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Statement List for Metamath Proof Explorer - 5901-6000 - Page 60 of 107
TypeLabelDescription
Statement
 
Theorempeano2nn 5901 Peano postulate: a successor of a natural number is a natural number.
|- (A e. NN -> (A + 1) e. NN)
 
Theoremdfnn2 5902 Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22.
|- NN = |^|{x | (x (_ RR /\ 1 e. x /\ A.y e. x (y + 1) e. x)}
 
Principle of mathematical induction
 
Theoremnnind 5903 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 hypothesis. See nnaddclt 5906 for an example of its use. See nn0ind 6178 for induction on nonnegative integers and uzind 6171, uzind4 6400 for induction on an arbitrary set of upper integers. See indstr 6411 for strong induction.
|- (x = 1 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. NN -> (ch -> th))   =>   |- (A e. NN -> ta)
 
TheoremnnindALT 5904 Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis. (This ALT version of nnind 5903 is easier to use with the Proof Assistant since 'assign last' will be applied to the substitution instances first. We may switch to it as the official version.)
|- (y e. NN -> (ch -> th))   &   |- ps   &   |- (x = 1 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   =>   |- (A e. NN -> ta)
 
Natural numbers (cont.)
 
Theoremnn1suc 5905 If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor.
|- (x = 1 -> (ph <-> ps))   &   |- (x = (y + 1) -> (ph <-> ch))   &   |- (x = A -> (ph <-> th))   &   |- ps   &   |- (y e. NN -> ch)   =>   |- (A e. NN -> th)
 
Theoremnnaddclt 5906 Closure of addition of natural numbers, proved by induction on the second addend.
|- ((A e. NN /\ B e. NN) -> (A + B) e. NN)
 
Theoremnnmulclt 5907 Closure of multiplication of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A x. B) e. NN)
 
Theoremnn2get 5908 There exists a natural number greater than or equal to any two others.
|- ((A e. NN /\ B e. NN) -> E.x e. NN (A <_ x /\ B <_ x))
 
Theoremnnge1t 5909 A natural number is one or greater.
|- (A e. NN -> 1 <_ A)
 
Theoremnngt1ne1t 5910 A natural number is greater than one iff it is not equal to one.
|- (A e. NN -> (1 < A <-> A =/= 1))
 
Theoremnnle1eq1t 5911 A natural number is less than or equal to one iff it is equal to one.
|- (A e. NN -> (A <_ 1 <-> A = 1))
 
Theoremnngt0t 5912 A natural number is positive.
|- (A e. NN -> 0 < A)
 
Theoremlt1nnn 5913 A number less than one is not a natural number.
|- ((A e. RR /\ A < 1) -> -. A e. NN)
 
Theorem0nnn 5914 Zero is not a natural number.
|- -. 0 e. NN
 
Theoremnnne0t 5915 A natural number is non-zero.
|- (A e. NN -> A =/= 0)
 
Theoremnngt0 5916 A natural number is positive (inference version).
|- A e. NN   =>   |- 0 < A
 
Theoremnnne0 5917 A natural number is non-zero (inference version).
|- A e. NN   =>   |- A =/= 0
 
Theoremnnrecgt0t 5918 The reciprocal of a natural number is positive.
|- (A e. NN -> 0 < (1 / A))
 
Theoremnnleltp1t 5919 Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A <_ B <-> A < (B + 1)))
 
Theoremnnltp1let 5920 Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A < B <-> (A + 1) <_ B))
 
Theoremnnsub 5921 Subtraction of natural numbers.
|- A e. NN   &   |- B e. NN   =>   |- (A < B <-> (B - A) e. NN)
 
Theoremnnsubt 5922 Subtraction of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A < B <-> (B - A) e. NN))
 
Theoremnnaddm1clt 5923 Closure of addition of natural numbers minus one.
|- ((A e. NN /\ B e. NN) -> ((A + B) - 1) e. NN)
 
