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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulle0r 8901 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
(((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„) โˆง (๐ด โ‰ค 0 โˆง 0 โ‰ค ๐ต)) โ†’ (๐ด ยท ๐ต) โ‰ค 0)
 
4.3.10  Suprema
 
Theoremlbreu 8902* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
((๐‘† โŠ† โ„ โˆง โˆƒ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ) โ†’ โˆƒ!๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ)
 
Theoremlbcl 8903* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((๐‘† โŠ† โ„ โˆง โˆƒ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ) โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ) โˆˆ ๐‘†)
 
Theoremlble 8904* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((๐‘† โŠ† โ„ โˆง โˆƒ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ โˆง ๐ด โˆˆ ๐‘†) โ†’ (โ„ฉ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ) โ‰ค ๐ด)
 
Theoremlbinf 8905* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((๐‘† โŠ† โ„ โˆง โˆƒ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ) โ†’ inf(๐‘†, โ„, < ) = (โ„ฉ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ))
 
Theoremlbinfcl 8906* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((๐‘† โŠ† โ„ โˆง โˆƒ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ) โ†’ inf(๐‘†, โ„, < ) โˆˆ ๐‘†)
 
Theoremlbinfle 8907* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((๐‘† โŠ† โ„ โˆง โˆƒ๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘† ๐‘ฅ โ‰ค ๐‘ฆ โˆง ๐ด โˆˆ ๐‘†) โ†’ inf(๐‘†, โ„, < ) โ‰ค ๐ด)
 
Theoremsuprubex 8908* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
(๐œ‘ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ (โˆ€๐‘ฆ โˆˆ ๐ด ยฌ ๐‘ฅ < ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ โ„ (๐‘ฆ < ๐‘ฅ โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ฆ < ๐‘ง)))    &   (๐œ‘ โ†’ ๐ด โŠ† โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ ๐ด)    โ‡’   (๐œ‘ โ†’ ๐ต โ‰ค sup(๐ด, โ„, < ))
 
Theoremsuprlubex 8909* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
(๐œ‘ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ (โˆ€๐‘ฆ โˆˆ ๐ด ยฌ ๐‘ฅ < ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ โ„ (๐‘ฆ < ๐‘ฅ โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ฆ < ๐‘ง)))    &   (๐œ‘ โ†’ ๐ด โŠ† โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (๐ต < sup(๐ด, โ„, < ) โ†” โˆƒ๐‘ง โˆˆ ๐ด ๐ต < ๐‘ง))
 
Theoremsuprnubex 8910* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
(๐œ‘ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ (โˆ€๐‘ฆ โˆˆ ๐ด ยฌ ๐‘ฅ < ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ โ„ (๐‘ฆ < ๐‘ฅ โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ฆ < ๐‘ง)))    &   (๐œ‘ โ†’ ๐ด โŠ† โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (ยฌ ๐ต < sup(๐ด, โ„, < ) โ†” โˆ€๐‘ง โˆˆ ๐ด ยฌ ๐ต < ๐‘ง))
 
Theoremsuprleubex 8911* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
(๐œ‘ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ (โˆ€๐‘ฆ โˆˆ ๐ด ยฌ ๐‘ฅ < ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ โ„ (๐‘ฆ < ๐‘ฅ โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ฆ < ๐‘ง)))    &   (๐œ‘ โ†’ ๐ด โŠ† โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (sup(๐ด, โ„, < ) โ‰ค ๐ต โ†” โˆ€๐‘ง โˆˆ ๐ด ๐‘ง โ‰ค ๐ต))
 
Theoremnegiso 8912 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
๐น = (๐‘ฅ โˆˆ โ„ โ†ฆ -๐‘ฅ)    โ‡’   (๐น Isom < , โ—ก < (โ„, โ„) โˆง โ—ก๐น = ๐น)
 
Theoremdfinfre 8913* The infimum of a set of reals ๐ด. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
(๐ด โŠ† โ„ โ†’ inf(๐ด, โ„, < ) = โˆช {๐‘ฅ โˆˆ โ„ โˆฃ (โˆ€๐‘ฆ โˆˆ ๐ด ๐‘ฅ โ‰ค ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ โ„ (๐‘ฅ < ๐‘ฆ โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ง < ๐‘ฆ))})
 
