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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnqercl 10801 Corollary of nqereu 10799: closure of [Q]. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ (N ร— N) โ†’ ([Q]โ€˜๐ด) โˆˆ Q)
 
Theoremnqerrel 10802 Any member of (N ร— N) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ (N ร— N) โ†’ ๐ด ~Q ([Q]โ€˜๐ด))
 
Theoremnqerid 10803 Corollary of nqereu 10799: the function [Q] acts as the identity on members of Q. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ ([Q]โ€˜๐ด) = ๐ด)
 
Theoremenqeq 10804 Corollary of nqereu 10799: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q โˆง ๐ด ~Q ๐ต) โ†’ ๐ด = ๐ต)
 
Theoremnqereq 10805 The function [Q] 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.)
((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N)) โ†’ (๐ด ~Q ๐ต โ†” ([Q]โ€˜๐ด) = ([Q]โ€˜๐ต)))
 
Theoremaddpipq2 10806 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N)) โ†’ (๐ด +pQ ๐ต) = โŸจ(((1st โ€˜๐ด) ยทN (2nd โ€˜๐ต)) +N ((1st โ€˜๐ต) ยทN (2nd โ€˜๐ด))), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต))โŸฉ)
 
Theoremaddpipq 10807 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(((๐ด โˆˆ N โˆง ๐ต โˆˆ N) โˆง (๐ถ โˆˆ N โˆง ๐ท โˆˆ N)) โ†’ (โŸจ๐ด, ๐ตโŸฉ +pQ โŸจ๐ถ, ๐ทโŸฉ) = โŸจ((๐ด ยทN ๐ท) +N (๐ถ ยทN ๐ต)), (๐ต ยทN ๐ท)โŸฉ)
 
Theoremaddpqnq 10808 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.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q) โ†’ (๐ด +Q ๐ต) = ([Q]โ€˜(๐ด +pQ ๐ต)))
 
Theoremmulpipq2 10809 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N)) โ†’ (๐ด ยทpQ ๐ต) = โŸจ((1st โ€˜๐ด) ยทN (1st โ€˜๐ต)), ((2nd โ€˜๐ด) ยทN (2nd โ€˜๐ต))โŸฉ)
 
Theoremmulpipq 10810 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.)
(((๐ด โˆˆ N โˆง ๐ต โˆˆ N) โˆง (๐ถ โˆˆ N โˆง ๐ท โˆˆ N)) โ†’ (โŸจ๐ด, ๐ตโŸฉ ยทpQ โŸจ๐ถ, ๐ทโŸฉ) = โŸจ(๐ด ยทN ๐ถ), (๐ต ยทN ๐ท)โŸฉ)
 
Theoremmulpqnq 10811 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.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q) โ†’ (๐ด ยทQ ๐ต) = ([Q]โ€˜(๐ด ยทpQ ๐ต)))
 
Theoremordpipq 10812 Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(โŸจ๐ด, ๐ตโŸฉ <pQ โŸจ๐ถ, ๐ทโŸฉ โ†” (๐ด ยทN ๐ท) <N (๐ถ ยทN ๐ต))
 
Theoremordpinq 10813 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.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q) โ†’ (๐ด <Q ๐ต โ†” ((1st โ€˜๐ด) ยทN (2nd โ€˜๐ต)) <N ((1st โ€˜๐ต) ยทN (2nd โ€˜๐ด))))
 
Theoremaddpqf 10814 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
+pQ :((N ร— N) ร— (N ร— N))โŸถ(N ร— N)
 
Theoremaddclnq 10815 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q) โ†’ (๐ด +Q ๐ต) โˆˆ Q)
 
Theoremmulpqf 10816 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
ยทpQ :((N ร— N) ร— (N ร— N))โŸถ(N ร— N)
 
Theoremmulclnq 10817 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q) โ†’ (๐ด ยทQ ๐ต) โˆˆ Q)
 
Theoremaddnqf 10818 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
+Q :(Q ร— Q)โŸถQ
 
Theoremmulnqf 10819 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
ยทQ :(Q ร— Q)โŸถQ
 
Theoremaddcompq 10820 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด +pQ ๐ต) = (๐ต +pQ ๐ด)
 
