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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnmulcl 8701 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)

Theoremnnmulcli 8702 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 · 𝐵) ∈ ℕ

Theoremnnge1 8703 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 1 ≤ 𝐴)

Theoremnnle1eq1 8704 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 8705 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(𝐴 ∈ ℕ → 0 < 𝐴)

Theoremnnnlt1 8706 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 8707 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
¬ 0 ∈ ℕ

Theoremnnne0 8708 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theoremnnap0 8709 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝐴 ∈ ℕ → 𝐴 # 0)

Theoremnngt0i 8710 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℕ       0 < 𝐴

Theoremnnap0i 8711 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
𝐴 ∈ ℕ       𝐴 # 0

Theoremnnne0i 8712 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ≠ 0

Theoremnn2ge 8713* 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 8714 A positive integer is either one or greater than one. This is for ; 0elnn 4500 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 8715 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 8716 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 8717 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)

Theoremnnrecgt0 8718 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 0 < (1 / 𝐴))

Theoremnnsub 8719 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 8720 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ)

Theoremnndiv 8721* 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 8722 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 8723 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)

Theoremnngt0d 8724 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)

Theoremnnne0d 8725 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ≠ 0)

Theoremnnap0d 8726 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 # 0)

Theoremnnrecred 8727 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremnnaddcld 8728 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ)

Theoremnnmulcld 8729 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ)

Theoremnndivred 8730 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 7591 through df-9 8746), 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 7591 and df-1 7592).

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 8731 Extend class notation to include the number 2.
class 2

Syntaxc3 8732 Extend class notation to include the number 3.
class 3

Syntaxc4 8733 Extend class notation to include the number 4.
class 4

Syntaxc5 8734 Extend class notation to include the number 5.
class 5

Syntaxc6 8735 Extend class notation to include the number 6.
class 6

Syntaxc7 8736 Extend class notation to include the number 7.
class 7

Syntaxc8 8737 Extend class notation to include the number 8.
class 8

Syntaxc9 8738 Extend class notation to include the number 9.
class 9

Definitiondf-2 8739 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)

Definitiondf-3 8740 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 8741 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 8742 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 8743 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 8744 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 8745 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 8746 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Theorem0ne1 8747 0 ≠ 1 (common case). See aso 1ap0 8315. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1ne0 8748 1 ≠ 0. See aso 1ap0 8315. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0

Theorem1m1e0 8749 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2re 8750 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ

Theorem2cn 8751 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 ∈ ℂ

Theorem2ex 8752 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V

Theorem2cnd 8753 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)

Theorem3re 8754 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ

Theorem3cn 8755 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 ∈ ℂ

Theorem3ex 8756 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V

Theorem4re 8757 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ

Theorem4cn 8758 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 ∈ ℂ

Theorem5re 8759 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ

Theorem5cn 8760 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 ∈ ℂ

Theorem6re 8761 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ

Theorem6cn 8762 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6 ∈ ℂ

Theorem7re 8763 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ

Theorem7cn 8764 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7 ∈ ℂ

Theorem8re 8765 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ

Theorem8cn 8766 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8 ∈ ℂ

Theorem9re 8767 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ

Theorem9cn 8768 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9 ∈ ℂ

Theorem0le0 8769 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 8770 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 8771 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 8772 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem2ap0 8773 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0

Theorem3pos 8774 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 8775 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem3ap0 8776 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0

Theorem4pos 8777 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 8778 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem4ap0 8779 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0

Theorem5pos 8780 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 8781 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 8782 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 8783 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 8784 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 8785 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 8786 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 8787 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 8788 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremneg1ap0 8789 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0

Theoremnegneg1e1 8790 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 8791 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 8792 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 8793 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 8794 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theoremfv0p1e1 8795 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))

Theorem1p0e1 8796 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 8797 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 8798 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8819. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 8799 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 8800 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2

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