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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnred 8701 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ)
 
Theoremnncnd 8702 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℂ)
 
Theorempeano2nnd 8703 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (𝐴 + 1) ∈ ℕ)
 
4.4.2  Principle of mathematical induction
 
Theoremnnind 8704* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8708 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
(𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ → (𝜒𝜃))       (𝐴 ∈ ℕ → 𝜏)
 
TheoremnnindALT 8705* 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 8704 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 → (𝜒𝜃))    &   𝜓    &   (𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))       (𝐴 ∈ ℕ → 𝜏)
 
Theoremnn1m1nn 8706 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
 
Theoremnn1suc 8707* 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 → (𝜑𝜓))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜑𝜃))    &   𝜓    &   (𝑦 ∈ ℕ → 𝜒)       (𝐴 ∈ ℕ → 𝜃)
 
Theoremnnaddcl 8708 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)
 
Theoremnnmulcl 8709 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
 
Theoremnnmulcli 8710 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 · 𝐵) ∈ ℕ
 
Theoremnnge1 8711 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 1 ≤ 𝐴)
 
Theoremnnle1eq1 8712 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))
 
Theoremnngt0 8713 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(𝐴 ∈ ℕ → 0 < 𝐴)
 
Theoremnnnlt1 8714 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)
 
Theorem0nnn 8715 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
¬ 0 ∈ ℕ
 
Theoremnnne0 8716 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)
 
Theoremnnap0 8717 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝐴 ∈ ℕ → 𝐴 # 0)
 
Theoremnngt0i 8718 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℕ       0 < 𝐴
 
Theoremnnap0i 8719 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
𝐴 ∈ ℕ       𝐴 # 0
 
Theoremnnne0i 8720 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ≠ 0
 
Theoremnn2ge 8721* 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 (𝐴𝑥𝐵𝑥))
 
Theoremnn1gt1 8722 A positive integer is either one or greater than one. This is for ; 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))
 
Theoremnngt1ne1 8723 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
(𝐴 ∈ ℕ → (1 < 𝐴𝐴 ≠ 1))
 
Theoremnndivre 8724 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ)
 
Theoremnnrecre 8725 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
 
Theoremnnrecgt0 8726 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 0 < (1 / 𝐴))
 
Theoremnnsub 8727 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ))
 
Theoremnnsubi 8728 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ)
 
Theoremnndiv 8729* 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))
 
Theoremnndivtr 8730 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)
 
Theoremnnge1d 8731 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)
 
Theoremnngt0d 8732 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)
 
Theoremnnne0d 8733 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ≠ 0)
 
Theoremnnap0d 8734 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 # 0)
 
Theoremnnrecred 8735 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (1 / 𝐴) ∈ ℝ)
 
Theoremnnaddcld 8736 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ)
 
Theoremnnmulcld 8737 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
 
Theoremnndivred 8738 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
 
4.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7595 through df-9 8754), 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.

 
Syntaxc2 8739 Extend class notation to include the number 2.
class 2
 
Syntaxc3 8740 Extend class notation to include the number 3.
class 3
 
Syntaxc4 8741 Extend class notation to include the number 4.
class 4
 
Syntaxc5 8742 Extend class notation to include the number 5.
class 5
 
Syntaxc6 8743 Extend class notation to include the number 6.
class 6
 
Syntaxc7 8744 Extend class notation to include the number 7.
class 7
 
Syntaxc8 8745 Extend class notation to include the number 8.
class 8
 
Syntaxc9 8746 Extend class notation to include the number 9.
class 9
 
Definitiondf-2 8747 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)
 
Definitiondf-3 8748 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)
 
Definitiondf-4 8749 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)
 
Definitiondf-5 8750 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)
 
Definitiondf-6 8751 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)
 
Definitiondf-7 8752 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)
 
Definitiondf-8 8753 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)
 
Definitiondf-9 8754 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)
 
Theorem0ne1 8755 0 ≠ 1 (common case). See aso 1ap0 8320. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1
 
Theorem1ne0 8756 1 ≠ 0. See aso 1ap0 8320. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0
 
Theorem1m1e0 8757 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0
 
Theorem2re 8758 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ
 
Theorem2cn 8759 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 ∈ ℂ
 
Theorem2ex 8760 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V
 
Theorem2cnd 8761 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)
 
Theorem3re 8762 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ
 
Theorem3cn 8763 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 ∈ ℂ
 
Theorem3ex 8764 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V
 
Theorem4re 8765 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ
 
Theorem4cn 8766 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 ∈ ℂ
 
Theorem5re 8767 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ
 
Theorem5cn 8768 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 ∈ ℂ
 
Theorem6re 8769 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ
 
Theorem6cn 8770 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6 ∈ ℂ
 
Theorem7re 8771 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ
 
Theorem7cn 8772 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7 ∈ ℂ
 
Theorem8re 8773 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ
 
Theorem8cn 8774 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8 ∈ ℂ
 
Theorem9re 8775 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ
 
Theorem9cn 8776 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9 ∈ ℂ
 
Theorem0le0 8777 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0
 
Theorem0le2 8778 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2
 
Theorem2pos 8779 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2
 
Theorem2ne0 8780 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0
 
Theorem2ap0 8781 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0
 
Theorem3pos 8782 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3
 
Theorem3ne0 8783 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0
 
Theorem3ap0 8784 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0
 
Theorem4pos 8785 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4
 
Theorem4ne0 8786 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0
 
Theorem4ap0 8787 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0
 
Theorem5pos 8788 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5
 
Theorem6pos 8789 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6
 
Theorem7pos 8790 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7
 
Theorem8pos 8791 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8
 
Theorem9pos 8792 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9
 
4.4.4  Some properties of specific numbers

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.

 
Theoremneg1cn 8793 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ
 
Theoremneg1rr 8794 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ
 
Theoremneg1ne0 8795 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0
 
Theoremneg1lt0 8796 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0
 
Theoremneg1ap0 8797 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0
 
Theoremnegneg1e1 8798 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1
 
Theorem1pneg1e0 8799 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0
 
Theorem0m0e0 8800 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0
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