Theoremsup3exmid 8914* If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
((๐‘ข โŠ† โ„ โˆง โˆƒ๐‘ค ๐‘ค โˆˆ ๐‘ข โˆง โˆƒ๐‘ฅ โˆˆ โ„ โˆ€๐‘ฆ โˆˆ ๐‘ข ๐‘ฆ โ‰ค ๐‘ฅ) โ†’ โˆƒ๐‘ฅ โˆˆ โ„ (โˆ€๐‘ฆ โˆˆ ๐‘ข ยฌ ๐‘ฅ < ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ โ„ (๐‘ฆ < ๐‘ฅ โ†’ โˆƒ๐‘ง โˆˆ ๐‘ข ๐‘ฆ < ๐‘ง)))    โ‡’   DECID ๐œ‘
 
4.3.11  Imaginary and complex number properties
 
Theoremcrap0 8915 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„) โ†’ ((๐ด # 0 โˆจ ๐ต # 0) โ†” (๐ด + (i ยท ๐ต)) # 0))
 
Theoremcreur 8916* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(๐ด โˆˆ โ„‚ โ†’ โˆƒ!๐‘ฅ โˆˆ โ„ โˆƒ๐‘ฆ โˆˆ โ„ ๐ด = (๐‘ฅ + (i ยท ๐‘ฆ)))
 
Theoremcreui 8917* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(๐ด โˆˆ โ„‚ โ†’ โˆƒ!๐‘ฆ โˆˆ โ„ โˆƒ๐‘ฅ โˆˆ โ„ ๐ด = (๐‘ฅ + (i ยท ๐‘ฆ)))
 
Theoremcju 8918* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
(๐ด โˆˆ โ„‚ โ†’ โˆƒ!๐‘ฅ โˆˆ โ„‚ ((๐ด + ๐‘ฅ) โˆˆ โ„ โˆง (i ยท (๐ด โˆ’ ๐‘ฅ)) โˆˆ โ„))
 
4.4  Integer sets
 
4.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 8919 Extend class notation to include the class of positive integers.
class โ„•
 
Definitiondf-inn 8920* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8921 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
โ„• = โˆฉ {๐‘ฅ โˆฃ (1 โˆˆ ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ ๐‘ฅ (๐‘ฆ + 1) โˆˆ ๐‘ฅ)}
 
Theoremdfnn2 8921* Definition of the set of positive integers. Another name for df-inn 8920. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
โ„• = โˆฉ {๐‘ฅ โˆฃ (1 โˆˆ ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ ๐‘ฅ (๐‘ฆ + 1) โˆˆ ๐‘ฅ)}
 
Theorempeano5nni 8922* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
((1 โˆˆ ๐ด โˆง โˆ€๐‘ฅ โˆˆ ๐ด (๐‘ฅ + 1) โˆˆ ๐ด) โ†’ โ„• โŠ† ๐ด)
 
Theoremnnssre 8923 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
โ„• โŠ† โ„
 
Theoremnnsscn 8924 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
โ„• โŠ† โ„‚
 
Theoremnnex 8925 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
โ„• โˆˆ V
 
Theoremnnre 8926 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
(๐ด โˆˆ โ„• โ†’ ๐ด โˆˆ โ„)
 
Theoremnncn 8927 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
(๐ด โˆˆ โ„• โ†’ ๐ด โˆˆ โ„‚)
 
Theoremnnrei 8928 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
๐ด โˆˆ โ„•    โ‡’   ๐ด โˆˆ โ„
 
Theoremnncni 8929 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
๐ด โˆˆ โ„•    โ‡’   ๐ด โˆˆ โ„‚
 
Theorem1nn 8930 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
1 โˆˆ โ„•
 
Theorempeano2nn 8931 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.)
(๐ด โˆˆ โ„• โ†’ (๐ด + 1) โˆˆ โ„•)
 
Theoremnnred 8932 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„)
 
Theoremnncnd 8933 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„‚)
 
Theorempeano2nnd 8934 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด + 1) โˆˆ โ„•)
 
4.4.2  Principle of mathematical induction
 
Theoremnnind 8935* 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 8939 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.)
(๐‘ฅ = 1 โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œƒ))    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œ))    &   ๐œ“    &   (๐‘ฆ โˆˆ โ„• โ†’ (๐œ’ โ†’ ๐œƒ))    โ‡’   (๐ด โˆˆ โ„• โ†’ ๐œ)
 
TheoremnnindALT 8936* 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 8935 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.)