Theoremaddcomnq 10821 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด +Q ๐ต) = (๐ต +Q ๐ด)
 
Theoremmulcompq 10822 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด ยทpQ ๐ต) = (๐ต ยทpQ ๐ด)
 
Theoremmulcomnq 10823 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด ยทQ ๐ต) = (๐ต ยทQ ๐ด)
 
Theoremadderpqlem 10824 Lemma for adderpq 10826. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N) โˆง ๐ถ โˆˆ (N ร— N)) โ†’ (๐ด ~Q ๐ต โ†” (๐ด +pQ ๐ถ) ~Q (๐ต +pQ ๐ถ)))
 
Theoremmulerpqlem 10825 Lemma for mulerpq 10827. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ (N ร— N) โˆง ๐ต โˆˆ (N ร— N) โˆง ๐ถ โˆˆ (N ร— N)) โ†’ (๐ด ~Q ๐ต โ†” (๐ด ยทpQ ๐ถ) ~Q (๐ต ยทpQ ๐ถ)))
 
Theoremadderpq 10826 Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(([Q]โ€˜๐ด) +Q ([Q]โ€˜๐ต)) = ([Q]โ€˜(๐ด +pQ ๐ต))
 
Theoremmulerpq 10827 Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(([Q]โ€˜๐ด) ยทQ ([Q]โ€˜๐ต)) = ([Q]โ€˜(๐ด ยทpQ ๐ต))
 
Theoremaddassnq 10828 Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
((๐ด +Q ๐ต) +Q ๐ถ) = (๐ด +Q (๐ต +Q ๐ถ))
 
Theoremmulassnq 10829 Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((๐ด ยทQ ๐ต) ยทQ ๐ถ) = (๐ด ยทQ (๐ต ยทQ ๐ถ))
 
Theoremmulcanenq 10830 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.)
((๐ด โˆˆ N โˆง ๐ต โˆˆ N โˆง ๐ถ โˆˆ N) โ†’ โŸจ(๐ด ยทN ๐ต), (๐ด ยทN ๐ถ)โŸฉ ~Q โŸจ๐ต, ๐ถโŸฉ)
 
Theoremdistrnq 10831 Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด ยทQ (๐ต +Q ๐ถ)) = ((๐ด ยทQ ๐ต) +Q (๐ด ยทQ ๐ถ))
 
Theorem1nqenq 10832 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด โˆˆ N โ†’ 1Q ~Q โŸจ๐ด, ๐ดโŸฉ)
 
Theoremmulidnq 10833 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ (๐ด ยทQ 1Q) = ๐ด)
 
Theoremrecmulnq 10834 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.)
(๐ด โˆˆ Q โ†’ ((*Qโ€˜๐ด) = ๐ต โ†” (๐ด ยทQ ๐ต) = 1Q))
 
Theoremrecidnq 10835 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.)
(๐ด โˆˆ Q โ†’ (๐ด ยทQ (*Qโ€˜๐ด)) = 1Q)
 
Theoremrecclnq 10836 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ (*Qโ€˜๐ด) โˆˆ Q)
 
Theoremrecrecnq 10837 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ (*Qโ€˜(*Qโ€˜๐ด)) = ๐ด)
 
Theoremdmrecnq 10838 Domain of reciprocal on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
dom *Q = Q
 
Theoremltsonq 10839 '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.)
<Q Or Q
 
Theoremlterpq 10840 Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
(๐ด <pQ ๐ต โ†” ([Q]โ€˜๐ด) <Q ([Q]โ€˜๐ต))
 
Theoremltanq 10841 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.)
(๐ถ โˆˆ Q โ†’ (๐ด <Q ๐ต โ†” (๐ถ +Q ๐ด) <Q (๐ถ +Q ๐ต)))
 
Theoremltmnq 10842 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.)
(๐ถ โˆˆ Q โ†’ (๐ด <Q ๐ต โ†” (๐ถ ยทQ ๐ด) <Q (๐ถ ยทQ ๐ต)))
 
Theorem1lt2nq 10843 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.)
1Q <Q (1Q +Q 1Q)
 
