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

Theorem8th4div3 7901 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)

Theoremhalfpm6th 7902 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
(((1 / 2) − (1 / 6)) = (1 / 3) ((1 / 2) + (1 / 6)) = (2 / 3))

Theoremit0e0 7903 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0

Theorem2mulicn 7904 (2 · i) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i)

Theoremiap0 7905 The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
i # 0

Theorem2muliap0 7906 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
(2 · i) # 0

Theorem2muline0 7907 (2 · i) ≠ 0. See also 2muliap0 7906. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0

3.4.5  Simple number properties

Theoremhalfcl 7908 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
(A ℂ → (A / 2) ℂ)

Theoremrehalfcl 7909 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(A ℝ → (A / 2) ℝ)

Theoremhalf0 7910 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
(A ℂ → ((A / 2) = 0 ↔ A = 0))

Theorem2halves 7911 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(A ℂ → ((A / 2) + (A / 2)) = A)

Theoremhalfpos2 7912 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(A ℝ → (0 < A ↔ 0 < (A / 2)))

Theoremhalfpos 7913 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(A ℝ → (0 < A ↔ (A / 2) < A))

Theoremhalfnneg2 7914 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(A ℝ → (0 ≤ A ↔ 0 ≤ (A / 2)))

Theoremhalfaddsubcl 7915 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((A B ℂ) → (((A + B) / 2) ((AB) / 2) ℂ))

Theoremhalfaddsub 7916 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((A B ℂ) → ((((A + B) / 2) + ((AB) / 2)) = A (((A + B) / 2) − ((AB) / 2)) = B))

Theoremlt2halves 7917 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
((A B 𝐶 ℝ) → ((A < (𝐶 / 2) B < (𝐶 / 2)) → (A + B) < 𝐶))

Theoremaddltmul 7918 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
(((A B ℝ) (2 < A 2 < B)) → (A + B) < (A · B))

Theoremnominpos 7919* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ x ℝ (0 < x ¬ y ℝ (0 < y y < x))

Theoremavglt1 7920 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (A < BA < ((A + B) / 2)))

Theoremavglt2 7921 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (A < B ↔ ((A + B) / 2) < B))

Theoremavgle1 7922 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (ABA ≤ ((A + B) / 2)))

Theoremavgle2 7923 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (AB ↔ ((A + B) / 2) ≤ B))

Theorem2timesd 7924 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (2 · A) = (A + A))

Theoremtimes2d 7925 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A · 2) = (A + A))

Theoremhalfcld 7926 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A / 2) ℂ)

Theorem2halvesd 7927 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → ((A / 2) + (A / 2)) = A)

Theoremrehalfcld 7928 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (A / 2) ℝ)

Theoremlt2halvesd 7929 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < (𝐶 / 2))    &   (φB < (𝐶 / 2))       (φ → (A + B) < 𝐶)

Theoremrehalfcli 7930 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
A        (A / 2)

Theoremadd1p1 7931 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))

Theoremsub1m1 7932 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))

Theoremcnm2m1cnm3 7933 Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(A ℂ → ((A − 2) − 1) = (A − 3))

3.4.6  The Archimedean property

Theoremarch 7934* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
(A ℝ → 𝑛 A < 𝑛)

Theoremnnrecl 7935* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
((A 0 < A) → 𝑛 ℕ (1 / 𝑛) < A)

Theorembndndx 7936* A bounded real sequence A(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
(x 𝑘 ℕ (A Ax) → 𝑘 A𝑘)

3.4.7  Nonnegative integers (as a subset of complex numbers)

Syntaxcn0 7937 Extend class notation to include the class of nonnegative integers.
class 0

Definitiondf-n0 7938 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})

Theoremelnn0 7939 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(A 0 ↔ (A A = 0))

Theoremnnssnn0 7940 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0

Theoremnn0ssre 7941 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ

Theoremnn0sscn 7942 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ

Theoremnn0ex 7943 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 V

Theoremnnnn0 7944 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(A ℕ → A 0)

Theoremnnnn0i 7945 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁        𝑁 0

Theoremnn0re 7946 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(A 0A ℝ)

Theoremnn0cn 7947 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(A 0A ℂ)

Theoremnn0rei 7948 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
A 0       A

Theoremnn0cni 7949 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
A 0       A

Theoremdfn2 7950 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
ℕ = (ℕ0 ∖ {0})

Theoremelnnne0 7951 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝑁 ℕ ↔ (𝑁 0 𝑁 ≠ 0))

