P. 64 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xpcomen 6301	Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- A e. _V   &    |- B e. _V   =>    |- ( A X. B ) ~~ ( B X. A )
Theorem	xpcomeng 6302	Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
|- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )
Theorem	xpsnen2g 6303	A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
|- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B )
Theorem	xpassen 6304	Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( ( A X. B ) X. C ) ~~ ( A X. ( B X. C ) )
Theorem	xpdom2 6305	Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- C e. _V   =>    |- ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) )
Theorem	xpdom2g 6306	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
Theorem	xpdom1g 6307	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )
Theorem	xpdom3m 6308*	A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
|- ( ( A e. V /\ B e. W /\ E. x x e. B ) -> A ~<_ ( A X. B ) )
Theorem	xpdom1 6309	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
|- C e. _V   =>    |- ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) )
Theorem	fopwdom 6310	Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )
Theorem	enen1 6311	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) )
Theorem	enen2 6312	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) )
Theorem	domen1 6313	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )
Theorem	domen2 6314	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) )
2.6.26  Pigeonhole Principle
Theorem	phplem1 6315	Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
|- ( ( A e. om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )
Theorem	phplem2 6316	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )
Theorem	phplem3 6317	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6319. (Contributed by NM, 26-May-1998.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )
Theorem	phplem4 6318	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A e. om /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )
Theorem	phplem3g 6319	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6317 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )
Theorem	nneneq 6320	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
|- ( ( A e. om /\ B e. om ) -> ( A ~~ B <-> A = B ) )
Theorem	php5 6321	A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
|- ( A e. om -> -. A ~~ suc A )
Theorem	snnen2og 6322	A singleton { A } is never equinumerous with the ordinal number 2. If A is a proper class, see snnen2oprc 6323. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. V -> -. { A } ~~ 2o )
Theorem	snnen2oprc 6323	A singleton { A } is never equinumerous with the ordinal number 2. If A is a set, see snnen2og 6322. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( -. A e. _V -> -. { A } ~~ 2o )
Theorem	phplem4dom 6324	Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( suc A ~<_ suc B -> A ~<_ B ) )
Theorem	php5dom 6325	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
Theorem	nndomo 6326	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
|- ( ( A e. om /\ B e. om ) -> ( A ~<_ B <-> A C_ B ) )
Theorem	phpm 6327*	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols E. x x e. ( A \ B ) (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6315 through phplem4 6318, nneneq 6320, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
|- ( ( A e. om /\ B C_ A /\ E. x x e. ( A \ B ) ) -> -. A ~~ B )
Theorem	phpelm 6328	Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
|- ( ( A e. om /\ B e. A ) -> -. A ~~ B )
Theorem	phplem4on 6329	Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
|- ( ( A e. On /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )
2.6.27  Finite sets
Theorem	fidceq 6330	Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that { B , C } is finite would require showing it is equinumerous to 1o or to 2o but to show that you'd need to know B = C or -. B = C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
|- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )
Theorem	fidifsnen 6331	All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
|- ( ( X e. Fin /\ A e. X /\ B e. X ) -> ( X \ { A } ) ~~ ( X \ { B } ) )
Theorem	fidifsnid 6332	If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3510 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
|- ( ( A e. Fin /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
Theorem	nnfi 6333	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- ( A e. om -> A e. Fin )
Theorem	enfi 6334	Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
|- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) )
Theorem	enfii 6335	A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
|- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin )
Theorem	ssfiexmid 6336*	If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
|- A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin )   =>    |- ( ph \/ -. ph )
Theorem	dif1en 6337	If a set A is equinumerous to the successor of a natural number M, then A with an element removed is equinumerous to M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
|- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( A \ { X } ) ~~ M )
Theorem	fiunsnnn 6338	Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
|- ( ( ( A e. Fin /\ B e. ( _V \ A ) ) /\ ( N e. om /\ A ~~ N ) ) -> ( A u. { B } ) ~~ suc N )
Theorem	php5fin 6339	A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
|- ( ( A e. Fin /\ B e. ( _V \ A ) ) -> -. A ~~ ( A u. { B } ) )
Theorem	fisbth 6340	Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
|- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Theorem	0fin 6341	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
|- (/) e. Fin
Theorem	fin0 6342*	A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
|- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )
Theorem	fin0or 6343*	A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )
Theorem	diffitest 6344*	If subtracting any set from a finite set gives a finite set, any proposition of the form -. ph is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove A e. Fin -> ( A \ B ) e. Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
|- A. a e. Fin A. b ( a \ b ) e. Fin   =>    |- ( -. ph \/ -. -. ph )
Theorem	findcard 6345*	Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = ( y \ { z } ) -> ( ph <-> ch ) )   &    |- ( x = y -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. Fin -> ( A. z e. y ch -> th ) )   =>    |- ( A e. Fin -> ta )
Theorem	findcard2 6346*	Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y u. { z } ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. Fin -> ( ch -> th ) )   =>    |- ( A e. Fin -> ta )
Theorem	findcard2s 6347*	Variation of findcard2 6346 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y u. { z } ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( ( y e. Fin /\ -. z e. y ) -> ( ch -> th ) )   =>    |- ( A e. Fin -> ta )
Theorem	findcard2d 6348*	Deduction version of findcard2 6346. If you also need y e. Fin (which doesn't come for free due to ssfiexmid 6336), use findcard2sd 6349 instead. (Contributed by SO, 16-Jul-2018.)
