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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addcomnqg 7001 | Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Theorem | addassnqg 7002 | Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.) |
Theorem | mulcomnqg 7003 | Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Theorem | mulassnqg 7004 | Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Theorem | mulcanenq 7005 | Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) |
Theorem | mulcanenqec 7006 | Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Theorem | distrnqg 7007 | Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Theorem | 1qec 7008 | The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) |
Theorem | mulidnq 7009 | Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) |
Theorem | recexnq 7010* | Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.) |
Theorem | recmulnqg 7011 | Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.) |
Theorem | recclnq 7012 | Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) |
Theorem | recidnq 7013 | A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) |
Theorem | recrecnq 7014 | Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) |
Theorem | rec1nq 7015 | Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.) |
Theorem | nqtri3or 7016 | Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Theorem | ltdcnq 7017 | Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
DECID | ||
Theorem | ltsonq 7018 | 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) |
Theorem | nqtric 7019 | Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Theorem | ltanqg 7020 | Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.) |
Theorem | ltmnqg 7021 | Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.) |
Theorem | ltanqi 7022 | Ordering property of addition for positive fractions. One direction of ltanqg 7020. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Theorem | ltmnqi 7023 | Ordering property of multiplication for positive fractions. One direction of ltmnqg 7021. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Theorem | lt2addnq 7024 | Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.) |
Theorem | lt2mulnq 7025 | Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.) |
Theorem | 1lt2nq 7026 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Theorem | ltaddnq 7027 | The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Theorem | ltexnqq 7028* | Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.) |
Theorem | ltexnqi 7029* | Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.) |
Theorem | halfnqq 7030* | One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.) |
Theorem | halfnq 7031* | 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.) |
Theorem | nsmallnqq 7032* | There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.) |
Theorem | nsmallnq 7033* | There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Theorem | subhalfnqq 7034* | There is a number which is less than half of any positive fraction. The case where is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 7030). (Contributed by Jim Kingdon, 25-Nov-2019.) |
Theorem | ltbtwnnqq 7035* | There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.) |
Theorem | ltbtwnnq 7036* | 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.) |
Theorem | archnqq 7037* | For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.) |
Theorem | prarloclemarch 7038* | A version of the Archimedean property. This variation is "stronger" than archnqq 7037 in the sense that we provide an integer which is larger than a given rational even after being multiplied by a second rational . (Contributed by Jim Kingdon, 30-Nov-2019.) |
Theorem | prarloclemarch2 7039* | Like prarloclemarch 7038 but the integer must be at least two, and there is also added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 7123. (Contributed by Jim Kingdon, 25-Nov-2019.) |
Theorem | ltrnqg 7040 | Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7041. (Contributed by Jim Kingdon, 29-Dec-2019.) |
Theorem | ltrnqi 7041 | Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7040. (Contributed by Jim Kingdon, 24-Sep-2019.) |
Theorem | nnnq 7042 | The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Theorem | ltnnnq 7043 | Ordering of positive integers via or is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.) |
Definition | df-enq0 7044* | Define equivalence relation for nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
~_{Q0} | ||
Definition | df-nq0 7045 | Define class of nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
Q_{0} ~_{Q0} | ||
Definition | df-0nq0 7046 | Define nonnegative fraction constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 5-Nov-2019.) |
0_{Q0} ~_{Q0} | ||
Definition | df-plq0 7047* | Define addition on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
+_{Q0} Q_{0} Q_{0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Definition | df-mq0 7048* | Define multiplication on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
·_{Q0} Q_{0} Q_{0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | dfmq0qs 7049* | Multiplication on nonnegative fractions. This definition is similar to df-mq0 7048 but expands Q_{0} (Contributed by Jim Kingdon, 22-Nov-2019.) |
·_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | dfplq0qs 7050* | Addition on nonnegative fractions. This definition is similar to df-plq0 7047 but expands Q_{0} (Contributed by Jim Kingdon, 24-Nov-2019.) |
+_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | enq0enq 7051 | Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.) |
~_{Q0} | ||
Theorem | enq0sym 7052 | The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7055. (Contributed by Jim Kingdon, 14-Nov-2019.) |
~_{Q0} ~_{Q0} | ||
Theorem | enq0ref 7053 | The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7055. (Contributed by Jim Kingdon, 14-Nov-2019.) |
~_{Q0} | ||
Theorem | enq0tr 7054 | The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7055. (Contributed by Jim Kingdon, 14-Nov-2019.) |
~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | enq0er 7055 | The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
~_{Q0} | ||
Theorem | enq0breq 7056 | Equivalence relation for nonnegative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.) |