Theorem0nn0 7952 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 0

Theorem1nn0 7953 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
1 0

Theorem2nn0 7954 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
2 0

Theorem3nn0 7955 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
3 0

Theorem4nn0 7956 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
4 0

Theorem5nn0 7957 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
5 0

Theorem6nn0 7958 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
6 0

Theorem7nn0 7959 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
7 0

Theorem8nn0 7960 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
8 0

Theorem9nn0 7961 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
9 0

Theorem10nn0 7962 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
10 0

Theoremnn0ge0 7963 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 0 → 0 ≤ 𝑁)

Theoremnn0nlt0 7964 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(A 0 → ¬ A < 0)

Theoremnn0ge0i 7965 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 0       0 ≤ 𝑁

Theoremnn0le0eq0 7966 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
(𝑁 0 → (𝑁 ≤ 0 ↔ 𝑁 = 0))

Theoremnn0p1gt0 7967 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑁 0 → 0 < (𝑁 + 1))

Theoremnnnn0addcl 7968 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 𝑁 0) → (𝑀 + 𝑁) ℕ)

Theoremnn0nnaddcl 7969 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
((𝑀 0 𝑁 ℕ) → (𝑀 + 𝑁) ℕ)

Theorem0mnnnnn0 7970 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
(𝑁 ℕ → (0 − 𝑁) ∉ ℕ0)

Theoremun0addcl 7971 If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(φ𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((φ (𝑀 𝑆 𝑁 𝑆)) → (𝑀 + 𝑁) 𝑆)       ((φ (𝑀 𝑇 𝑁 𝑇)) → (𝑀 + 𝑁) 𝑇)

Theoremun0mulcl 7972 If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(φ𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((φ (𝑀 𝑆 𝑁 𝑆)) → (𝑀 · 𝑁) 𝑆)       ((φ (𝑀 𝑇 𝑁 𝑇)) → (𝑀 · 𝑁) 𝑇)

Theoremnn0addcl 7973 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 0 𝑁 0) → (𝑀 + 𝑁) 0)

Theoremnn0mulcl 7974 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 0 𝑁 0) → (𝑀 · 𝑁) 0)

Theoremnn0addcli 7975 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 0    &   𝑁 0       (𝑀 + 𝑁) 0

Theoremnn0mulcli 7976 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 0    &   𝑁 0       (𝑀 · 𝑁) 0

Theoremnn0p1nn 7977 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 0 → (𝑁 + 1) ℕ)

Theorempeano2nn0 7978 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 0 → (𝑁 + 1) 0)

Theoremnnm1nn0 7979 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ℕ → (𝑁 − 1) 0)

Theoremelnn0nn 7980 The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
(𝑁 0 ↔ (𝑁 (𝑁 + 1) ℕ))

Theoremelnnnn0 7981 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
(𝑁 ℕ ↔ (𝑁 (𝑁 − 1) 0))

Theoremelnnnn0b 7982 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
(𝑁 ℕ ↔ (𝑁 0 0 < 𝑁))

Theoremelnnnn0c 7983 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
(𝑁 ℕ ↔ (𝑁 0 1 ≤ 𝑁))

Theoremnn0addge1 7984 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
((A 𝑁 0) → A ≤ (A + 𝑁))

Theoremnn0addge2 7985 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
((A 𝑁 0) → A ≤ (𝑁 + A))

Theoremnn0addge1i 7986 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
A     &   𝑁 0       A ≤ (A + 𝑁)

Theoremnn0addge2i 7987 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
A     &   𝑁 0       A ≤ (𝑁 + A)

Theoremnn0le2xi 7988 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 0       𝑁 ≤ (2 · 𝑁)

Theoremnn0lele2xi 7989 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 0    &   𝑁 0       (𝑁𝑀𝑁 ≤ (2 · 𝑀))

Theoremnn0supp 7990 Two ways to write the support of a function on 0. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐹:𝐼⟶ℕ0 → (𝐹 “ (V ∖ {0})) = (𝐹 “ ℕ))

Theoremnnnn0d 7991 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA 0)

Theoremnn0red 7992 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)       (φA ℝ)

Theoremnn0cnd 7993 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)       (φA ℂ)

Theoremnn0ge0d 7994 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)       (φ → 0 ≤ A)

Theoremnn0addcld 7995 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)    &   (φB 0)       (φ → (A + B) 0)

Theoremnn0mulcld 7996 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)    &   (φB 0)       (φ → (A · B) 0)

Theoremnn0readdcl 7997 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((A 0 B 0) → (A + B) ℝ)

Theoremnn0ge2m1nn 7998 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
((𝑁 0 2 ≤ 𝑁) → (𝑁 − 1) ℕ)

Theoremnn0ge2m1nn0 7999 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
((𝑁 0 2 ≤ 𝑁) → (𝑁 − 1) 0)

Theoremnn0nndivcl 8000 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 0 𝐿 ℕ) → (𝐾 / 𝐿) ℝ)

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