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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1nn 7901 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
1 ∈ ℕ

Theorempeano2nn 7902 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 7903 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ)

Theoremnncnd 7904 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℂ)

Theorempeano2nnd 7905 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 7906* 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 7910 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 7907* 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 7906 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 7908 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))

Theoremnn1suc 7909* 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 7910 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 7911 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)

Theoremnnmulcli 7912 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 · 𝐵) ∈ ℕ

Theoremnnge1 7913 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 1 ≤ 𝐴)

Theoremnnle1eq1 7914 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 7915 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(𝐴 ∈ ℕ → 0 < 𝐴)

Theoremnnnlt1 7916 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 7917 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
¬ 0 ∈ ℕ

Theoremnnne0 7918 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)

Theoremnnap0 7919 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝐴 ∈ ℕ → 𝐴 # 0)

Theoremnngt0i 7920 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℕ       0 < 𝐴

Theoremnnne0i 7921 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℕ       𝐴 ≠ 0

Theoremnn2ge 7922* 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 7923 A positive integer is either one or greater than one. This is for ; 0elnn 4327 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 7924 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 7925 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 7926 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)

Theoremnnrecgt0 7927 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(𝐴 ∈ ℕ → 0 < (1 / 𝐴))

Theoremnnsub 7928 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 7929 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ       (𝐴 < 𝐵 ↔ (𝐵𝐴) ∈ ℕ)

Theoremnndiv 7930* 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 7931 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 7932 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 1 ≤ 𝐴)

Theoremnngt0d 7933 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → 0 < 𝐴)

Theoremnnne0d 7934 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ≠ 0)

Theoremnnap0d 7935 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 # 0)

Theoremnnrecred 7936 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremnnaddcld 7937 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ)

Theoremnnmulcld 7938 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ)

Theoremnndivred 7939 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 6877 and df-1 6878).

Only the digits 0 through 9 (df-0 6877 through df-9 7956) and the number 10 (df-10 7957) 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 7940 Extend class notation to include the number 2.
class 2

Syntaxc3 7941 Extend class notation to include the number 3.
class 3

Syntaxc4 7942 Extend class notation to include the number 4.
class 4

Syntaxc5 7943 Extend class notation to include the number 5.
class 5

Syntaxc6 7944 Extend class notation to include the number 6.
class 6

Syntaxc7 7945 Extend class notation to include the number 7.
class 7

Syntaxc8 7946 Extend class notation to include the number 8.
class 8

Syntaxc9 7947 Extend class notation to include the number 9.
class 9

Syntaxc10 7948 Extend class notation to include the number 10.
class 10

Definitiondf-2 7949 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)

Definitiondf-3 7950 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 7951 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 7952 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 7953 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 7954 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 7955 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 7956 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Definitiondf-10 7957 Define the number 10. See remarks under df-2 7949. (Contributed by NM, 5-Feb-2007.)
10 = (9 + 1)

Theorem0ne1 7958 0 ≠ 1 (common case). See aso 1ap0 7557. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1ne0 7959 1 ≠ 0. See aso 1ap0 7557. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0

Theorem1m1e0 7960 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2re 7961 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ

Theorem2cn 7962 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 ∈ ℂ

Theorem2ex 7963 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V

Theorem2cnd 7964 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)

Theorem3re 7965 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ

Theorem3cn 7966 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 ∈ ℂ

Theorem3ex 7967 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V

Theorem4re 7968 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ

Theorem4cn 7969 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 ∈ ℂ

Theorem5re 7970 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ

Theorem5cn 7971 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 ∈ ℂ

Theorem6re 7972 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ

Theorem6cn 7973 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6 ∈ ℂ

Theorem7re 7974 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ

Theorem7cn 7975 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7 ∈ ℂ

Theorem8re 7976 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ

Theorem8cn 7977 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8 ∈ ℂ

Theorem9re 7978 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ

Theorem9cn 7979 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9 ∈ ℂ

Theorem10re 7980 The number 10 is real. (Contributed by NM, 5-Feb-2007.)
10 ∈ ℝ

Theorem0le0 7981 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 7982 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 7983 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 7984 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem2ap0 7985 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0

Theorem3pos 7986 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 7987 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem4pos 7988 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 7989 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem5pos 7990 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 7991 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 7992 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 7993 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 7994 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9

Theorem10pos 7995 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 7996 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 7997 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 7998 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 7999 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremneg1ap0 8000 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0

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