Theoremltaddnq 10844 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.)
((๐ด โˆˆ Q โˆง ๐ต โˆˆ Q) โ†’ ๐ด <Q (๐ด +Q ๐ต))
 
Theoremltexnq 10845* 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.)
(๐ต โˆˆ Q โ†’ (๐ด <Q ๐ต โ†” โˆƒ๐‘ฅ(๐ด +Q ๐‘ฅ) = ๐ต))
 
Theoremhalfnq 10846* 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.)
(๐ด โˆˆ Q โ†’ โˆƒ๐‘ฅ(๐‘ฅ +Q ๐‘ฅ) = ๐ด)
 
Theoremnsmallnq 10847* The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ โˆƒ๐‘ฅ ๐‘ฅ <Q ๐ด)
 
Theoremltbtwnnq 10848* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(๐ด <Q ๐ต โ†” โˆƒ๐‘ฅ(๐ด <Q ๐‘ฅ โˆง ๐‘ฅ <Q ๐ต))
 
Theoremltrnq 10849 Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(๐ด <Q ๐ต โ†” (*Qโ€˜๐ต) <Q (*Qโ€˜๐ด))
 
Theoremarchnq 10850* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ โˆƒ๐‘ฅ โˆˆ N ๐ด <Q โŸจ๐‘ฅ, 1oโŸฉ)
 
Definitiondf-np 10851* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 10991, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
P = {๐‘ฅ โˆฃ ((โˆ… โŠŠ ๐‘ฅ โˆง ๐‘ฅ โŠŠ Q) โˆง โˆ€๐‘ฆ โˆˆ ๐‘ฅ (โˆ€๐‘ง(๐‘ง <Q ๐‘ฆ โ†’ ๐‘ง โˆˆ ๐‘ฅ) โˆง โˆƒ๐‘ง โˆˆ ๐‘ฅ ๐‘ฆ <Q ๐‘ง))}
 
Definitiondf-1p 10852 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 10991, and is intended to be used only by the construction. Definition of [Gleason] p. 122. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
1P = {๐‘ฅ โˆฃ ๐‘ฅ <Q 1Q}
 
Definitiondf-plp 10853* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 10991, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
+P = (๐‘ฅ โˆˆ P, ๐‘ฆ โˆˆ P โ†ฆ {๐‘ค โˆฃ โˆƒ๐‘ฃ โˆˆ ๐‘ฅ โˆƒ๐‘ข โˆˆ ๐‘ฆ ๐‘ค = (๐‘ฃ +Q ๐‘ข)})
 
Definitiondf-mp 10854* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 10991, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
ยทP = (๐‘ฅ โˆˆ P, ๐‘ฆ โˆˆ P โ†ฆ {๐‘ค โˆฃ โˆƒ๐‘ฃ โˆˆ ๐‘ฅ โˆƒ๐‘ข โˆˆ ๐‘ฆ ๐‘ค = (๐‘ฃ ยทQ ๐‘ข)})
 
Definitiondf-ltp 10855* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 10991, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<P = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ((๐‘ฅ โˆˆ P โˆง ๐‘ฆ โˆˆ P) โˆง ๐‘ฅ โŠŠ ๐‘ฆ)}
 
Theoremnpex 10856 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
P โˆˆ V
 
Theoremelnp 10857* Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
(๐ด โˆˆ P โ†” ((โˆ… โŠŠ ๐ด โˆง ๐ด โŠŠ Q) โˆง โˆ€๐‘ฅ โˆˆ ๐ด (โˆ€๐‘ฆ(๐‘ฆ <Q ๐‘ฅ โ†’ ๐‘ฆ โˆˆ ๐ด) โˆง โˆƒ๐‘ฆ โˆˆ ๐ด ๐‘ฅ <Q ๐‘ฆ)))
 
Theoremelnpi 10858* Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ P โ†” ((๐ด โˆˆ V โˆง โˆ… โŠŠ ๐ด โˆง ๐ด โŠŠ Q) โˆง โˆ€๐‘ฅ โˆˆ ๐ด (โˆ€๐‘ฆ(๐‘ฆ <Q ๐‘ฅ โ†’ ๐‘ฆ โˆˆ ๐ด) โˆง โˆƒ๐‘ฆ โˆˆ ๐ด ๐‘ฅ <Q ๐‘ฆ)))
 
