P. 99 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltrnq 9801	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.)
|- ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	archnq 9802*	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.)
|- ( A e. Q. -> E. x e. N. A <Q <. x , 1o >. )
Definition	df-np 9803*	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 9942, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
|- P. = { x | ( ( (/) C. x /\ x C. Q. ) /\ A. y e. x ( A. z ( z <Q y -> z e. x ) /\ E. z e. x y <Q z ) ) }
Definition	df-1p 9804	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9942, 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 = { x | x <Q 1Q }
Definition	df-plp 9805*	Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, 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. = ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v +Q u ) } )
Definition	df-mp 9806*	Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, 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.)
|- .P. = ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v .Q u ) } )
Definition	df-ltp 9807*	Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, 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 = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ x C. y ) }
Theorem	npex 9808	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
|- P. e. _V
Theorem	elnp 9809*	Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
|- ( A e. P. <-> ( ( (/) C. A /\ A C. Q. ) /\ A. x e. A ( A. y ( y <Q x -> y e. A ) /\ E. y e. A x <Q y ) ) )
Theorem	elnpi 9810*	Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
|- ( A e. P. <-> ( ( A e. _V /\ (/) C. A /\ A C. Q. ) /\ A. x e. A ( A. y ( y <Q x -> y e. A ) /\ E. y e. A x <Q y ) ) )
Theorem	prn0 9811	A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
|- ( A e. P. -> A =/= (/) )
Theorem	prpssnq 9812	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.)
|- ( A e. P. -> A C. Q. )
Theorem	elprnq 9813	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.)
|- ( ( A e. P. /\ B e. A ) -> B e. Q. )
Theorem	0npr 9814	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
|- -. (/) e. P.
Theorem	prcdnq 9815	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.)
|- ( ( A e. P. /\ B e. A ) -> ( C <Q B -> C e. A ) )
Theorem	prub 9816	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.)
|- ( ( ( A e. P. /\ B e. A ) /\ C e. Q. ) -> ( -. C e. A -> B <Q C ) )
Theorem	prnmax 9817*	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.)
|- ( ( A e. P. /\ B e. A ) -> E. x e. A B <Q x )
Theorem	npomex 9818	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 RR, 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 9815 and nsmallnq 9799). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
|- ( A e. P. -> om e. _V )
Theorem	prnmadd 9819*	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.)
|- ( ( A e. P. /\ B e. A ) -> E. x ( B +Q x ) e. A )
Theorem	ltrelpr 9820	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
|- <P C_ ( P. X. P. )
Theorem	genpv 9821*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = { f | E. g e. A E. h e. B f = ( g G h ) } )
Theorem	genpelv 9822*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( A F B ) <-> E. g e. A E. h e. B C = ( g G h ) ) )
Theorem	genpprecl 9823*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( ( C e. A /\ D e. B ) -> ( C G D ) e. ( A F B ) ) )
Theorem	genpdm 9824*	Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- dom F = ( P. X. P. )
Theorem	genpn0 9825*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> (/) C. ( A F B ) )
Theorem	genpss 9826*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) C_ Q. )
Theorem	genpnnp 9827*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( z e. Q. -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )   &    |- ( x G y ) = ( y G x )   =>    |- ( ( A e. P. /\ B e. P. ) -> -. ( A F B ) = Q. )
Theorem	genpcd 9828*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( A F B ) ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( A F B ) -> ( x <Q f -> x e. ( A F B ) ) ) )
Theorem	genpnmax 9829*	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.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( v e. Q. -> ( z <Q w <-> ( v G z ) <Q ( v G w ) ) )   &    |- ( z G w ) = ( w G z )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( A F B ) -> E. x e. ( A F B ) f <Q x ) )
Theorem	genpcl 9830*	Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( h e. Q. -> ( f <Q g <-> ( h G f ) <Q ( h G g ) ) )   &    |- ( x G y ) = ( y G x )   &    |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( A F B ) ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
Theorem	genpass 9831*	Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- dom F = ( P. X. P. )   &    |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )   &    |- ( ( f G g ) G h ) = ( f G ( g G h ) )   =>    |- ( ( A F B ) F C ) = ( A F ( B F C ) )
Theorem	plpv 9832*	Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = { x | E. y e. A E. z e. B x = ( y +Q z ) } )
Theorem	mpv 9833*	Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = { x | E. y e. A E. z e. B x = ( y .Q z ) } )
Theorem	dmplp 9834	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
|- dom +P. = ( P. X. P. )
Theorem	dmmp 9835	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
|- dom .P. = ( P. X. P. )
Theorem	nqpr 9836*	The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- ( A e. Q. -> { x | x <Q A } e. P. )
Theorem	1pr 9837	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- 1P e. P.
