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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelni2 8701 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)

Theorempinn 8702 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Theorempion 8703 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Theorempiord 8704 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)

Theoremniex 8705 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Theorem0npi 8706 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Theorem1pi 8707 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)

Theoremaddpiord 8708 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremmulpiord 8709 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremmulidpi 8710 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)

Theoremltpiord 8711 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremltsopi 8712 Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)

Theoremltrelpi 8713 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)

Theoremdmaddpi 8714 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Theoremdmmulpi 8715 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Theoremaddclpi 8716 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Theoremmulclpi 8717 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Theoremaddcompi 8718 Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremaddasspi 8719 Addition of positive integers is associative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremmulcompi 8720 Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

Theoremmulasspi 8721 Multiplication of positive integers is associative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

Theoremdistrpi 8722 Multiplication of positive integers is distributive. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

Theoremaddcanpi 8723 Addition cancellation law for positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulcanpi 8724 Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theoremaddnidpi 8725 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)

Theoremltexpi 8726* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)

Theoremltapi 8727 Ordering property of addition for positive integers. (Contributed by NM, 7-Mar-1996.) (New usage is discouraged.)

Theoremltmpi 8728 Ordering property of multiplication for positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)

Theorem1lt2pi 8729 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)

Theoremnlt1pi 8730 No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Theoremindpi 8731* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Definitiondf-plpq 8732* Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition (df-plq 8738) works with the equivalence classes of these ordered pairs determined by the equivalence relation (df-enq 8735). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Definitiondf-mpq 8733* Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Definitiondf-ltpq 8734* Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Definitiondf-enq 8735* Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Definitiondf-nq 8736* Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.) (New usage is discouraged.)

Definitiondf-erq 8737 Define a convenience function that "reduces" a fraction to lowest terms. Note that in this form, it is not obviously a function; we prove this in nqerf 8754. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Definitiondf-plq 8738 Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.) (New usage is discouraged.)

Definitiondf-mq 8739 Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.) (New usage is discouraged.)

Definitiondf-1nq 8740 Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)

Definitiondf-rq 8741 Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by NM, 6-Mar-1996.) (New usage is discouraged.)

Definitiondf-ltnq 8742 Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8943, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) (New usage is discouraged.)

Theoremenqbreq 8743 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Theoremenqbreq2 8744 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremenqer 8745 The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)

Theoremenqex 8746 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Theoremnqex 8747 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Theorem0nnq 8748 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Theoremelpqn 8749 Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremltrelnq 8750 Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Theorempinq 8751 The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Theorem1nq 8752 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremnqereu 8753* There is a unique element of equivalent to each element of . (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremnqerf 8754 Corollary of nqereu 8753: the function is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Theoremnqercl 8755 Corollary of nqereu 8753: closure of . (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Theoremnqerrel 8756 Any member of relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Theoremnqerid 8757 Corollary of nqereu 8753: the function acts as the identity on members of . (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Theoremenqeq 8758 Corollary of nqereu 8753: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Theoremnqereq 8759 The function acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Theoremaddpipq2 8760 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremaddpipq 8761 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremaddpqnq 8762 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)

Theoremmulpipq2 8763 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulpipq 8764 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulpqnq 8765 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)

Theoremordpipq 8766 Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremordpinq 8767 Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremaddpqf 8768 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremaddclnq 8769 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulpqf 8770 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulclnq 8771 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremaddnqf 8772 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)

Theoremmulnqf 8773 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)

Theoremaddcompq 8774 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremaddcomnq 8775 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremmulcompq 8776 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremmulcomnq 8777 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremadderpqlem 8778 Lemma for adderpq 8780. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulerpqlem 8779 Lemma for mulerpq 8781. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremadderpq 8780 Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulerpq 8781 Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremaddassnq 8782 Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremmulassnq 8783 Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremmulcanenq 8784 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremdistrnq 8785 Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theorem1nqenq 8786 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremmulidnq 8787 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Theoremrecmulnq 8788 Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremrecidnq 8789 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremrecclnq 8790 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Theoremrecrecnq 8791 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)

Theoremdmrecnq 8792 Domain of reciprocal on positive fractions. (Contributed by Mario Carneiro, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)

Theoremltsonq 8793 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)

Theoremlterpq 8794 Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)

Theoremltanq 8795 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theoremltmnq 8796 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theorem1lt2nq 8797 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theoremltaddnq 8798 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theoremltexnq 8799* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Theoremhalfnq 8800* One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

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