Theoremnndivt 5924 Two ways to express "A divides B" for natural numbers.
|- ((A e. NN /\ B e. NN) -> (E.x e. NN (A x. x) = B <-> (B / A) e. NN))
 
Theoremnndivtrt 5925 Transitive property of divisibility: if A divides B and B divides C, then A divides C.
|- (((A e. NN /\ B e. NN /\ C e. CC) /\ ((B / A) e. NN /\ (C / B) e. NN)) -> (C / A) e. NN)
 
Decimal representation of numbers
 
Syntaxc2 5926 Extend class notation to include the number 2.
class 2
 
Syntaxc3 5927 Extend class notation to include the number 3.
class 3
 
Syntaxc4 5928 Extend class notation to include the number 4.
class 4
 
Syntaxc5 5929 Extend class notation to include the number 5.
class 5
 
Syntaxc6 5930 Extend class notation to include the number 6.
class 6
 
Syntaxc7 5931 Extend class notation to include the number 7.
class 7
 
Syntaxc8 5932 Extend class notation to include the number 8.
class 8
 
Syntaxc9 5933 Extend class notation to include the number 9.
class 9
 
Syntaxc10 5934 Extend class notation to include the number 10.
class 10
 
Definitiondf-2 5935 Define the number 2.

Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 5231 and df-1 5232).

Note: Only the digits 0 through 9 (df-0 5231 through df-9 5942) and the number 10 (df-10 5943) are explicitly defined. Integers can be exhibited as sums of powers of 10 or as some other expression built from operations on the numbers 0 through 10. For example, the prime number 823541 can be expressed as (7^7) - 2. Decimals can be expressed as ratios of integers, as in cos2bnd 7435. (Fortunately, 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.)

A decimal representation of numbers may be added at some point in the future if it is deemed useful. Ideas for a clean, eliminable definition are welcome. (An awkward earlier definition was deleted from the database on 18-Sep-1999.)

|- 2 = (1 + 1)
 
Definitiondf-3 5936 Define the number 3.
|- 3 = (2 + 1)
 
Definitiondf-4 5937 Define the number 4.
|- 4 = (3 + 1)
 
Definitiondf-5 5938 Define the number 5.
|- 5 = (4 + 1)
 
Definitiondf-6 5939 Define the number 6.
|- 6 = (5 + 1)
 
Definitiondf-7 5940 Define the number 7.
|- 7 = (6 + 1)
 
Definitiondf-8 5941 Define the number 8.
|- 8 = (7 + 1)
 
Definitiondf-9 5942 Define the number 9.
|- 9 = (8 + 1)
 
Definitiondf-10 5943 Define the number 10. See remarks under df-2 5935.
|- 10 = (9 + 1)
 
Theorem2re 5944 The number 2 is real.
|- 2 e. RR
 
Theorem2cn 5945 The number 2 is a complex number.
|- 2 e. CC
 
Theorem3re 5946 The number 3 is real.
|- 3 e. RR
 
Theorem4re 5947 The number 4 is real.
|- 4 e. RR
 
Theorem5re 5948 The number 5 is real.
|- 5 e. RR
 
Theorem6re 5949 The number 6 is real.
|- 6 e. RR
 
Theorem7re 5950 The number 7 is real.
|- 7 e. RR
 
Theorem8re 5951 The number 8 is real.
|- 8 e. RR
 
Theorem9re 5952 The number 9 is real.
|- 9 e. RR
 
Theorem10re 5953 The number 10 is real.
|- 10 e. RR
 
Theorem2pos 5954 The number 2 is positive.
|- 0 < 2
 
Theorem2ne0 5955 The number 2 is nonzero.
|- 2 =/= 0
 
Theorem3pos 5956 The number 3 is positive.
|- 0 < 3
 
Theorem4pos 5957 The number 4 is positive.
|- 0 < 4
 
Theorem5pos 5958 The number 5 is positive.
|- 0 < 5
 
Theorem6pos 5959 The number 6 is positive.
|- 0 < 6
 
Theorem7pos 5960 The number 7 is positive.
|- 0 < 7
 
Theorem8pos 5961 The number 8 is positive.
|- 0 < 8
 
Theorem9pos 5962 The number 9 is positive.
|- 0 < 9
 
Theorem10pos 5963 The number 10 is positive.
|- 0 < 10
 
Theorem2nn 5964 2 is a natural number.
|- 2 e. NN
 
Theorem3nn 5965 3 is a natural number.
|- 3 e. NN
 
Some properties of specific numbers
 
Theorem2p2e4 5966 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpegif/mmset.html#trivia.
|- (2 + 2) = 4
 
