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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mulneg12 11701 | Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = (𝐴 · -𝐵)) | ||
| Theorem | mul2neg 11702 | Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | ||
| Theorem | submul2 11703 | Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 · 𝐶)) = (𝐴 + (𝐵 · -𝐶))) | ||
| Theorem | mulm1 11704 | Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) |
| ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | ||
| Theorem | addneg1mul 11705 | Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (-1 · 𝐵)) = (𝐴 − 𝐵)) | ||
| Theorem | mulsub 11706 | Product of two differences. (Contributed by NM, 14-Jan-2006.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
| Theorem | mulsub2 11707 | Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) · (𝐶 − 𝐷)) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))) | ||
| Theorem | mulm1i 11708 | Product with minus one is negative. (Contributed by NM, 31-Jul-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (-1 · 𝐴) = -𝐴 | ||
| Theorem | mulneg1i 11709 | Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 · 𝐵) = -(𝐴 · 𝐵) | ||
| Theorem | mulneg2i 11710 | Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · -𝐵) = -(𝐴 · 𝐵) | ||
| Theorem | mul2negi 11711 | Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 · -𝐵) = (𝐴 · 𝐵) | ||
| Theorem | subdii 11712 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)) | ||
| Theorem | subdiri 11713 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)) | ||
| Theorem | muladdi 11714 | Product of two sums. (Contributed by NM, 17-May-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))) | ||
| Theorem | mulm1d 11715 | Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) | ||
| Theorem | mulneg1d 11716 | Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) | ||
| Theorem | mulneg2d 11717 | Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | ||
| Theorem | mul2negd 11718 | Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | ||
| Theorem | subdid 11719 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) | ||
| Theorem | subdird 11720 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))) | ||
| Theorem | muladdd 11721 | Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
| Theorem | mulsubd 11722 | Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
| Theorem | muls1d 11723 | Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 − 1)) = ((𝐴 · 𝐵) − 𝐴)) | ||
| Theorem | mulsubfacd 11724 | Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) − 𝐵) = ((𝐴 − 1) · 𝐵)) | ||
| Theorem | addmulsub 11725 | The product of a sum and a difference. (Contributed by AV, 5-Mar-2023.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐵 · 𝐶)) − ((𝐴 · 𝐷) + (𝐵 · 𝐷)))) | ||
| Theorem | subaddmulsub 11726 | The difference with a product of a sum and a difference. (Contributed by AV, 5-Mar-2023.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ 𝐸 ∈ ℂ) → (𝐸 − ((𝐴 + 𝐵) · (𝐶 − 𝐷))) = (((𝐸 − (𝐴 · 𝐶)) − (𝐵 · 𝐶)) + ((𝐴 · 𝐷) + (𝐵 · 𝐷)))) | ||
| Theorem | mulsubaddmulsub 11727 | A special difference of a product with a product of a sum and a difference. (Contributed by AV, 5-Mar-2023.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐵 · 𝐶) − ((𝐴 + 𝐵) · (𝐶 − 𝐷))) = (((𝐴 · 𝐷) + (𝐵 · 𝐷)) − (𝐴 · 𝐶))) | ||
| Theorem | gt0ne0 11728 | Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | ||
| Theorem | lt0ne0 11729 | A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0) | ||
| Theorem | ltadd1 11730 | Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))) | ||
| Theorem | leadd1 11731 | Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))) | ||
| Theorem | leadd2 11732 | Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) | ||
| Theorem | ltsubadd 11733 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵))) | ||
| Theorem | ltsubadd2 11734 | '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 | lesubadd 11735 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | ||
| Theorem | lesubadd2 11736 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶))) | ||
| Theorem | ltaddsub 11737 | 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) | ||
| Theorem | ltaddsub2 11738 | 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) | ||
| Theorem | leaddsub 11739 | 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵))) | ||
| Theorem | leaddsub2 11740 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 − 𝐴))) | ||
| Theorem | suble 11741 | Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ (𝐴 − 𝐶) ≤ 𝐵)) | ||
| Theorem | lesub 11742 | Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ (𝐵 − 𝐶) ↔ 𝐶 ≤ (𝐵 − 𝐴))) | ||
| Theorem | ltsub23 11743 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ (𝐴 − 𝐶) < 𝐵)) | ||
| Theorem | ltsub13 11744 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵 − 𝐶) ↔ 𝐶 < (𝐵 − 𝐴))) | ||
| Theorem | le2add 11745 | Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | ||
| Theorem | ltleadd 11746 | Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | ||
