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Type | Label | Description |
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Statement | ||
Theorem | carden2bex 7201* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | pm54.43 7202 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
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Theorem | pr2nelem 7203 | Lemma for pr2ne 7204. (Contributed by FL, 17-Aug-2008.) |
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Theorem | pr2ne 7204 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
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Theorem | exmidonfinlem 7205* | Lemma for exmidonfin 7206. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | exmidonfin 7206 | If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6885 and nnon 4621. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | en2eleq 7207 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Theorem | en2other2 7208 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Theorem | dju1p1e2 7209 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | infpwfidom 7210 |
The collection of finite subsets of a set dominates the set. (We use
the weaker sethood assumption ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | exmidfodomrlemeldju 7211 | Lemma for exmidfodomr 7216. A variant of djur 7081. (Contributed by Jim Kingdon, 2-Jul-2022.) |
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Theorem | exmidfodomrlemreseldju 7212 | Lemma for exmidfodomrlemrALT 7215. A variant of eldju 7080. (Contributed by Jim Kingdon, 9-Jul-2022.) |
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Theorem | exmidfodomrlemim 7213* | Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | exmidfodomrlemr 7214* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | exmidfodomrlemrALT 7215* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7214. In particular, this proof uses eldju 7080 instead of djur 7081 and avoids djulclb 7067. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.) |
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Theorem | exmidfodomr 7216* | Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Syntax | wac 7217 | Formula for an abbreviation of the axiom of choice. |
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Definition | df-ac 7218* |
The expression CHOICE will be used as a readable shorthand for
any
form of the axiom of choice; all concrete forms are long, cryptic, have
dummy variables, or all three, making it useful to have a short name.
Similar to the Axiom of Choice (first form) of [Enderton] p. 49.
There are some decisions about how to write this definition especially around whether ax-setind 4548 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.) |
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Theorem | acfun 7219* | A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.) |
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Theorem | exmidaclem 7220* | Lemma for exmidac 7221. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.) |
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Theorem | exmidac 7221 | The axiom of choice implies excluded middle. See acexmid 5887 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.) |
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Theorem | endjudisj 7222 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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Theorem | djuen 7223 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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Theorem | djuenun 7224 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
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Theorem | dju1en 7225 | Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | dju0en 7226 | Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | xp2dju 7227 | Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | djucomen 7228 | Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | djuassen 7229 | Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | xpdjuen 7230 | Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | djudoml 7231 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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Theorem | djudomr 7232 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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Theorem | exmidontriimlem1 7233 | Lemma for exmidontriim 7237. A variation of r19.30dc 2634. (Contributed by Jim Kingdon, 12-Aug-2024.) |
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Theorem | exmidontriimlem2 7234* | Lemma for exmidontriim 7237. (Contributed by Jim Kingdon, 12-Aug-2024.) |
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Theorem | exmidontriimlem3 7235* |
Lemma for exmidontriim 7237. What we get to do based on induction on
both
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Theorem | exmidontriimlem4 7236* |
Lemma for exmidontriim 7237. The induction step for the induction on
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Theorem | exmidontriim 7237* | Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.) |
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Theorem | pw1on 7238 |
The power set of ![]() |
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Theorem | pw1dom2 7239 |
The power set of ![]() ![]() |
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Theorem | pw1ne0 7240 |
The power set of ![]() |
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Theorem | pw1ne1 7241 |
The power set of ![]() |
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Theorem | pw1ne3 7242 |
The power set of ![]() |
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Theorem | pw1nel3 7243 |
Negated excluded middle implies that the power set of ![]() ![]() |
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Theorem | sucpw1ne3 7244 |
Negated excluded middle implies that the successor of the power set of
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Theorem | sucpw1nel3 7245 |
The successor of the power set of ![]() ![]() |
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Theorem | 3nelsucpw1 7246 |
Three is not an element of the successor of the power set of ![]() |
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Theorem | sucpw1nss3 7247 |
Negated excluded middle implies that the successor of the power set of
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Theorem | 3nsssucpw1 7248 |
Negated excluded middle implies that ![]() ![]() |
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Theorem | onntri35 7249* |
Double negated ordinal trichotomy.
