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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | recreclt 10801 | Given a positive number 𝐴, construct a new positive number less than both 𝐴 and 1. (Contributed by NM, 28-Dec-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((1 / (1 + (1 / 𝐴))) < 1 ∧ (1 / (1 + (1 / 𝐴))) < 𝐴)) | ||
Theorem | le2msq 10802 | The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵))) | ||
Theorem | msq11 10803 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | ledivp1 10804 | Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ 𝐴) | ||
Theorem | squeeze0 10805* | If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) | ||
Theorem | ltp1i 10806 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 < (𝐴 + 1) | ||
Theorem | recgt0i 10807 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 < 𝐴 → 0 < (1 / 𝐴)) | ||
Theorem | recgt0ii 10808 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 0 < (1 / 𝐴) | ||
Theorem | prodgt0i 10809 | Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵)) → 0 < 𝐵) | ||
Theorem | prodge0i 10810 | Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵)) → 0 ≤ 𝐵) | ||
Theorem | divgt0i 10811 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 / 𝐵)) | ||
Theorem | divge0i 10812 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 ≤ (𝐴 / 𝐵)) | ||
Theorem | ltreci 10813 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))) | ||
Theorem | lereci 10814 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) | ||
Theorem | lt2msqi 10815 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵))) | ||
Theorem | le2msqi 10816 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵))) | ||
Theorem | msq11i 10817 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | divgt0i2i 10818 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 < 𝐵 ⇒ ⊢ (0 < 𝐴 → 0 < (𝐴 / 𝐵)) | ||
Theorem | ltrecii 10819 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 < 𝐴 & ⊢ 0 < 𝐵 ⇒ ⊢ (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)) | ||
Theorem | divgt0ii 10820 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 < 𝐴 & ⊢ 0 < 𝐵 ⇒ ⊢ 0 < (𝐴 / 𝐵) | ||
Theorem | ltmul1i 10821 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) | ||
Theorem | ltdiv1i 10822 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) | ||
Theorem | ltmuldivi 10823 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (0 < 𝐶 → ((𝐴 · 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶))) | ||
Theorem | ltmul2i 10824 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) | ||
Theorem | lemul1i 10825 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (0 < 𝐶 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) | ||
Theorem | lemul2i 10826 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (0 < 𝐶 → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) | ||
Theorem | ltdiv23i 10827 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((0 < 𝐵 ∧ 0 < 𝐶) → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) | ||
Theorem | ledivp1i 10828 | Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐶 ∧ 𝐴 ≤ (𝐵 / (𝐶 + 1))) → (𝐴 · 𝐶) ≤ 𝐵) | ||
Theorem | ltdivp1i 10829 | Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐶 ∧ 𝐴 < (𝐵 / (𝐶 + 1))) → (𝐴 · 𝐶) < 𝐵) | ||
Theorem | ltdiv23ii 10830 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 0 < 𝐵 & ⊢ 0 < 𝐶 ⇒ ⊢ ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵) | ||
Theorem | ltmul1ii 10831 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 0 < 𝐶 ⇒ ⊢ (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)) | ||
Theorem | ltdiv1ii 10832 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 0 < 𝐶 ⇒ ⊢ (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)) | ||
Theorem | ltp1d 10833 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) | ||
Theorem | lep1d 10834 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐴 + 1)) | ||
Theorem | ltm1d 10835 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 − 1) < 𝐴) | ||
Theorem | lem1d 10836 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 − 1) ≤ 𝐴) | ||
Theorem | recgt0d 10837 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → 0 < (1 / 𝐴)) | ||
Theorem | divgt0d 10838 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 / 𝐵)) | ||
Theorem | mulgt1d 10839 | The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 1 < 𝐵) ⇒ ⊢ (𝜑 → 1 < (𝐴 · 𝐵)) | ||
Theorem | lemulge11d 10840 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 1 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐴 · 𝐵)) | ||
Theorem | lemulge12d 10841 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 1 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐵 · 𝐴)) | ||
Theorem | lemul1ad 10842 | Multiplication of both sides of 'less than or equal to' by 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 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | ||
Theorem | lemul2ad 10843 | Multiplication of both sides of 'less than or equal to' by 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 · 𝐴) ≤ (𝐶 · 𝐵)) | ||
