HomeHome Intuitionistic Logic Explorer
Theorem List (p. 80 of 94)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremavglt2 7901 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (A < B ↔ ((A + B) / 2) < B))
 
Theoremavgle1 7902 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((A B ℝ) → (ABA ≤ ((A + B) / 2)))
 
Theoremavgle2 7903 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 7904 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (2 · A) = (A + A))
 
Theoremtimes2d 7905 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A · 2) = (A + A))
 
Theoremhalfcld 7906 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A / 2) ℂ)
 
Theorem2halvesd 7907 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → ((A / 2) + (A / 2)) = A)
 
Theoremrehalfcld 7908 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (A / 2) ℝ)
 
Theoremlt2halvesd 7909 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 7910 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
A        (A / 2)
 
Theoremadd1p1 7911 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
 
Theoremsub1m1 7912 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
 
Theoremcnm2m1cnm3 7913 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 7914* 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 7915* 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 7916* 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 7917 Extend class notation to include the class of nonnegative integers.
class 0
 
Definitiondf-n0 7918 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})
 
Theoremelnn0 7919 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(A 0 ↔ (A A = 0))
 
Theoremnnssnn0 7920 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0
 
Theoremnn0ssre 7921 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ
 
Theoremnn0sscn 7922 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ
 
Theoremnn0ex 7923 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 V
 
Theoremnnnn0 7924 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(A ℕ → A 0)
 
Theoremnnnn0i 7925 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁        𝑁 0
 
Theoremnn0re 7926 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(A 0A ℝ)
 
Theoremnn0cn 7927 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(A 0A ℂ)
 
Theoremnn0rei 7928 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
A 0       A
 
Theoremnn0cni 7929 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
A 0       A
 
Theoremdfn2 7930 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 7931 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 7932 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 0
 
Theorem1nn0 7933 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
1 0
 
Theorem2nn0 7934 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
2 0
 
Theorem3nn0 7935 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
3 0
 
Theorem4nn0 7936 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
4 0
 
Theorem5nn0 7937 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
5 0
 
Theorem6nn0 7938 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
6 0
 
Theorem7nn0 7939 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
7 0
 
Theorem8nn0 7940 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
8 0
 
Theorem9nn0 7941 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
9 0
 
Theorem10nn0 7942 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
10 0
 
Theoremnn0ge0 7943 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 7944 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 7945 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 0       0 ≤ 𝑁
 
Theoremnn0le0eq0 7946 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 7947 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 7948 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 7949 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 7950 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 7951 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 7952 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 7953 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 7954 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 7955 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 0    &   𝑁 0       (𝑀 + 𝑁) 0
 
Theoremnn0mulcli 7956 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 0    &   𝑁 0       (𝑀 · 𝑁) 0
 
Theoremnn0p1nn 7957 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 7958 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 0 → (𝑁 + 1) 0)
 
Theoremnnm1nn0 7959 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 7960 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 7961 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
(𝑁 ℕ ↔ (𝑁 (𝑁 − 1) 0))
 
Theoremelnnnn0b 7962 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
(𝑁 ℕ ↔ (𝑁 0 0 < 𝑁))
 
Theoremelnnnn0c 7963 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
(𝑁 ℕ ↔ (𝑁 0 1 ≤ 𝑁))
 
Theoremnn0addge1 7964 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 7965 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 7966 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 7967 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 7968 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 0       𝑁 ≤ (2 · 𝑁)
 
Theoremnn0lele2xi 7969 '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 7970 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 7971 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℕ)       (φA 0)
 
Theoremnn0red 7972 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)       (φA ℝ)
 
Theoremnn0cnd 7973 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)       (φA ℂ)
 
Theoremnn0ge0d 7974 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)       (φ → 0 ≤ A)
 
Theoremnn0addcld 7975 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)    &   (φB 0)       (φ → (A + B) 0)
 
Theoremnn0mulcld 7976 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(φA 0)    &   (φB 0)       (φ → (A · B) 0)
 
Theoremnn0readdcl 7977 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 7978 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 7979 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 7980 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)
 
3.4.8  Integers (as a subset of complex numbers)
 
Syntaxcz 7981 Extend class notation to include the class of integers.
class
 
Definitiondf-z 7982 Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)
ℤ = {𝑛 ℝ ∣ (𝑛 = 0 𝑛 -𝑛 ℕ)}
 
Theoremelz 7983 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ℤ ↔ (𝑁 (𝑁 = 0 𝑁 -𝑁 ℕ)))
 
Theoremnnnegz 7984 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
(𝑁 ℕ → -𝑁 ℤ)
 
Theoremzre 7985 An integer is a real. (Contributed by NM, 8-Jan-2002.)
(𝑁 ℤ → 𝑁 ℝ)
 
Theoremzcn 7986 An integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝑁 ℤ → 𝑁 ℂ)
 
Theoremzrei 7987 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
A        A
 
Theoremzssre 7988 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℝ
 
Theoremzsscn 7989 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℂ
 
Theoremzex 7990 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
V
 
Theoremelnnz 7991 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ℕ ↔ (𝑁 0 < 𝑁))
 
Theorem0z 7992 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
0
 
Theorem0zd 7993 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(φ → 0 ℤ)
 
Theoremelnn0z 7994 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
(𝑁 0 ↔ (𝑁 0 ≤ 𝑁))
 
Theoremelznn0nn 7995 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
(𝑁 ℤ ↔ (𝑁 0 (𝑁 -𝑁 ℕ)))
 
Theoremelznn0 7996 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ℤ ↔ (𝑁 (𝑁 0 -𝑁 0)))
 
Theoremelznn 7997 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
(𝑁 ℤ ↔ (𝑁 (𝑁 -𝑁 0)))
 
Theoremnnssz 7998 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
ℕ ⊆ ℤ
 
Theoremnn0ssz 7999 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
0 ⊆ ℤ
 
Theoremnnz 8000 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ℕ → 𝑁 ℤ)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9381
  Copyright terms: Public domain < Previous  Next >