Theoremprn0 10859 A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ P โ†’ ๐ด โ‰  โˆ…)
 
Theoremprpssnq 10860 A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
(๐ด โˆˆ P โ†’ ๐ด โŠŠ Q)
 
Theoremelprnq 10861 A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ ๐ด) โ†’ ๐ต โˆˆ Q)
 
Theorem0npr 10862 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
ยฌ โˆ… โˆˆ P
 
Theoremprcdnq 10863 A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ ๐ด) โ†’ (๐ถ <Q ๐ต โ†’ ๐ถ โˆˆ ๐ด))
 
Theoremprub 10864 A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ ๐ด) โˆง ๐ถ โˆˆ Q) โ†’ (ยฌ ๐ถ โˆˆ ๐ด โ†’ ๐ต <Q ๐ถ))
 
Theoremprnmax 10865* A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ ๐ด) โ†’ โˆƒ๐‘ฅ โˆˆ ๐ด ๐ต <Q ๐‘ฅ)
 
Theoremnpomex 10866 A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of P is an infinite set, the negation of Infinity implies that P, and hence โ„, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 10863 and nsmallnq 10847). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
(๐ด โˆˆ P โ†’ ฯ‰ โˆˆ V)
 
Theoremprnmadd 10867* A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ ๐ด) โ†’ โˆƒ๐‘ฅ(๐ต +Q ๐‘ฅ) โˆˆ ๐ด)
 
Theoremltrelpr 10868 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<P โŠ† (P ร— P)
 
Theoremgenpv 10869* Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด๐น๐ต) = {๐‘“ โˆฃ โˆƒ๐‘” โˆˆ ๐ด โˆƒโ„Ž โˆˆ ๐ต ๐‘“ = (๐‘”๐บโ„Ž)})
 
Theoremgenpelv 10870* Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ถ โˆˆ (๐ด๐น๐ต) โ†” โˆƒ๐‘” โˆˆ ๐ด โˆƒโ„Ž โˆˆ ๐ต ๐ถ = (๐‘”๐บโ„Ž)))
 
Theoremgenpprecl 10871* Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ ((๐ถ โˆˆ ๐ด โˆง ๐ท โˆˆ ๐ต) โ†’ (๐ถ๐บ๐ท) โˆˆ (๐ด๐น๐ต)))
 
Theoremgenpdm 10872* Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    โ‡’   dom ๐น = (P ร— P)
 
Theoremgenpn0 10873* The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ โˆ… โŠŠ (๐ด๐น๐ต))
 
Theoremgenpss 10874* The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด๐น๐ต) โŠ† Q)
 
Theoremgenpnnp 10875* The result of an operation on positive reals is different from the set of positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   (๐‘ง โˆˆ Q โ†’ (๐‘ฅ <Q ๐‘ฆ โ†” (๐‘ง๐บ๐‘ฅ) <Q (๐‘ง๐บ๐‘ฆ)))    &   (๐‘ฅ๐บ๐‘ฆ) = (๐‘ฆ๐บ๐‘ฅ)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ ยฌ (๐ด๐น๐ต) = Q)
 
Theoremgenpcd 10876* Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   ((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘”๐บโ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด๐น๐ต)))    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐‘“ โˆˆ (๐ด๐น๐ต) โ†’ (๐‘ฅ <Q ๐‘“ โ†’ ๐‘ฅ โˆˆ (๐ด๐น๐ต))))
 
Theoremgenpnmax 10877* An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   (๐‘ฃ โˆˆ Q โ†’ (๐‘ง <Q ๐‘ค โ†” (๐‘ฃ๐บ๐‘ง) <Q (๐‘ฃ๐บ๐‘ค)))    &   (๐‘ง๐บ๐‘ค) = (๐‘ค๐บ๐‘ง)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐‘“ โˆˆ (๐ด๐น๐ต) โ†’ โˆƒ๐‘ฅ โˆˆ (๐ด๐น๐ต)๐‘“ <Q ๐‘ฅ))
 
