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Type | Label | Description |
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Statement | ||
Theorem | cc4f 7101* |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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Theorem | cc4 7102* |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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Theorem | cc4n 7103* |
Countable choice with a simpler restriction on how every set in the
countable collection needs to be inhabited. That is, compared with
cc4 7102, the hypotheses only require an A(n) for each
value of ![]() ![]() ![]() ![]() ![]() |
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This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6378 and similar theorems ), going from there to positive integers (df-ni 7136) and then positive rational numbers (df-nqqs 7180) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". | ||
Syntax | cnpi 7104 |
The set of positive integers, which is the set of natural numbers ![]() Note: This is the start of the Dedekind-cut construction of real and complex numbers. |
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Syntax | cpli 7105 | Positive integer addition. |
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Syntax | cmi 7106 | Positive integer multiplication. |
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Syntax | clti 7107 | Positive integer ordering relation. |
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Syntax | cplpq 7108 | Positive pre-fraction addition. |
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Syntax | cmpq 7109 | Positive pre-fraction multiplication. |
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Syntax | cltpq 7110 | Positive pre-fraction ordering relation. |
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Syntax | ceq 7111 | Equivalence class used to construct positive fractions. |
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Syntax | cnq 7112 | Set of positive fractions. |
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Syntax | c1q 7113 | The positive fraction constant 1. |
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Syntax | cplq 7114 | Positive fraction addition. |
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Syntax | cmq 7115 | Positive fraction multiplication. |
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Syntax | crq 7116 | Positive fraction reciprocal operation. |
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Syntax | cltq 7117 | Positive fraction ordering relation. |
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Syntax | ceq0 7118 | Equivalence class used to construct nonnegative fractions. |
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Syntax | cnq0 7119 | Set of nonnegative fractions. |
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Syntax | c0q0 7120 | The nonnegative fraction constant 0. |
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Syntax | cplq0 7121 | Nonnegative fraction addition. |
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Syntax | cmq0 7122 | Nonnegative fraction multiplication. |
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Syntax | cnp 7123 | Set of positive reals. |
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Syntax | c1p 7124 | Positive real constant 1. |
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Syntax | cpp 7125 | Positive real addition. |
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Syntax | cmp 7126 | Positive real multiplication. |
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Syntax | cltp 7127 | Positive real ordering relation. |
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Syntax | cer 7128 | Equivalence class used to construct signed reals. |
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Syntax | cnr 7129 | Set of signed reals. |
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Syntax | c0r 7130 | The signed real constant 0. |
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Syntax | c1r 7131 | The signed real constant 1. |
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Syntax | cm1r 7132 | The signed real constant -1. |
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Syntax | cplr 7133 | Signed real addition. |
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Syntax | cmr 7134 | Signed real multiplication. |
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Syntax | cltr 7135 | Signed real ordering relation. |
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Definition | df-ni 7136 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) |
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Definition | df-pli 7137 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
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Definition | df-mi 7138 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
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Definition | df-lti 7139 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) |
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Theorem | elni 7140 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
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Theorem | pinn 7141 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
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Theorem | pion 7142 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
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Theorem | piord 7143 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
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Theorem | niex 7144 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
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Theorem | 0npi 7145 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
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Theorem | elni2 7146 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
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Theorem | 1pi 7147 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
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Theorem | addpiord 7148 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
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Theorem | mulpiord 7149 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
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Theorem | mulidpi 7150 | 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) |
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Theorem | ltpiord 7151 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
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Theorem | ltsopi 7152 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) |
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Theorem | pitric 7153 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
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Theorem | pitri3or 7154 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
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Theorem | ltdcpi 7155 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | ltrelpi 7156 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) |
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Theorem | dmaddpi 7157 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
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Theorem | dmmulpi 7158 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
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Theorem | addclpi 7159 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
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Theorem | mulclpi 7160 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
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Theorem | addcompig 7161 | Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
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Theorem | addasspig 7162 | Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
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Theorem | mulcompig 7163 | Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
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Theorem | mulasspig 7164 | Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
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Theorem | distrpig 7165 | Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.) |
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Theorem | addcanpig 7166 | Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.) |
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Theorem | mulcanpig 7167 | Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.) |
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Theorem | addnidpig 7168 | There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
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Theorem | ltexpi 7169* | Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
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Theorem | ltapig 7170 | Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
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Theorem | ltmpig 7171 | Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
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Theorem | 1lt2pi 7172 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
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Theorem | nlt1pig 7173 | No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
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Theorem | indpi 7174* | Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) |
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Theorem | nnppipi 7175 | A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.) |
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Definition | df-plpq 7176* |
Define pre-addition on positive fractions. This is a "temporary" set
used in the construction of complex numbers, 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
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Definition | df-mpq 7177* | Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
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Definition | df-ltpq 7178* | Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) |
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Definition | df-enq 7179* | Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
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Definition | df-nqqs 7180 | Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
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Definition | df-plqqs 7181* | Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
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Definition | df-mqqs 7182* | Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.) |
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Definition | df-1nqqs 7183 | Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers, 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.) |
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Definition | df-rq 7184* | 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, 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 Jim Kingdon, 20-Sep-2019.) |
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Definition | df-ltnqqs 7185* | Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) |
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Theorem | dfplpq2 7186* | Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.) |
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Theorem | dfmpq2 7187* | Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.) |
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Theorem | enqbreq 7188 | Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) |
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Theorem | enqbreq2 7189 | Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) |
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Theorem | enqer 7190 | 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.) |
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Theorem | enqeceq 7191 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
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Theorem | enqex 7192 | The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) |
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Theorem | enqdc 7193 | The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
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Theorem | enqdc1 7194 | The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
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Theorem | nqex 7195 | The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) |
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Theorem | 0nnq 7196 | The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) |
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Theorem | ltrelnq 7197 | Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) |
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Theorem | 1nq 7198 | The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) |
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Theorem | addcmpblnq 7199 | Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.) |
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Theorem | mulcmpblnq 7200 | Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.) |
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