Theorem	addclprlem1 9838	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.)
|- ( ( ( A e. P. /\ g e. A ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A ) )
Theorem	addclprlem2 9839*	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.)
|- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> x e. ( A +P. B ) ) )
Theorem	addclpr 9840	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.)
|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
Theorem	mulclprlem 9841*	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.)
|- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g .Q h ) -> x e. ( A .P. B ) ) )
Theorem	mulclpr 9842	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.)
|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
Theorem	addcompr 9843	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.)
|- ( A +P. B ) = ( B +P. A )
Theorem	addasspr 9844	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.)
|- ( ( A +P. B ) +P. C ) = ( A +P. ( B +P. C ) )
Theorem	mulcompr 9845	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.)
|- ( A .P. B ) = ( B .P. A )
Theorem	mulasspr 9846	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.)
|- ( ( A .P. B ) .P. C ) = ( A .P. ( B .P. C ) )
Theorem	distrlem1pr 9847	Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. ( B +P. C ) ) C_ ( ( A .P. B ) +P. ( A .P. C ) ) )
Theorem	distrlem4pr 9848*	Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. A /\ y e. B ) /\ ( f e. A /\ z e. C ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( A .P. ( B +P. C ) ) )
Theorem	distrlem5pr 9849	Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) C_ ( A .P. ( B +P. C ) ) )
Theorem	distrpr 9850	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.)
|- ( A .P. ( B +P. C ) ) = ( ( A .P. B ) +P. ( A .P. C ) )
Theorem	1idpr 9851	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.)
|- ( A e. P. -> ( A .P. 1P ) = A )
Theorem	ltprord 9852	Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> A C. B ) )
Theorem	psslinpr 9853	Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. ) -> ( A C. B \/ A = B \/ B C. A ) )
Theorem	ltsopr 9854	Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
|- <P Or P.
Theorem	prlem934 9855*	Lemma 9-3.4 of [Gleason] p. 122. (Contributed by NM, 25-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
|- B e. _V   =>    |- ( A e. P. -> E. x e. A -. ( x +Q B ) e. A )
Theorem	ltaddpr 9856	The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )
Theorem	ltaddpr2 9857	The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
|- ( C e. P. -> ( ( A +P. B ) = C -> A <P C ) )
Theorem	ltexprlem1 9858*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( B e. P. -> ( A C. B -> C =/= (/) ) )
Theorem	ltexprlem2 9859*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( B e. P. -> C C. Q. )
Theorem	ltexprlem3 9860*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( B e. P. -> ( x e. C -> A. z ( z <Q x -> z e. C ) ) )
Theorem	ltexprlem4 9861*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( B e. P. -> ( x e. C -> E. z ( z e. C /\ x <Q z ) ) )
Theorem	ltexprlem5 9862*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( ( B e. P. /\ A C. B ) -> C e. P. )
Theorem	ltexprlem6 9863*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( ( ( A e. P. /\ B e. P. ) /\ A C. B ) -> ( A +P. C ) C_ B )
Theorem	ltexprlem7 9864*	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- C = { x | E. y ( -. y e. A /\ ( y +Q x ) e. B ) }   =>    |- ( ( ( A e. P. /\ B e. P. ) /\ A C. B ) -> B C_ ( A +P. C ) )
Theorem	ltexpri 9865*	Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
|- ( A <P B -> E. x e. P. ( A +P. x ) = B )
Theorem	ltaprlem 9866	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
|- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
Theorem	ltapr 9867	Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
|- ( C e. P. -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )
Theorem	addcanpr 9868	Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
|- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. B ) = ( A +P. C ) -> B = C ) )
Theorem	prlem936 9869*	Lemma 9-3.6 of [Gleason] p. 124. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- ( ( A e. P. /\ 1Q <Q B ) -> E. x e. A -. ( x .Q B ) e. A )
Theorem	reclem2pr 9870*	Lemma for Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- B = { x | E. y ( x <Q y /\ -. ( *Q ` y ) e. A ) }   =>    |- ( A e. P. -> B e. P. )
Theorem	reclem3pr 9871*	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- B = { x | E. y ( x <Q y /\ -. ( *Q ` y ) e. A ) }   =>    |- ( A e. P. -> 1P C_ ( A .P. B ) )
Theorem	reclem4pr 9872*	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)
|- B = { x | E. y ( x <Q y /\ -. ( *Q ` y ) e. A ) }   =>    |- ( A e. P. -> ( A .P. B ) = 1P )
Theorem	recexpr 9873*	The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P )
Theorem	suplem1pr 9874*	The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- ( ( A =/= (/) /\ E. x e. P. A. y e. A y <P x ) -> U. A e. P. )
Theorem	suplem2pr 9875*	The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
|- ( A C_ P. -> ( ( y e. A -> -. U. A <P y ) /\ ( y <P U. A -> E. z e. A y <P z ) ) )
Theorem	supexpr 9876*	The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.)