Theorem | ltmul12ad 10844 | Comparison of product of two positive 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 → 0 ≤ 𝐶) & ⊢ (𝜑 → 𝐶 < 𝐷) ⇒ ⊢ (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐷)) | ||
Theorem | lemul12ad 10845 | Comparison of product 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 → 0 ≤ 𝐶) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)) | ||
Theorem | lemul12bd 10846 | Comparison of product 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 → 0 ≤ 𝐷) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)) | ||
Theorem | fimaxre 10847* | A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | ||
Theorem | fimaxre2 10848* | A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | ||
Theorem | fimaxre3 10849* | A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝐵 ≤ 𝑥) | ||
Theorem | negfi 10850* | The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ∈ Fin) | ||
Theorem | fiminre 10851* | A nonempty finite set of real numbers has a minimum. Analogous to fimaxre 10847. (Contributed by AV, 9-Aug-2020.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | ||
Theorem | lbreu 10852* | If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) | ||
Theorem | lbcl 10853* | If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∈ 𝑆) | ||
Theorem | lble 10854* | If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴) | ||
Theorem | lbinf 10855* | If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → inf(𝑆, ℝ, < ) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)) | ||
Theorem | lbinfcl 10856* | If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → inf(𝑆, ℝ, < ) ∈ 𝑆) | ||
Theorem | lbinfle 10857* | If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴) | ||
Theorem | sup2 10858* | A nonempty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
Theorem | sup3 10859* | A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
Theorem | infm3lem 10860* | Lemma for infm3 10861. (Contributed by NM, 14-Jun-2005.) |
⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑥 = -𝑦) | ||
Theorem | infm3 10861* | The completeness axiom for reals in terms of infimum: a nonempty, bounded-below set of reals has an infimum. (This theorem is the dual of sup3 10859.) (Contributed by NM, 14-Jun-2005.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | ||
Theorem | suprcl 10862* | Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Oct-2004.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | ||
Theorem | suprub 10863* | A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < )) | ||
Theorem | suprlub 10864* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) | ||
Theorem | suprnub 10865* | An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) | ||
Theorem | suprleub 10866* | The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) | ||
Theorem | supaddc 10867* | The supremum function distributes over addition in a sense similar to that in supmul1 10869. (Contributed by Brendan Leahy, 25-Sep-2017.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 + 𝐵)} ⇒ ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + 𝐵) = sup(𝐶, ℝ, < )) | ||
Theorem | supadd 10868* | The supremum function distributes over addition in a sense similar to that in supmul 10872. (Contributed by Brendan Leahy, 26-Sep-2017.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥) & ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 + 𝑏)} ⇒ ⊢ (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < )) | ||
Theorem | supmul1 10869* | The supremum function distributes over multiplication, in the sense that 𝐴 · (sup𝐵) = sup(𝐴 · 𝐵), where 𝐴 · 𝐵 is shorthand for {𝐴 · 𝑏 ∣ 𝑏 ∈ 𝐵} and is defined as 𝐶 below. This is the simple version, with only one set argument; see supmul 10872 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.) |
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐵 𝑧 = (𝐴 · 𝑣)} & ⊢ (𝜑 ↔ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥))) ⇒ ⊢ (𝜑 → (𝐴 · sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < )) | ||
Theorem | supmullem1 10870* | Lemma for supmul 10872. (Contributed by Mario Carneiro, 5-Jul-2013.) |
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)} & ⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥))) ⇒ ⊢ (𝜑 → ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < ))) | ||
Theorem | supmullem2 10871* | Lemma for supmul 10872. (Contributed by Mario Carneiro, 5-Jul-2013.) |
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)} & ⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥))) ⇒ ⊢ (𝜑 → (𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝐶 𝑤 ≤ 𝑥)) | ||
Theorem | supmul 10872* | The supremum function distributes over multiplication, in the sense that (sup𝐴) · (sup𝐵) = sup(𝐴 · 𝐵), where 𝐴 · 𝐵 is shorthand for {𝑎 · 𝑏 ∣ 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} and is defined as 𝐶 below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 9685). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)} & ⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥))) ⇒ ⊢ (𝜑 → (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < )) | ||