|- ( x = (/) -> ( ps <-> ch ) )   &    |- ( x = y -> ( ps <-> th ) )   &    |- ( x = ( y u. { z } ) -> ( ps <-> ta ) )   &    |- ( x = A -> ( ps <-> et ) )   &    |- ( ph -> ch )   &    |- ( ( ph /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( th -> ta ) )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> et )
Theorem	findcard2sd 6349*	Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
|- ( x = (/) -> ( ps <-> ch ) )   &    |- ( x = y -> ( ps <-> th ) )   &    |- ( x = ( y u. { z } ) -> ( ps <-> ta ) )   &    |- ( x = A -> ( ps <-> et ) )   &    |- ( ph -> ch )   &    |- ( ( ( ph /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( th -> ta ) )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> et )
Theorem	diffisn 6350	Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
|- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )
Theorem	diffifi 6351	Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> ( A \ B ) e. Fin )
Theorem	ac6sfi 6352*	Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
|- ( y = ( f ` x ) -> ( ph <-> ps ) )   =>    |- ( ( A e. Fin /\ A. x e. A E. y e. B ph ) -> E. f ( f : A --> B /\ A. x e. A ps ) )
Theorem	fientri3 6353	Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
Theorem	nnwetri 6354*	A natural number is well-ordered by _E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
|- ( A e. om -> ( _E We A /\ A. x e. A A. y e. A ( x _E y \/ x = y \/ y _E x ) ) )
Theorem	onunsnss 6355	Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
|- ( ( B e. V /\ ( A u. { B } ) e. On ) -> B C_ A )
Theorem	snon0 6356	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
2.6.28  Ordinal isomorphism
Theorem	ordiso2 6357	Generalize ordiso 6358 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
Theorem	ordiso 6358*	Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
|- ( ( A e. On /\ B e. On ) -> ( A = B <-> E. f f Isom _E , _E ( A , B ) ) )
2.6.29  Cardinal numbers
Syntax	ccrd 6359	Extend class definition to include the cardinal size function.
class card
Definition	df-card 6360*	Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
|- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
Theorem	cardcl 6361*	The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. y e. On y ~~ A -> ( card ` A ) e. On )
Theorem	isnumi 6362	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. On /\ A ~~ B ) -> B e. dom card )
Theorem	finnum 6363	Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A e. Fin -> A e. dom card )
Theorem	onenon 6364	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( A e. On -> A e. dom card )
Theorem	cardval3ex 6365*	The value of ( card ` A ). (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
Theorem	oncardval 6366*	The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
Theorem	cardonle 6367	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
|- ( A e. On -> ( card ` A ) C_ A )
Theorem	card0 6368	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
|- ( card ` (/) ) = (/)
Theorem	carden2bex 6369*	If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )
PART 3  REAL AND COMPLEX NUMBERS 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 6053 and similar theorems ), going from there to positive integers (df-ni 6402) and then positive rational numbers (df-nqqs 6446) 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".
3.1  Construction and axiomatization of real and complex numbers
3.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 6370	The set of positive integers, which is the set of natural numbers om with 0 removed. Note: This is the start of the Dedekind-cut construction of real and _complex numbers.
class N.
Syntax	cpli 6371	Positive integer addition.
class +N
Syntax	cmi 6372	Positive integer multiplication.
class .N
Syntax	clti 6373	Positive integer ordering relation.
class <N
Syntax	cplpq 6374	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 6375	Positive pre-fraction multiplication.
class .pQ
Syntax	cltpq 6376	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 6377	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 6378	Set of positive fractions.
class Q.
Syntax	c1q 6379	The positive fraction constant 1.
class 1Q
Syntax	cplq 6380	Positive fraction addition.
class +Q
Syntax	cmq 6381	Positive fraction multiplication.
class .Q
Syntax	crq 6382	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 6383	Positive fraction ordering relation.
class <Q
Syntax	ceq0 6384	Equivalence class used to construct non-negative fractions.
class ~Q0
Syntax	cnq0 6385	Set of non-negative fractions.
class Q0
Syntax	c0q0 6386	The non-negative fraction constant 0.
class 0Q0
Syntax	cplq0 6387	Non-negative fraction addition.
class +Q0
Syntax	cmq0 6388	Non-negative fraction multiplication.
class ·Q0
Syntax	cnp 6389	Set of positive reals.
class P.
Syntax	c1p 6390	Positive real constant 1.
class 1P
Syntax	cpp 6391	Positive real addition.
class +P.
Syntax	cmp 6392	Positive real multiplication.
class .P.
Syntax	cltp 6393	Positive real ordering relation.
class <P
Syntax	cer 6394	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 6395	Set of signed reals.
class R.
Syntax	c0r 6396	The signed real constant 0.
class 0R
Syntax	c1r 6397	The signed real constant 1.
class 1R
Syntax	cm1r 6398	The signed real constant -1.
class -1R
Syntax	cplr 6399	Signed real addition.
class +R
Syntax	cmr 6400	Signed real multiplication.
class .R