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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnindALT 8701* |
Principle of Mathematical Induction (inference schema). The last four
hypotheses give us the substitution instances we need; the first two are
the induction step and the basis.
This ALT version of nnind 8700 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) & ⊢ 𝜓 & ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝐴 ∈ ℕ → 𝜏) | ||
Theorem | nn1m1nn 8702 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) | ||
Theorem | nn1suc 8703* | If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ ℕ → 𝜒) ⇒ ⊢ (𝐴 ∈ ℕ → 𝜃) | ||
Theorem | nnaddcl 8704 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ) | ||
Theorem | nnmulcl 8705 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) | ||
Theorem | nnmulcli 8706 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℕ | ||
Theorem | nnge1 8707 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | ||
Theorem | nnle1eq1 8708 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ 𝐴 = 1)) | ||
Theorem | nngt0 8709 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | ||
Theorem | nnnlt1 8710 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1) | ||
Theorem | 0nnn 8711 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
⊢ ¬ 0 ∈ ℕ | ||
Theorem | nnne0 8712 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) | ||
Theorem | nnap0 8713 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
⊢ (𝐴 ∈ ℕ → 𝐴 # 0) | ||
Theorem | nngt0i 8714 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ 0 < 𝐴 | ||
Theorem | nnap0i 8715 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ 𝐴 # 0 | ||
Theorem | nnne0i 8716 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ 𝐴 ≠ 0 | ||
Theorem | nn2ge 8717* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) | ||
Theorem | nn1gt1 8718 | A positive integer is either one or greater than one. This is for ℕ; 0elnn 4502 is a similar theorem for ω (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) | ||
Theorem | nngt1ne1 8719 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1)) | ||
Theorem | nndivre 8720 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) | ||
Theorem | nnrecre 8721 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ) | ||
Theorem | nnrecgt0 8722 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴)) | ||
Theorem | nnsub 8723 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ)) | ||
Theorem | nnsubi 8724 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ) | ||
Theorem | nndiv 8725* | Two ways to express "𝐴 divides 𝐵 " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ)) | ||
Theorem | nndivtr 8726 | Transitive property of divisibility: if 𝐴 divides 𝐵 and 𝐵 divides 𝐶, then 𝐴 divides 𝐶. Typically, 𝐶 would be an integer, although the theorem holds for complex 𝐶. (Contributed by NM, 3-May-2005.) |
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ) | ||
Theorem | nnge1d 8727 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 1 ≤ 𝐴) | ||
Theorem | nngt0d 8728 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 0 < 𝐴) | ||
Theorem | nnne0d 8729 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | nnap0d 8730 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 # 0) | ||
Theorem | nnrecred 8731 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) | ||
Theorem | nnaddcld 8732 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ) | ||
Theorem | nnmulcld 8733 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) | ||
Theorem | nndivred 8734 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) | ||
The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7595 through df-9 8750), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 7595 and df-1 7596). Integers can also be exhibited as sums of powers of 10 (e.g. the number 103 can be expressed as ((;10↑2) + 3)) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as (7↑7) − 2. Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12. | ||
Syntax | c2 8735 | Extend class notation to include the number 2. |
class 2 | ||
Syntax | c3 8736 | Extend class notation to include the number 3. |
class 3 | ||
Syntax | c4 8737 | Extend class notation to include the number 4. |
class 4 | ||
Syntax | c5 8738 | Extend class notation to include the number 5. |
class 5 | ||
Syntax | c6 8739 | Extend class notation to include the number 6. |
class 6 | ||
Syntax | c7 8740 | Extend class notation to include the number 7. |
class 7 | ||
Syntax | c8 8741 | Extend class notation to include the number 8. |
class 8 | ||
Syntax | c9 8742 | Extend class notation to include the number 9. |
class 9 | ||
Definition | df-2 8743 | Define the number 2. (Contributed by NM, 27-May-1999.) |
⊢ 2 = (1 + 1) | ||
Definition | df-3 8744 | Define the number 3. (Contributed by NM, 27-May-1999.) |
⊢ 3 = (2 + 1) | ||
Definition | df-4 8745 | Define the number 4. (Contributed by NM, 27-May-1999.) |
⊢ 4 = (3 + 1) | ||
Definition | df-5 8746 | Define the number 5. (Contributed by NM, 27-May-1999.) |
⊢ 5 = (4 + 1) | ||