There are five equivalent statements: (1)
Another way of stating this is that EXMID is equivalent
to
trichotomy, either the (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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Theorem | onntri13 7250 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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Theorem | exmidontri 7251* | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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Theorem | onntri51 7252* | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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Theorem | onntri45 7253* | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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Theorem | onntri24 7254 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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Theorem | exmidontri2or 7255* | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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Theorem | onntri52 7256* | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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Theorem | onntri3or 7257* | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
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Theorem | onntri2or 7258* | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
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Syntax | wap 7259 | Apartness predicate symbol. |
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Definition | df-pap 7260* |
Apartness predicate. A relation ![]() |
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Syntax | wtap 7261 | Tight apartness predicate symbol. |
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Definition | df-tap 7262* |
Tight apartness predicate. A relation ![]() |
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Theorem | dftap2 7263* | Tight apartness with the apartness properties from df-pap 7260 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.) |
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Theorem | tapeq1 7264 | Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.) |
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Theorem | tapeq2 7265 | Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.) |
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Theorem | netap 7266* | Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.) |
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Theorem | 2onetap 7267* |
Negated equality is a tight apartness on ![]() |
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Theorem | 2oneel 7268* |
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Theorem | 2omotaplemap 7269* | Lemma for 2omotap 7271. (Contributed by Jim Kingdon, 6-Feb-2025.) |
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Theorem | 2omotaplemst 7270* | Lemma for 2omotap 7271. (Contributed by Jim Kingdon, 6-Feb-2025.) |
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Theorem | 2omotap 7271 |
If there is at most one tight apartness on ![]() |
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Theorem | exmidapne 7272* | Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.) |
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Theorem | exmidmotap 7273* | The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.) |
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We have already introduced the full Axiom of Choice df-ac 7218 but since it implies excluded middle as shown at exmidac 7221, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle. | ||
Syntax | wacc 7274 | Formula for an abbreviation of countable choice. |
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Definition | df-cc 7275* | The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7218 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.) |
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Theorem | ccfunen 7276* | Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.) |
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Theorem | cc1 7277* | Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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Theorem | cc2lem 7278* | Lemma for cc2 7279. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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Theorem | cc2 7279* | Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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Theorem | cc3 7280* | Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.) |
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Theorem | cc4f 7281* |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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Theorem | cc4 7282* |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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Theorem | cc4n 7283* |
Countable choice with a simpler restriction on how every set in the
countable collection needs to be inhabited. That is, compared with
cc4 7282, 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 6488 and similar theorems ), going from there to positive integers (df-ni 7316) and then positive rational numbers (df-nqqs 7360) 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 or choice principles. With excluded middle, it is natural to define a cut as the lower set only (as Metamath Proof Explorer does), but here 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". When working constructively, there are several possible definitions of real numbers. Here we adopt the most common definition, as two-sided Dedekind cuts with the properties described at df-inp 7478. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7944 and the MacNeille reals fail to satisfy axltwlin 8038, and we do not develop them here. For more on differing definitions of the reals, see the introduction to Chapter 11 in [HoTT] or Section 1.2 of [BauerHanson]. | ||
Syntax | cnpi 7284 |
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 7285 | Positive integer addition. |
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Syntax | cmi 7286 | Positive integer multiplication. |
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Syntax | clti 7287 | Positive integer ordering relation. |
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Syntax | cplpq 7288 | Positive pre-fraction addition. |
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Syntax | cmpq 7289 | Positive pre-fraction multiplication. |
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Syntax | cltpq 7290 | Positive pre-fraction ordering relation. |
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Syntax | ceq 7291 | Equivalence class used to construct positive fractions. |
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Syntax | cnq 7292 | Set of positive fractions. |
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Syntax | c1q 7293 | The positive fraction constant 1. |
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Syntax | cplq 7294 | Positive fraction addition. |
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Syntax | cmq 7295 | Positive fraction multiplication. |
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Syntax | crq 7296 | Positive fraction reciprocal operation. |
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Syntax | cltq 7297 | Positive fraction ordering relation. |
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Syntax | ceq0 7298 | Equivalence class used to construct nonnegative fractions. |
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Syntax | cnq0 7299 | Set of nonnegative fractions. |
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Syntax | c0q0 7300 | The nonnegative fraction constant 0. |
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