(๐‘ฆ โˆˆ โ„• โ†’ (๐œ’ โ†’ ๐œƒ))    &   ๐œ“    &   (๐‘ฅ = 1 โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œƒ))    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œ))    โ‡’   (๐ด โˆˆ โ„• โ†’ ๐œ)
 
Theoremnn1m1nn 8937 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(๐ด โˆˆ โ„• โ†’ (๐ด = 1 โˆจ (๐ด โˆ’ 1) โˆˆ โ„•))
 
Theoremnn1suc 8938* 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.)
(๐‘ฅ = 1 โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œƒ))    &   ๐œ“    &   (๐‘ฆ โˆˆ โ„• โ†’ ๐œ’)    โ‡’   (๐ด โˆˆ โ„• โ†’ ๐œƒ)
 
Theoremnnaddcl 8939 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด + ๐ต) โˆˆ โ„•)
 
Theoremnnmulcl 8940 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด ยท ๐ต) โˆˆ โ„•)
 
Theoremnnmulcli 8941 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
๐ด โˆˆ โ„•    &   ๐ต โˆˆ โ„•    โ‡’   (๐ด ยท ๐ต) โˆˆ โ„•
 
Theoremnnge1 8942 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(๐ด โˆˆ โ„• โ†’ 1 โ‰ค ๐ด)
 
Theoremnnle1eq1 8943 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
(๐ด โˆˆ โ„• โ†’ (๐ด โ‰ค 1 โ†” ๐ด = 1))
 
Theoremnngt0 8944 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(๐ด โˆˆ โ„• โ†’ 0 < ๐ด)
 
Theoremnnnlt1 8945 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(๐ด โˆˆ โ„• โ†’ ยฌ ๐ด < 1)
 
Theorem0nnn 8946 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
ยฌ 0 โˆˆ โ„•
 
Theoremnnne0 8947 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
(๐ด โˆˆ โ„• โ†’ ๐ด โ‰  0)
 
Theoremnnap0 8948 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(๐ด โˆˆ โ„• โ†’ ๐ด # 0)
 
Theoremnngt0i 8949 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
๐ด โˆˆ โ„•    โ‡’   0 < ๐ด
 
Theoremnnap0i 8950 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
๐ด โˆˆ โ„•    โ‡’   ๐ด # 0
 
Theoremnnne0i 8951 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
๐ด โˆˆ โ„•    โ‡’   ๐ด โ‰  0
 
Theoremnn2ge 8952* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ โˆƒ๐‘ฅ โˆˆ โ„• (๐ด โ‰ค ๐‘ฅ โˆง ๐ต โ‰ค ๐‘ฅ))
 
Theoremnn1gt1 8953 A positive integer is either one or greater than one. This is for โ„•; 0elnn 4619 is a similar theorem for ฯ‰ (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
(๐ด โˆˆ โ„• โ†’ (๐ด = 1 โˆจ 1 < ๐ด))
 
Theoremnngt1ne1 8954 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
(๐ด โˆˆ โ„• โ†’ (1 < ๐ด โ†” ๐ด โ‰  1))
 
Theoremnndivre 8955 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((๐ด โˆˆ โ„ โˆง ๐‘ โˆˆ โ„•) โ†’ (๐ด / ๐‘) โˆˆ โ„)
 
Theoremnnrecre 8956 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(๐‘ โˆˆ โ„• โ†’ (1 / ๐‘) โˆˆ โ„)
 
Theoremnnrecgt0 8957 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(๐ด โˆˆ โ„• โ†’ 0 < (1 / ๐ด))
 
Theoremnnsub 8958 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด < ๐ต โ†” (๐ต โˆ’ ๐ด) โˆˆ โ„•))
 
