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

Theoremdivgt0d 7901 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulgt1d 7902 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge11d 7903 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge12d 7904 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul1ad 7905 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2ad 7906 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul12ad 7907 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12ad 7908 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12bd 7909 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

3.3.10  Imaginary and complex number properties

Theoremcrap0 7910 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.)
# # #

Theoremcreur 7911* 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 7912* 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 7913* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

3.4  Integer sets

3.4.1  Positive integers (as a subset of complex numbers)

Syntaxcn 7914 Extend class notation to include the class of positive integers.

Definitiondf-inn 7915* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7916 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)

Theoremdfnn2 7916* Definition of the set of positive integers. Another name for df-inn 7915. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5nni 7917* 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 7918 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremnnsscn 7919 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremnnex 7920 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnre 7921 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncn 7922 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)

Theoremnnrei 7923 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncni 7924 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)

Theorem1nn 7925 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)

Theorempeano2nn 7926 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.)

Theoremnnred 7927 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncnd 7928 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempeano2nnd 7929 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)

3.4.2  Principle of mathematical induction

Theoremnnind 7930* 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 7934 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.)

TheoremnnindALT 7931* 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 7930 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.)

Theoremnn1m1nn 7932 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn1suc 7933* 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.)

Theoremnnaddcl 7934 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcl 7935 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcli 7936 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnnge1 7937 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)

Theoremnnle1eq1 7938 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)

Theoremnngt0 7939 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)

Theoremnnnlt1 7940 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0nnn 7941 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)

Theoremnnne0 7942 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)

Theoremnnap0 7943 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
#

Theoremnngt0i 7944 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)

Theoremnnne0i 7945 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)

Theoremnn2ge 7946* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)

Theoremnn1gt1 7947 A positive integer is either one or greater than one. This is for ; 0elnn 4340 is a similar theorem for (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)

Theoremnngt1ne1 7948 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)

Theoremnndivre 7949 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)

Theoremnnrecre 7950 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)

Theoremnnrecgt0 7951 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)

Theoremnnsub 7952 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnsubi 7953 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)

Theoremnndiv 7954* Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnndivtr 7955 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 7956 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnngt0d 7957 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnne0d 7958 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnap0d 7959 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
#

Theoremnnrecred 7960 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnaddcld 7961 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnmulcld 7962 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnndivred 7963 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

3.4.3  Decimal representation of numbers

Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 6896 and df-1 6897).

Only the digits 0 through 9 (df-0 6896 through df-9 7980) and the number 10 (df-10 7981) are explicitly defined.

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 7964 Extend class notation to include the number 2.

Syntaxc3 7965 Extend class notation to include the number 3.

Syntaxc4 7966 Extend class notation to include the number 4.

Syntaxc5 7967 Extend class notation to include the number 5.

Syntaxc6 7968 Extend class notation to include the number 6.

Syntaxc7 7969 Extend class notation to include the number 7.

Syntaxc8 7970 Extend class notation to include the number 8.

Syntaxc9 7971 Extend class notation to include the number 9.

Syntaxc10 7972 Extend class notation to include the number 10.

Definitiondf-2 7973 Define the number 2. (Contributed by NM, 27-May-1999.)

Definitiondf-3 7974 Define the number 3. (Contributed by NM, 27-May-1999.)

Definitiondf-4 7975 Define the number 4. (Contributed by NM, 27-May-1999.)

Definitiondf-5 7976 Define the number 5. (Contributed by NM, 27-May-1999.)

Definitiondf-6 7977 Define the number 6. (Contributed by NM, 27-May-1999.)

Definitiondf-7 7978 Define the number 7. (Contributed by NM, 27-May-1999.)

Definitiondf-8 7979 Define the number 8. (Contributed by NM, 27-May-1999.)

Definitiondf-9 7980 Define the number 9. (Contributed by NM, 27-May-1999.)

Definitiondf-10 7981 Define the number 10. See remarks under df-2 7973. (Contributed by NM, 5-Feb-2007.)

Theorem0ne1 7982 (common case). See aso 1ap0 7581. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem1ne0 7983 . See aso 1ap0 7581. (Contributed by Jim Kingdon, 9-Mar-2020.)

Theorem1m1e0 7984 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2re 7985 The number 2 is real. (Contributed by NM, 27-May-1999.)

Theorem2cn 7986 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)

Theorem2ex 7987 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem2cnd 7988 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem3re 7989 The number 3 is real. (Contributed by NM, 27-May-1999.)

Theorem3cn 7990 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)

Theorem3ex 7991 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem4re 7992 The number 4 is real. (Contributed by NM, 27-May-1999.)

Theorem4cn 7993 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem5re 7994 The number 5 is real. (Contributed by NM, 27-May-1999.)

Theorem5cn 7995 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem6re 7996 The number 6 is real. (Contributed by NM, 27-May-1999.)

Theorem6cn 7997 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem7re 7998 The number 7 is real. (Contributed by NM, 27-May-1999.)

Theorem7cn 7999 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem8re 8000 The number 8 is real. (Contributed by NM, 27-May-1999.)

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