Theoremgenpcl 10878* Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   (โ„Ž โˆˆ Q โ†’ (๐‘“ <Q ๐‘” โ†” (โ„Ž๐บ๐‘“) <Q (โ„Ž๐บ๐‘”)))    &   (๐‘ฅ๐บ๐‘ฆ) = (๐‘ฆ๐บ๐‘ฅ)    &   ((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘”๐บโ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด๐น๐ต)))    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด๐น๐ต) โˆˆ P)
 
Theoremgenpass 10879* Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   dom ๐น = (P ร— P)    &   ((๐‘“ โˆˆ P โˆง ๐‘” โˆˆ P) โ†’ (๐‘“๐น๐‘”) โˆˆ P)    &   ((๐‘“๐บ๐‘”)๐บโ„Ž) = (๐‘“๐บ(๐‘”๐บโ„Ž))    โ‡’   ((๐ด๐น๐ต)๐น๐ถ) = (๐ด๐น(๐ต๐น๐ถ))
 
Theoremplpv 10880* Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด +P ๐ต) = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐ด โˆƒ๐‘ง โˆˆ ๐ต ๐‘ฅ = (๐‘ฆ +Q ๐‘ง)})
 
Theoremmpv 10881* Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด ยทP ๐ต) = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐ด โˆƒ๐‘ง โˆˆ ๐ต ๐‘ฅ = (๐‘ฆ ยทQ ๐‘ง)})
 
Theoremdmplp 10882 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom +P = (P ร— P)
 
Theoremdmmp 10883 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom ยทP = (P ร— P)
 
Theoremnqpr 10884* The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ {๐‘ฅ โˆฃ ๐‘ฅ <Q ๐ด} โˆˆ P)
 
Theorem1pr 10885 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
1P โˆˆ P
 
Theoremaddclprlem1 10886 Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘” +Q โ„Ž) โ†’ ((๐‘ฅ ยทQ (*Qโ€˜(๐‘” +Q โ„Ž))) ยทQ ๐‘”) โˆˆ ๐ด))
 
Theoremaddclprlem2 10887* Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘” +Q โ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด +P ๐ต)))
 
Theoremaddclpr 10888 Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด +P ๐ต) โˆˆ P)
 
Theoremmulclprlem 10889* Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)
((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘” ยทQ โ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด ยทP ๐ต)))
 
Theoremmulclpr 10890 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด ยทP ๐ต) โˆˆ P)
 
Theoremaddcompr 10891 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
(๐ด +P ๐ต) = (๐ต +P ๐ด)
 
Theoremaddasspr 10892 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
((๐ด +P ๐ต) +P ๐ถ) = (๐ด +P (๐ต +P ๐ถ))
 
Theoremmulcompr 10893 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
(๐ด ยทP ๐ต) = (๐ต ยทP ๐ด)
 
Theoremmulasspr 10894 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
((๐ด ยทP ๐ต) ยทP ๐ถ) = (๐ด ยทP (๐ต ยทP ๐ถ))
 
Theoremdistrlem1pr 10895 Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P โˆง ๐ถ โˆˆ P) โ†’ (๐ด ยทP (๐ต +P ๐ถ)) โŠ† ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)))
 
Theoremdistrlem4pr 10896* Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ P โˆง ๐ถ โˆˆ P) โˆง ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง (๐‘“ โˆˆ ๐ด โˆง ๐‘ง โˆˆ ๐ถ))) โ†’ ((๐‘ฅ ยทQ ๐‘ฆ) +Q (๐‘“ ยทQ ๐‘ง)) โˆˆ (๐ด ยทP (๐ต +P ๐ถ)))
 
Theoremdistrlem5pr 10897 Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P โˆง ๐ถ โˆˆ P) โ†’ ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)) โŠ† (๐ด ยทP (๐ต +P ๐ถ)))
 
Theoremdistrpr 10898 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(๐ด ยทP (๐ต +P ๐ถ)) = ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ))
 
Theorem1idpr 10899 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
(๐ด โˆˆ P โ†’ (๐ด ยทP 1P) = ๐ด)
 
Theoremltprord 10900 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด<P ๐ต โ†” ๐ด โŠŠ ๐ต))
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