Theorem | sup3ii 10873* | A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.) |
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) | ||
Theorem | suprclii 10874* | Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Sep-1999.) |
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ sup(𝐴, ℝ, < ) ∈ ℝ | ||
Theorem | suprubii 10875* | A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999.) |
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 ≤ sup(𝐴, ℝ, < )) | ||
Theorem | suprlubii 10876* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ (𝐵 ∈ ℝ → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) | ||
Theorem | suprnubii 10877* | An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ (𝐵 ∈ ℝ → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) | ||
Theorem | suprleubii 10878* | The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ (𝐵 ∈ ℝ → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) | ||
Theorem | riotaneg 10879* | The negative of the unique real such that 𝜑. (Contributed by NM, 13-Jun-2005.) |
⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ ℝ 𝜑 → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) | ||
Theorem | negiso 10880 | Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥) ⇒ ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹) | ||
Theorem | dfinfre 10881* | The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))}) | ||
Theorem | infrecl 10882* | Closure of infimum of a nonempty bounded set of reals. (Contributed by NM, 8-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) ∈ ℝ) | ||
Theorem | infrenegsup 10883* | The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. The antecedent ensures that 𝐴 is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < )) | ||
Theorem | infregelb 10884* | Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by AV, 4-Sep-2020.) (Proof modification is discouraged.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)) | ||
Theorem | infrelb 10885* | If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by AV, 4-Sep-2020.) |
⊢ ((𝐵 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴) | ||
Theorem | supfirege 10886 | The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.) |
⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < )) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝑆) | ||
Theorem | inelr 10887 | The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.) |
⊢ ¬ i ∈ ℝ | ||
Theorem | rimul 10888 | A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0) | ||
Theorem | cru 10889 | The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | crne0 10890 | The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ (𝐴 + (i · 𝐵)) ≠ 0)) | ||
Theorem | creur 10891* | The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | creui 10892* | The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | cju 10893* | The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ)) | ||
Theorem | ofsubeq0 10894 | Function analogue of subeq0 10186. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘𝑓 − 𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺)) | ||
Theorem | ofnegsub 10895 | Function analogue of negsub 10208. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹 ∘𝑓 + ((𝐴 × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) | ||
Theorem | ofsubge0 10896 | Function analogue of subge0 10420. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ) → ((𝐴 × {0}) ∘𝑟 ≤ (𝐹 ∘𝑓 − 𝐺) ↔ 𝐺 ∘𝑟 ≤ 𝐹)) | ||
Syntax | cn 10897 | Extend class notation to include the class of positive integers. |
class ℕ | ||
Definition | df-nn 10898 |
Define the set of positive integers. Some authors, especially in analysis
books, call these the natural numbers, whereas other authors choose to
include 0 in their definition of natural numbers. Note that ℕ is a
subset of complex numbers (nnsscn 10902), in contrast to the more elementary
ordinal natural numbers ω, df-om 6958). See nnind 10915 for the
principle of mathematical induction. See df-n0 11170 for the set of
nonnegative integers ℕ0. See dfn2 11182
for ℕ defined in terms of
ℕ0.
This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 8421 in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing 1 as well as the successor of every member") see dfnn3 10911 (or its slight variant dfnn2 10910). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.) |
⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | ||
Theorem | nnexALT 10899 | Alternate proof of nnex 10903, more direct, that makes use of ax-rep 4699. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℕ ∈ V | ||
Theorem | peano5nni 10900* | Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴) |
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