| Theorem | leltadd 11747 | Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | ||
| Theorem | lt2add 11748 | Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | ||
| Theorem | addgt0 11749 | The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵)) | ||
| Theorem | addgegt0 11750 | The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵)) | ||
| Theorem | addgtge0 11751 | The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ 𝐵)) → 0 < (𝐴 + 𝐵)) | ||
| Theorem | addge0 11752 | The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵)) | ||
| Theorem | ltaddpos 11753 | Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) | ||
| Theorem | ltaddpos2 11754 | Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐴 + 𝐵))) | ||
| Theorem | ltsubpos 11755 | Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵 − 𝐴) < 𝐵)) | ||
| Theorem | posdif 11756 | Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | ||
| Theorem | lesub1 11757 | Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))) | ||
| Theorem | lesub2 11758 | Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐴))) | ||
| Theorem | ltsub1 11759 | Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 − 𝐶) < (𝐵 − 𝐶))) | ||
| Theorem | ltsub2 11760 | Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 − 𝐵) < (𝐶 − 𝐴))) | ||
| Theorem | lt2sub 11761 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐷 < 𝐵) → (𝐴 − 𝐵) < (𝐶 − 𝐷))) | ||
| Theorem | le2sub 11762 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) | ||
| Theorem | ltneg 11763 | Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)) | ||
| Theorem | ltnegcon1 11764 | Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)) | ||
| Theorem | ltnegcon2 11765 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < -𝐵 ↔ 𝐵 < -𝐴)) | ||
| Theorem | leneg 11766 | Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) | ||
| Theorem | lenegcon1 11767 | Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴)) | ||
| Theorem | lenegcon2 11768 | Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴)) | ||
| Theorem | lt0neg1 11769 | Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
| ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | ||
| Theorem | lt0neg2 11770 | Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.) |
| ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0)) | ||
| Theorem | le0neg1 11771 | Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.) |
| ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴)) | ||
| Theorem | le0neg2 11772 | Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.) |
| ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0)) | ||
| Theorem | addge01 11773 | A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) | ||
| Theorem | addge02 11774 | A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + 𝐴))) | ||
| Theorem | add20 11775 | Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | ||
| Theorem | subge0 11776 | Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | ||
| Theorem | suble0 11777 | Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵)) | ||
| Theorem | leaddle0 11778 | The sum of a real number and a second real number is less than the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐴 ↔ 𝐵 ≤ 0)) | ||
| Theorem | subge02 11779 | Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (𝐴 − 𝐵) ≤ 𝐴)) | ||
| Theorem | lesub0 11780 | 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 11781 | 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 11782 | 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 11783 | 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 11784 | A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴)) | ||
| Theorem | 0lt1 11785 | 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) |
| ⊢ 0 < 1 | ||
| Theorem | 0le1 11786 | 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ 0 ≤ 1 | ||
| Theorem | relin01 11787 | 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 11788* | Lemma for ltord1 11789. (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 11789* | 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 11790* | 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 11791* | 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 11792* | 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 11793* | 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 11794* | 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 11795* | Form of wlogle 11796 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 11796* | 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 11797 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ≤ 𝐴 | ||
| Theorem | gt0ne0i 11798 | Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 < 𝐴 → 𝐴 ≠ 0) | ||
| Theorem | gt0ne0ii 11799 | Positive implies nonzero. (Contributed by NM, 15-May-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 𝐴 ≠ 0 | ||
| Theorem | msqgt0i 11800 | 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)) | ||
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