Definition | df-6 8747 | Define the number 6. (Contributed by NM, 27-May-1999.) |
⊢ 6 = (5 + 1) | ||
Definition | df-7 8748 | Define the number 7. (Contributed by NM, 27-May-1999.) |
⊢ 7 = (6 + 1) | ||
Definition | df-8 8749 | Define the number 8. (Contributed by NM, 27-May-1999.) |
⊢ 8 = (7 + 1) | ||
Definition | df-9 8750 | Define the number 9. (Contributed by NM, 27-May-1999.) |
⊢ 9 = (8 + 1) | ||
Theorem | 0ne1 8751 | 0 ≠ 1 (common case). See aso 1ap0 8319. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≠ 1 | ||
Theorem | 1ne0 8752 | 1 ≠ 0. See aso 1ap0 8319. (Contributed by Jim Kingdon, 9-Mar-2020.) |
⊢ 1 ≠ 0 | ||
Theorem | 1m1e0 8753 | (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (1 − 1) = 0 | ||
Theorem | 2re 8754 | The number 2 is real. (Contributed by NM, 27-May-1999.) |
⊢ 2 ∈ ℝ | ||
Theorem | 2cn 8755 | The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) |
⊢ 2 ∈ ℂ | ||
Theorem | 2ex 8756 | 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 2 ∈ V | ||
Theorem | 2cnd 8757 | 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 2 ∈ ℂ) | ||
Theorem | 3re 8758 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
⊢ 3 ∈ ℝ | ||
Theorem | 3cn 8759 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) |
⊢ 3 ∈ ℂ | ||
Theorem | 3ex 8760 | 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ V | ||
Theorem | 4re 8761 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
⊢ 4 ∈ ℝ | ||
Theorem | 4cn 8762 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 4 ∈ ℂ | ||
Theorem | 5re 8763 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
⊢ 5 ∈ ℝ | ||
Theorem | 5cn 8764 | The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 5 ∈ ℂ | ||
Theorem | 6re 8765 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
⊢ 6 ∈ ℝ | ||
Theorem | 6cn 8766 | The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 6 ∈ ℂ | ||
Theorem | 7re 8767 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
⊢ 7 ∈ ℝ | ||
Theorem | 7cn 8768 | The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 7 ∈ ℂ | ||
Theorem | 8re 8769 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
⊢ 8 ∈ ℝ | ||
Theorem | 8cn 8770 | The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 8 ∈ ℂ | ||
Theorem | 9re 8771 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
⊢ 9 ∈ ℝ | ||
Theorem | 9cn 8772 | The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 9 ∈ ℂ | ||
Theorem | 0le0 8773 | Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ≤ 0 | ||
Theorem | 0le2 8774 | 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ 0 ≤ 2 | ||
Theorem | 2pos 8775 | The number 2 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 2 | ||
Theorem | 2ne0 8776 | The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) |
⊢ 2 ≠ 0 | ||
Theorem | 2ap0 8777 | The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
⊢ 2 # 0 | ||
Theorem | 3pos 8778 | The number 3 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 3 | ||
Theorem | 3ne0 8779 | The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ 3 ≠ 0 | ||
Theorem | 3ap0 8780 | The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ 3 # 0 | ||
Theorem | 4pos 8781 | The number 4 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 4 | ||
Theorem | 4ne0 8782 | The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ 4 ≠ 0 | ||
Theorem | 4ap0 8783 | The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ 4 # 0 | ||
Theorem | 5pos 8784 | The number 5 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 5 | ||
Theorem | 6pos 8785 | The number 6 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 6 | ||
Theorem | 7pos 8786 | The number 7 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 7 | ||
Theorem | 8pos 8787 | The number 8 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 8 | ||
Theorem | 9pos 8788 | The number 9 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 9 | ||
This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10. | ||
Theorem | neg1cn 8789 | -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ -1 ∈ ℂ | ||
Theorem | neg1rr 8790 | -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℝ | ||
Theorem | neg1ne0 8791 | -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 ≠ 0 | ||
Theorem | neg1lt0 8792 | -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 < 0 | ||
Theorem | neg1ap0 8793 | -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ -1 # 0 | ||
Theorem | negneg1e1 8794 | --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ --1 = 1 | ||
Theorem | 1pneg1e0 8795 | 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + -1) = 0 | ||
Theorem | 0m0e0 8796 | 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0 − 0) = 0 | ||
Theorem | 1m0e1 8797 | 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 − 0) = 1 | ||
Theorem | 0p1e1 8798 | 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (0 + 1) = 1 | ||
Theorem | fv0p1e1 8799 | Function value at 𝑁 + 1 with 𝑁 replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1)) | ||
Theorem | 1p0e1 8800 | 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + 0) = 1 |
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