Theoremnnsubi 8959 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
๐ด โˆˆ โ„•    &   ๐ต โˆˆ โ„•    โ‡’   (๐ด < ๐ต โ†” (๐ต โˆ’ ๐ด) โˆˆ โ„•)
 
Theoremnndiv 8960* Two ways to express "๐ด divides ๐ต " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (โˆƒ๐‘ฅ โˆˆ โ„• (๐ด ยท ๐‘ฅ) = ๐ต โ†” (๐ต / ๐ด) โˆˆ โ„•))
 
Theoremnndivtr 8961 Transitive property of divisibility: if ๐ด divides ๐ต and ๐ต divides ๐ถ, then ๐ด divides ๐ถ. Typically, ๐ถ would be an integer, although the theorem holds for complex ๐ถ. (Contributed by NM, 3-May-2005.)
(((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„• โˆง ๐ถ โˆˆ โ„‚) โˆง ((๐ต / ๐ด) โˆˆ โ„• โˆง (๐ถ / ๐ต) โˆˆ โ„•)) โ†’ (๐ถ / ๐ด) โˆˆ โ„•)
 
Theoremnnge1d 8962 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ 1 โ‰ค ๐ด)
 
Theoremnngt0d 8963 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ 0 < ๐ด)
 
Theoremnnne0d 8964 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด โ‰  0)
 
Theoremnnap0d 8965 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด # 0)
 
Theoremnnrecred 8966 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (1 / ๐ด) โˆˆ โ„)
 
Theoremnnaddcld 8967 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด + ๐ต) โˆˆ โ„•)
 
Theoremnnmulcld 8968 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด ยท ๐ต) โˆˆ โ„•)
 
Theoremnndivred 8969 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด / ๐ต) โˆˆ โ„)
 
4.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7818 through df-9 8985), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 7818 and df-1 7819).

Integers can also be exhibited as sums of powers of 10 (e.g., the number 103 can be expressed as ((10โ†‘2) + 3)) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as (7โ†‘7) โˆ’ 2.

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 8970 Extend class notation to include the number 2.
class 2
 
Syntaxc3 8971 Extend class notation to include the number 3.
class 3
 
Syntaxc4 8972 Extend class notation to include the number 4.
class 4
 
Syntaxc5 8973 Extend class notation to include the number 5.
class 5
 
Syntaxc6 8974 Extend class notation to include the number 6.
class 6
 
Syntaxc7 8975 Extend class notation to include the number 7.
class 7
 
Syntaxc8 8976 Extend class notation to include the number 8.
class 8
 
Syntaxc9 8977 Extend class notation to include the number 9.
class 9
 
Definitiondf-2 8978 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)
 
Definitiondf-3 8979 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)
 
Definitiondf-4 8980 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)
 
Definitiondf-5 8981 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)
 
Definitiondf-6 8982 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)
 
Definitiondf-7 8983 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)
 
Definitiondf-8 8984 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)
 
Definitiondf-9 8985 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)
 
Theorem0ne1 8986 0 โ‰  1 (common case). See aso 1ap0 8547. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 โ‰  1
 
Theorem1ne0 8987 1 โ‰  0. See aso 1ap0 8547. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 โ‰  0
 
Theorem1m1e0 8988 (1 โˆ’ 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 โˆ’ 1) = 0
 
Theorem2re 8989 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 โˆˆ โ„
 
Theorem2cn 8990 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 โˆˆ โ„‚
 
Theorem2ex 8991 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 โˆˆ V
 
Theorem2cnd 8992 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(๐œ‘ โ†’ 2 โˆˆ โ„‚)
 
Theorem3re 8993 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 โˆˆ โ„
 
Theorem3cn 8994 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 โˆˆ โ„‚
 
Theorem3ex 8995 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 โˆˆ V
 
Theorem4re 8996 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 โˆˆ โ„
 
Theorem4cn 8997 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 โˆˆ โ„‚
 
Theorem5re 8998 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 โˆˆ โ„
 
Theorem5cn 8999 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 โˆˆ โ„‚
 
Theorem6re 9000 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 โˆˆ โ„
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