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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnnlt1 7701 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 7702 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
¬ 0

Theoremnnne0 7703 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(A ℕ → A ≠ 0)

Theoremnnap0 7704 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(A ℕ → A # 0)

Theoremnngt0i 7705 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
A        0 < A

Theoremnnne0i 7706 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
A        A ≠ 0

Theoremnn2ge 7707* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
((A B ℕ) → x ℕ (Ax Bx))

Theoremnn1gt1 7708 A positive integer is either one or greater than one. This is for ; 0elnn 4283 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 7709 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 7710 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((A 𝑁 ℕ) → (A / 𝑁) ℝ)

Theoremnnrecre 7711 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ℕ → (1 / 𝑁) ℝ)

Theoremnnrecgt0 7712 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(A ℕ → 0 < (1 / A))

Theoremnnsub 7713 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
((A B ℕ) → (A < B ↔ (BA) ℕ))

Theoremnnsubi 7714 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
A     &   B        (A < B ↔ (BA) ℕ)

Theoremnndiv 7715* 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 B ℕ) → (x ℕ (A · x) = B ↔ (B / A) ℕ))

Theoremnndivtr 7716 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 B 𝐶 ℂ) ((B / A) (𝐶 / B) ℕ)) → (𝐶 / A) ℕ)

Theoremnnge1d 7717 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → 1 ≤ A)

Theoremnngt0d 7718 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → 0 < A)

Theoremnnne0d 7719 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA ≠ 0)

Theoremnnrecred 7720 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → (1 / A) ℝ)

Theoremnnaddcld 7721 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)    &   (φB ℕ)       (φ → (A + B) ℕ)

Theoremnnmulcld 7722 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)    &   (φB ℕ)       (φ → (A · B) ℕ)

Theoremnndivred 7723 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℕ)       (φ → (A / B) ℝ)

3.4.3  Decimal representation of numbers

Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 6698 and df-1 6699).

Only the digits 0 through 9 (df-0 6698 through df-9 7740) and the number 10 (df-10 7741) are explicitly defined.

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 7724 Extend class notation to include the number 2.
class 2

Syntaxc3 7725 Extend class notation to include the number 3.
class 3

Syntaxc4 7726 Extend class notation to include the number 4.
class 4

Syntaxc5 7727 Extend class notation to include the number 5.
class 5

Syntaxc6 7728 Extend class notation to include the number 6.
class 6

Syntaxc7 7729 Extend class notation to include the number 7.
class 7

Syntaxc8 7730 Extend class notation to include the number 8.
class 8

Syntaxc9 7731 Extend class notation to include the number 9.
class 9

Syntaxc10 7732 Extend class notation to include the number 10.
class 10

Definitiondf-2 7733 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)

Definitiondf-3 7734 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 7735 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 7736 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 7737 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 7738 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 7739 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 7740 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Definitiondf-10 7741 Define the number 10. See remarks under df-2 7733. (Contributed by NM, 5-Feb-2007.)
10 = (9 + 1)

Theorem0ne1 7742 0 ≠ 1 (common case). See aso 1ap0 7354. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1ne0 7743 1 ≠ 0. See aso 1ap0 7354. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0

Theorem1m1e0 7744 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2re 7745 The number 2 is real. (Contributed by NM, 27-May-1999.)
2

Theorem2cn 7746 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2

Theorem2ex 7747 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 V

Theorem2cnd 7748 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(φ → 2 ℂ)

Theorem3re 7749 The number 3 is real. (Contributed by NM, 27-May-1999.)
3

Theorem3cn 7750 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3

Theorem3ex 7751 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 V

Theorem4re 7752 The number 4 is real. (Contributed by NM, 27-May-1999.)
4

Theorem4cn 7753 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4

Theorem5re 7754 The number 5 is real. (Contributed by NM, 27-May-1999.)
5

Theorem5cn 7755 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5

Theorem6re 7756 The number 6 is real. (Contributed by NM, 27-May-1999.)
6

Theorem6cn 7757 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6

Theorem7re 7758 The number 7 is real. (Contributed by NM, 27-May-1999.)
7

Theorem7cn 7759 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7

Theorem8re 7760 The number 8 is real. (Contributed by NM, 27-May-1999.)
8

Theorem8cn 7761 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8

Theorem9re 7762 The number 9 is real. (Contributed by NM, 27-May-1999.)
9

Theorem9cn 7763 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9

Theorem10re 7764 The number 10 is real. (Contributed by NM, 5-Feb-2007.)
10

Theorem0le0 7765 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 7766 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 7767 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 7768 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem2ap0 7769 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0

Theorem3pos 7770 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 7771 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem4pos 7772 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 7773 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem5pos 7774 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 7775 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 7776 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 7777 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 7778 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9

Theorem10pos 7779 The number 10 is positive. (Contributed by NM, 5-Feb-2007.)
0 < 10

3.4.4  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 7780 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1

Theoremneg1rr 7781 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1

Theoremneg1ne0 7782 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 7783 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremneg1ap0 7784 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0

Theoremnegneg1e1 7785 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 7786 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 7787 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 7788 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 7789 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theorem1p0e1 7790 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 7791 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 7792 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 7814. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 7793 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 7794 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2

Theorem2p2e4 7795 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4

Theorem2times 7796 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
(A ℂ → (2 · A) = (A + A))

Theoremtimes2 7797 A number times 2. (Contributed by NM, 16-Oct-2007.)
(A ℂ → (A · 2) = (A + A))

Theorem2timesi 7798 Two times a number. (Contributed by NM, 1-Aug-1999.)
A        (2 · A) = (A + A)

Theoremtimes2i 7799 A number times 2. (Contributed by NM, 11-May-2004.)
A        (A · 2) = (A + A)

Theorem2div2e1 7800 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1

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