Theorem4nn 5967 4 is a natural number.
|- 4 e. NN
 
Theorem2times 5968 Two times a number.
|- A e. CC   =>   |- (2 x. A) = (A + A)
 
Theorem2timest 5969 Two times a number.
|- (A e. CC -> (2 x. A) = (A + A))
 
Theoremtimes2t 5970 A number times 2.
|- (A e. CC -> (A x. 2) = (A + A))
 
Theoremtimes2 5971 A number times 2.
|- A e. CC   =>   |- (A x. 2) = (A + A)
 
Theorem3p2e5 5972 3 + 2 = 5.
|- (3 + 2) = 5
 
Theorem3p3e6 5973 3 + 3 = 6.
|- (3 + 3) = 6
 
Theorem4p2e6 5974 4 + 2 = 6.
|- (4 + 2) = 6
 
Theorem4p3e7 5975 4 + 3 = 7.
|- (4 + 3) = 7
 
Theorem4p4e8 5976 4 + 4 = 8.
|- (4 + 4) = 8
 
Theorem5p2e7 5977 5 + 2 = 7.
|- (5 + 2) = 7
 
Theorem5p3e8 5978 5 + 3 = 8.
|- (5 + 3) = 8
 
Theorem5p4e9 5979 5 + 4 = 9.
|- (5 + 4) = 9
 
Theorem5p5e10 5980 5 + 5 = 10.
|- (5 + 5) = 10
 
Theorem6p2e8 5981 6 + 2 = 8.
|- (6 + 2) = 8
 
Theorem6p3e9 5982 6 + 3 = 9.
|- (6 + 3) = 9
 
Theorem6p4e10 5983 6 + 4 = 10.
|- (6 + 4) = 10
 
Theorem7p2e9 5984 7 + 2 = 9.
|- (7 + 2) = 9
 
Theorem7p3e10 5985 7 + 3 = 10.
|- (7 + 3) = 10
 
Theorem8p2e10 5986 8 + 2 = 10.
|- (8 + 2) = 10
 
Theorem2t2e4 5987 2 times 2 equals 4.
|- (2 x. 2) = 4
 
Theorem3t2e6 5988 3 times 2 equals 6.
|- (3 x. 2) = 6
 
Theorem3t3e9 5989 3 times 3 equals 9.
|- (3 x. 3) = 9
 
Theorem4t2e8 5990 4 times 2 equals 8.
|- (4 x. 2) = 8
 
Theorem5t2e10 5991 5 times 2 equals 10.
|- (5 x. 2) = 10
 
Theorem4d2e2 5992 One half of four is two.
|- (4 / 2) = 2
 
Theorem1lt2 5993 1 is less than 2.
|- 1 < 2
 
Theoremhalfgt0 5994 One-half is greater than zero.
|- 0 < (1 / 2)
 
Theoremhalflt1 5995 One-half is less than one.
|- (1 / 2) < 1
 
Theorem8th4div3 5996 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ((1 / 8) x. (4 / 3)) = (1 / 6)
 
Theoremhalfpm6th 5997 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))
 
Theoremhalfclt 5998 Closure of half of a number (frequently used special case).
|- (A e. CC -> (A / 2) e. CC)
 
Theoremrehalfclt 5999 Real closure of half.
|- (A e. RR -> (A / 2) e. RR)
 
Theoremhalf0t 6000 Half of a number is zero iff the number is zero.
|- (A e. CC -> ((A / 2) = 0 <-> A = 0))
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