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Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrecgt0d 9901 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivgt0d 9902 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulgt1d 9903 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge11d 9904 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge12d 9905 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul1ad 9906 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2ad 9907 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul12ad 9908 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12ad 9909 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12bd 9910 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

5.3.8  Completeness Axiom and Suprema

Theoremfimaxre 9911* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfimaxre2 9912* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)

Theoremfimaxre3 9913* A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremlbreu 9914* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)

Theoremlbcl 9915* 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.)

Theoremlble 9916* 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.)

Theoremlbinfm 9917* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremlbinfmcl 9918* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.)

Theoremlbinfmle 9919* If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.)

Theoremsup2 9920* A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)

Theoremsup3 9921* A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.)

Theoreminfm3lem 9922* Lemma for infm3 9923. (Contributed by NM, 14-Jun-2005.)

Theoreminfm3 9923* The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has an infimum. (This theorem is the dual of sup3 9921.) (Contributed by NM, 14-Jun-2005.)

Theoremsuprcl 9924* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Oct-2004.)

Theoremsuprub 9925* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.)

Theoremsuprlub 9926* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprnub 9927* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprleub 9928* The supremum of a non-empty 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.)

Theoremsupmul1 9929* The supremum function distributes over multiplication, in the sense that , where is shorthand for and is defined as below. This is the simple version, with only one set argument; see supmul 9932 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmullem1 9930* Lemma for supmul 9932. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmullem2 9931* Lemma for supmul 9932. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmul 9932* The supremum function distributes over multiplication, in the sense that , where is shorthand for and is defined as below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 8817). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsup3ii 9933* A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.)

Theoremsuprclii 9934* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Sep-1999.)

Theoremsuprubii 9935* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999.)

Theoremsuprlubii 9936* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprnubii 9937* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprleubii 9938* The supremum of a non-empty 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.)

Theoremriotaneg 9939* The negative of the unique real such that . (Contributed by NM, 13-Jun-2005.)

Theoremnegiso 9940 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremdfinfmr 9941* The infimum (expressed as supremum with converse 'less-than') of a set of reals . (Contributed by NM, 9-Oct-2005.)

Theoreminfmsup 9942* The infimum (expressed as supremum with converse 'less-than') of a set of reals is the negative of the supremum of the negatives of its elements. The antecedent ensures that is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoreminfmrcl 9943* Closure of infimum of a non-empty bounded set of reals. (Contributed by NM, 8-Oct-2005.)

Theoreminfmrgelb 9944* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoreminfmrlb 9945* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.)

5.3.9  Imaginary and complex number properties

Theoreminelr 9946 The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.)

Theoremrimul 9947 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremcru 9948 The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcrne0 9949 The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcreur 9950* 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.)

Theoremcreui 9951* 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.)

Theoremcju 9952* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

5.3.10  Function operation analogue theorems

Theoremofsubeq0 9953 Function analog of subeq0 9283. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofnegsub 9954 Function analog of negsub 9305. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofsubge0 9955 Function analog of subge0 9497. (Contributed by Mario Carneiro, 24-Jul-2014.)

5.4  Integer sets

5.4.1  Natural numbers (as a subset of complex numbers)

Syntaxcn 9956 Extend class notation to include the class of positive integers.

Definitiondf-nn 9957 The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set , df-om 4805, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 9974 for the principle of mathematical induction. See dfnn2 9969 for a slight variant. See df-n0 10178 for the set of nonnegative integers starting at zero. See dfn2 10190 for defined in terms of .

This is a technical definition that helps us avoid the Axiom of Infinity in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing as well as the successor of every member") see dfnn3 9970. (Contributed by NM, 10-Jan-1997.)

TheoremnnexALT 9958 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempeano5nni 9959* 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.)

Theoremnnssre 9960 The natural numbers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremnnsscn 9961 The natural numbers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremnnex 9962 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnre 9963 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncn 9964 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)

Theoremnnrei 9965 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncni 9966 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)

Theorem1nn 9967 Peano postulate: 1 is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theorempeano2nn 9968 Peano postulate: a successor of a natural number is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremdfnn2 9969* Alternate definition of the set of natural numbers. This was our original definition, before the current df-nn 9957 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Theoremdfnn3 9970* Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)

Theoremnnred 9971 A natural number is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncnd 9972 A natural number is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempeano2nnd 9973 Peano postulate: a successor of a natural number is a natural number. (Contributed by Mario Carneiro, 27-May-2016.)

5.4.2  Principle of mathematical induction

Theoremnnind 9974* 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 nnaddcl 9978 for an example of its use. See nn0ind 10322 for induction on nonnegative integers and uzind 10317, uzind4 10490 for induction on an arbitrary set of upper integers. See indstr 10501 for strong induction. See also nnindALT 9975. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

TheoremnnindALT 9975* 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 9974 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.)

Theoremnn1m1nn 9976 Every natural number is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn1suc 9977* 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. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnaddcl 9978 Closure of addition of natural numbers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcl 9979 Closure of multiplication of natural numbers. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcli 9980 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn2ge 9981* There exists a natural number greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)

Theoremnnge1 9982 A natural number is one or greater. (Contributed by NM, 25-Aug-1999.)

Theoremnngt1ne1 9983 A natural number is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)

Theoremnnle1eq1 9984 A natural number is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)

Theoremnngt0 9985 A natural number is positive. (Contributed by NM, 26-Sep-1999.)

Theoremnnnlt1 9986 A natural number is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0nnn 9987 Zero is not a natural number. (Contributed by NM, 25-Aug-1999.)

Theoremnnne0 9988 A natural number is nonzero. (Contributed by NM, 27-Sep-1999.)

Theoremnngt0i 9989 A natural number is positive (inference version). (Contributed by NM, 17-Sep-1999.)

Theoremnnne0i 9990 A natural number is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)

Theoremnndivre 9991 The quotient of a real and a natural number is real. (Contributed by NM, 28-Nov-2008.)

Theoremnnrecre 9992 The reciprocal of a natural number is real. (Contributed by NM, 8-Feb-2008.)

Theoremnnrecgt0 9993 The reciprocal of a natural number is positive. (Contributed by NM, 25-Aug-1999.)

Theoremnnsub 9994 Subtraction of natural numbers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnsubi 9995 Subtraction of natural numbers. (Contributed by NM, 19-Aug-2001.)

Theoremnndiv 9996* Two ways to express " divides " for natural numbers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnndivtr 9997 Transitive property of divisibility: if divides and divides , then divides . Typically, would be an integer, although the theorem holds for complex . (Contributed by NM, 3-May-2005.)

Theoremnnge1d 9998 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnngt0d 9999 A natural number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnne0d 10000 A natural number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

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