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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnex 7701 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
V
 
Theoremnnre 7702 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
(A ℕ → A ℝ)
 
Theoremnncn 7703 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
(A ℕ → A ℂ)
 
Theoremnnrei 7704 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
A        A
 
Theoremnncni 7705 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
A        A
 
Theorem1nn 7706 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
1
 
Theorempeano2nn 7707 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
(A ℕ → (A + 1) ℕ)
 
Theoremnnred 7708 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA ℝ)
 
Theoremnncnd 7709 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA ℂ)
 
Theorempeano2nnd 7710 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)
 
3.4.2  Principle of mathematical induction
 
Theoremnnind 7711* 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 7715 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 7712* 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 7711 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 7713 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(A ℕ → (A = 1 (A − 1) ℕ))
 
Theoremnn1suc 7714* 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 7715 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
((A B ℕ) → (A + B) ℕ)
 
Theoremnnmulcl 7716 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((A B ℕ) → (A · B) ℕ)
 
Theoremnnmulcli 7717 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
A     &   B        (A · B)
 
Theoremnnge1 7718 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(A ℕ → 1 ≤ A)
 
Theoremnnle1eq1 7719 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 7720 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(A ℕ → 0 < A)
 
Theoremnnnlt1 7721 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 7722 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
¬ 0
 
Theoremnnne0 7723 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(A ℕ → A ≠ 0)
 
Theoremnnap0 7724 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(A ℕ → A # 0)
 
Theoremnngt0i 7725 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
A        0 < A
 
Theoremnnne0i 7726 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
A        A ≠ 0
 
Theoremnn2ge 7727* 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 7728 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 7729 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 7730 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((A 𝑁 ℕ) → (A / 𝑁) ℝ)
 
Theoremnnrecre 7731 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ℕ → (1 / 𝑁) ℝ)
 
Theoremnnrecgt0 7732 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(A ℕ → 0 < (1 / A))
 
Theoremnnsub 7733 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 7734 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
A     &   B        (A < B ↔ (BA) ℕ)
 
Theoremnndiv 7735* 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 7736 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 7737 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → 1 ≤ A)
 
Theoremnngt0d 7738 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → 0 < A)
 
Theoremnnne0d 7739 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA ≠ 0)
 
Theoremnnrecred 7740 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φ → (1 / A) ℝ)
 
Theoremnnaddcld 7741 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)    &   (φB ℕ)       (φ → (A + B) ℕ)
 
Theoremnnmulcld 7742 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)    &   (φB ℕ)       (φ → (A · B) ℕ)
 
Theoremnndivred 7743 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 6718 and df-1 6719).

Only the digits 0 through 9 (df-0 6718 through df-9 7760) and the number 10 (df-10 7761) 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 7744 Extend class notation to include the number 2.
class 2
 
Syntaxc3 7745 Extend class notation to include the number 3.
class 3
 
Syntaxc4 7746 Extend class notation to include the number 4.
class 4
 
Syntaxc5 7747 Extend class notation to include the number 5.
class 5
 
Syntaxc6 7748 Extend class notation to include the number 6.
class 6
 
Syntaxc7 7749 Extend class notation to include the number 7.
class 7
 
Syntaxc8 7750 Extend class notation to include the number 8.
class 8
 
Syntaxc9 7751 Extend class notation to include the number 9.
class 9
 
Syntaxc10 7752 Extend class notation to include the number 10.
class 10
 
Definitiondf-2 7753 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)
 
Definitiondf-3 7754 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)
 
Definitiondf-4 7755 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)
 
Definitiondf-5 7756 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)
 
Definitiondf-6 7757 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)
 
Definitiondf-7 7758 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)
 
Definitiondf-8 7759 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)
 
Definitiondf-9 7760 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)
 
Definitiondf-10 7761 Define the number 10. See remarks under df-2 7753. (Contributed by NM, 5-Feb-2007.)
10 = (9 + 1)
 
Theorem0ne1 7762 0 ≠ 1 (common case). See aso 1ap0 7374. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1
 
Theorem1ne0 7763 1 ≠ 0. See aso 1ap0 7374. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0
 
Theorem1m1e0 7764 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0
 
Theorem2re 7765 The number 2 is real. (Contributed by NM, 27-May-1999.)
2
 
Theorem2cn 7766 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2
 
Theorem2ex 7767 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 V
 
Theorem2cnd 7768 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(φ → 2 ℂ)
 
Theorem3re 7769 The number 3 is real. (Contributed by NM, 27-May-1999.)
3
 
Theorem3cn 7770 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3
 
Theorem3ex 7771 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 V
 
Theorem4re 7772 The number 4 is real. (Contributed by NM, 27-May-1999.)
4
 
Theorem4cn 7773 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4
 
Theorem5re 7774 The number 5 is real. (Contributed by NM, 27-May-1999.)
5
 
Theorem5cn 7775 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5
 
Theorem6re 7776 The number 6 is real. (Contributed by NM, 27-May-1999.)
6
 
Theorem6cn 7777 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6
 
Theorem7re 7778 The number 7 is real. (Contributed by NM, 27-May-1999.)
7
 
Theorem7cn 7779 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7
 
Theorem8re 7780 The number 8 is real. (Contributed by NM, 27-May-1999.)
8
 
Theorem8cn 7781 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8
 
Theorem9re 7782 The number 9 is real. (Contributed by NM, 27-May-1999.)
9
 
Theorem9cn 7783 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9
 
Theorem10re 7784 The number 10 is real. (Contributed by NM, 5-Feb-2007.)
10
 
Theorem0le0 7785 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0
 
Theorem0le2 7786 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2
 
Theorem2pos 7787 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2
 
Theorem2ne0 7788 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0
 
Theorem2ap0 7789 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0
 
Theorem3pos 7790 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3
 
Theorem3ne0 7791 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0
 
Theorem4pos 7792 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4
 
Theorem4ne0 7793 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0
 
Theorem5pos 7794 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5
 
Theorem6pos 7795 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6
 
Theorem7pos 7796 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7
 
Theorem8pos 7797 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8
 
Theorem9pos 7798 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9
 
Theorem10pos 7799 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 7800 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1
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