~_{Q0} | ||
Theorem | enq0eceq 7057 | Equivalence class equality of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.) |
~_{Q0} ~_{Q0} | ||
Theorem | nqnq0pi 7058 | A nonnegative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.) |
~_{Q0} | ||
Theorem | enq0ex 7059 | The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
~_{Q0} | ||
Theorem | nq0ex 7060 | The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
Q_{0} | ||
Theorem | nqnq0 7061 | A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
Q_{0} | ||
Theorem | nq0nn 7062* | Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.) |
Q_{0} ~_{Q0} | ||
Theorem | addcmpblnq0 7063 | Lemma showing compatibility of addition on nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.) |
~_{Q0} | ||
Theorem | mulcmpblnq0 7064 | Lemma showing compatibility of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) |
~_{Q0} | ||
Theorem | mulcanenq0ec 7065 | Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.) |
~_{Q0} ~_{Q0} | ||
Theorem | nnnq0lem1 7066* | Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7069 and mulnnnq0 7070. (Contributed by Jim Kingdon, 23-Nov-2019.) |
~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | addnq0mo 7067* | There is at most one result from adding nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.) |
~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | mulnq0mo 7068* | There is at most one result from multiplying nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) |
~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | addnnnq0 7069 | Addition of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.) |
~_{Q0} +_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | mulnnnq0 7070 | Multiplication of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.) |
~_{Q0} ·_{Q0} ~_{Q0} ~_{Q0} | ||
Theorem | addclnq0 7071 | Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Q_{0} Q_{0} +_{Q0} Q_{0} | ||
Theorem | mulclnq0 7072 | Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
Q_{0} Q_{0} ·_{Q0} Q_{0} | ||
Theorem | nqpnq0nq 7073 | A positive fraction plus a nonnegative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.) |
Q_{0} +_{Q0} | ||
Theorem | nqnq0a 7074 | Addition of positive fractions is equal with or +_{Q0}. (Contributed by Jim Kingdon, 10-Nov-2019.) |
+_{Q0} | ||
Theorem | nqnq0m 7075 | Multiplication of positive fractions is equal with or ·_{Q0}. (Contributed by Jim Kingdon, 10-Nov-2019.) |
·_{Q0} | ||
Theorem | nq0m0r 7076 | Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Q_{0} 0_{Q0} ·_{Q0} 0_{Q0} | ||
Theorem | nq0a0 7077 | Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Q_{0} +_{Q0} 0_{Q0} | ||
Theorem | nnanq0 7078 | Addition of nonnegative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.) |
~_{Q0} ~_{Q0} +_{Q0} ~_{Q0} | ||
Theorem | distrnq0 7079 | Multiplication of nonnegative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.) |
Q_{0} Q_{0} Q_{0} ·_{Q0} +_{Q0} ·_{Q0} +_{Q0} ·_{Q0} | ||
Theorem | mulcomnq0 7080 | Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.) |
Q_{0} Q_{0} ·_{Q0} ·_{Q0} | ||
Theorem | addassnq0lemcl 7081 | A natural number closure law. Lemma for addassnq0 7082. (Contributed by Jim Kingdon, 3-Dec-2019.) |
Theorem | addassnq0 7082 | Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Q_{0} Q_{0} Q_{0} +_{Q0} +_{Q0} +_{Q0} +_{Q0} | ||
Theorem | distnq0r 7083 | Multiplication of nonnegative fractions is distributive. Version of distrnq0 7079 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Q_{0} Q_{0} Q_{0} +_{Q0} ·_{Q0} ·_{Q0} +_{Q0} ·_{Q0} | ||
Theorem | addpinq1 7084 | Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Theorem | nq02m 7085 | Multiply a nonnegative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Q_{0} ~_{Q0} ·_{Q0} +_{Q0} | ||
Definition | df-inp 7086* |
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.
Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers. A Dedekind cut is an ordered pair of a lower set and an upper set which is inhabited ( ), rounded ( and likewise for ), disjoint ( ) and located ( ). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts. (Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.) |
Definition | df-i1p 7087* | Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by Jim Kingdon, 25-Sep-2019.) |
Definition | df-iplp 7088* |
Define addition on positive reals. From Section 11.2.1 of [HoTT], p.
(varies). We write this definition to closely resemble the definition
in HoTT although some of the conditions are redundant (for example,
implies ) and can be simplified
as
shown at genpdf 7128.
This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.) |
Definition | df-imp 7089* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 7088 or the definition of
multiplication on positive reals in Metamath Proof Explorer. This is as
opposed to the more complicated definition of multiplication given in
Section 11.2.1 of [HoTT], p. (varies),
which appears to be motivated by
handling negative numbers or handling modified Dedekind cuts in which
locatedness is omitted.
This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Definition | df-iltp 7090* |
Define ordering on positive reals. We define
if there is a
positive fraction which is an element of the upper cut of
and the lower cut of . From the definition of < in Section 11.2.1
of [HoTT], p. (varies).
This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Theorem | npsspw 7091 | Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
Theorem | preqlu 7092 | Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Theorem | npex 7093 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
Theorem | elinp 7094* | Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
Theorem | prop 7095 | A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.) |
Theorem | elnp1st2nd 7096* | Membership in positive reals, using and to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.) |
Theorem | prml 7097* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
Theorem | prmu 7098* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
Theorem | prssnql 7099 | The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Theorem | prssnqu 7100 | The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
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