theorem | subge0i 11537 | Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴) | ||
Theorem | ltadd1i 11538 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)) | ||
Theorem | leadd1i 11539 | Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)) | ||
Theorem | leadd2i 11540 | Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)) | ||
Theorem | ltsubaddi 11541 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵)) | ||
Theorem | lesubaddi 11542 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵)) | ||
Theorem | ltsubadd2i 11543 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)) | ||
Theorem | lesubadd2i 11544 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶)) | ||
Theorem | ltaddsubi 11545 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵)) | ||
Theorem | lt2addi 11546 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 𝐷 ∈ ℝ ⇒ ⊢ ((𝐴 < 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)) | ||
Theorem | le2addi 11547 | Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 𝐷 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) | ||
Theorem | gt0ne0d 11548 | Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | lt0ne0d 11549 | Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | leidd 11550 | 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐴) | ||
Theorem | msqgt0d 11551 | A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → 0 < (𝐴 · 𝐴)) | ||
Theorem | msqge0d 11552 | A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐴)) | ||
Theorem | lt0neg1d 11553 | Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) | ||
Theorem | lt0neg2d 11554 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0)) | ||
Theorem | le0neg1d 11555 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴)) | ||
Theorem | le0neg2d 11556 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0)) | ||
Theorem | addgegt0d 11557 | Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | addgtge0d 11558 | Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | addgt0d 11559 | Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | addge0d 11560 | Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) | ||
Theorem | mulge0d 11561 | The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) | ||
Theorem | ltnegd 11562 | Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)) | ||
Theorem | lenegd 11563 | Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) | ||
Theorem | ltnegcon1d 11564 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → -𝐴 < 𝐵) ⇒ ⊢ (𝜑 → -𝐵 < 𝐴) | ||
Theorem | ltnegcon2d 11565 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < -𝐵) ⇒ ⊢ (𝜑 → 𝐵 < -𝐴) | ||
Theorem | lenegcon1d 11566 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → -𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → -𝐵 ≤ 𝐴) | ||
Theorem | lenegcon2d 11567 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ -𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ -𝐴) | ||
Theorem | ltaddposd 11568 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) | ||
Theorem | ltaddpos2d 11569 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ 𝐵 < (𝐴 + 𝐵))) | ||
Theorem | ltsubposd 11570 | Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ (𝐵 − 𝐴) < 𝐵)) | ||
Theorem | posdifd 11571 | Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | ||
Theorem | addge01d 11572 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) | ||
Theorem | addge02d 11573 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + 𝐴))) | ||
Theorem | subge0d 11574 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | suble0d 11575 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵)) | ||
Theorem | subge02d 11576 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ (𝐴 − 𝐵) ≤ 𝐴)) | ||
Theorem | ltadd1d 11577 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))) | ||
Theorem | leadd1d 11578 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))) | ||
Theorem | leadd2d 11579 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) | ||
Theorem | ltsubaddd 11580 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵))) | ||
Theorem | lesubaddd 11581 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | ||
Theorem | ltsubadd2d 11582 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶))) | ||
Theorem | lesubadd2d 11583 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶))) | ||
Theorem | ltaddsubd 11584 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) | ||
Theorem | ltaddsub2d 11585 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) | ||
Theorem | leaddsub2d 11586 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 − 𝐴))) | ||
Theorem | subled 11587 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝐴 − 𝐵) ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) ≤ 𝐵) | ||
Theorem | lesubd 11588 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ≤ (𝐵 − 𝐴)) | ||
Theorem | ltsub23d 11589 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐶) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) < 𝐵) | ||
Theorem | ltsub13d 11590 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐶 < (𝐵 − 𝐴)) | ||
Theorem | lesub1d 11591 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))) | ||
Theorem | lesub2d 11592 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐴))) | ||
Theorem | ltsub1d 11593 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 − 𝐶) < (𝐵 − 𝐶))) | ||
Theorem | ltsub2d 11594 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 − 𝐵) < (𝐶 − 𝐴))) | ||
Theorem | ltadd1dd 11595 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (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 | ltsub1dd 11596 | Subtraction from both sides of 'less than'. (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 | ltsub2dd 11597 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐶 − 𝐵) < (𝐶 − 𝐴)) | ||
Theorem | leadd1dd 11598 | Addition to both sides of 'less than or equal to'. (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 | leadd2dd 11599 | Addition to both sides of 'less than or equal to'. (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 | lesub1dd 11600 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) |
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