|- ( ( A =/= (/) /\ E. x e. P. A. y e. A y <P x ) -> E. x e. P. ( A. y e. A -. x <P y /\ A. y e. P. ( y <P x -> E. z e. A y <P z ) ) )
Definition	df-enr 9877*	Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
|- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) }
Definition	df-nr 9878	Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
|- R. = ( ( P. X. P. ) /. ~R )
Definition	df-plr 9879*	Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
|- +R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) ) }
Definition	df-mr 9880*	Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
|- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. f ) ) , ( ( w .P. f ) +P. ( v .P. u ) ) >. ] ~R ) ) }
Definition	df-ltr 9881*	Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
|- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
Definition	df-0r 9882	Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
|- 0R = [ <. 1P , 1P >. ] ~R
Definition	df-1r 9883	Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
|- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
Definition	df-m1r 9884	Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
|- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
Theorem	enrbreq 9885	Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( <. A , B >. ~R <. C , D >. <-> ( A +P. D ) = ( B +P. C ) ) )
Theorem	enrer 9886	The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
|- ~R Er ( P. X. P. )
Theorem	enreceq 9887	Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R = [ <. C , D >. ] ~R <-> ( A +P. D ) = ( B +P. C ) ) )
Theorem	enrex 9888	The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
|- ~R e. _V
Theorem	ltrelsr 9889	Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
|- <R C_ ( R. X. R. )
Theorem	addcmpblnr 9890	Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( A +P. F ) , ( B +P. G ) >. ~R <. ( C +P. R ) , ( D +P. S ) >. ) )
Theorem	mulcmpblnrlem 9891	Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
|- ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> ( ( D .P. F ) +P. ( ( ( A .P. F ) +P. ( B .P. G ) ) +P. ( ( C .P. S ) +P. ( D .P. R ) ) ) ) = ( ( D .P. F ) +P. ( ( ( A .P. G ) +P. ( B .P. F ) ) +P. ( ( C .P. R ) +P. ( D .P. S ) ) ) ) )
Theorem	mulcmpblnr 9892	Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
|- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( ( F e. P. /\ G e. P. ) /\ ( R e. P. /\ S e. P. ) ) ) -> ( ( ( A +P. D ) = ( B +P. C ) /\ ( F +P. S ) = ( G +P. R ) ) -> <. ( ( A .P. F ) +P. ( B .P. G ) ) , ( ( A .P. G ) +P. ( B .P. F ) ) >. ~R <. ( ( C .P. R ) +P. ( D .P. S ) ) , ( ( C .P. S ) +P. ( D .P. R ) ) >. ) )
Theorem	prsrlem1 9893*	Decomposing signed reals into positive reals. Lemma for addsrpr 9896 and mulsrpr 9897. (Contributed by Jim Kingdon, 30-Dec-2019.)
|- ( ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) /\ ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ ( A = [ <. s , f >. ] ~R /\ B = [ <. g , h >. ] ~R ) ) ) -> ( ( ( ( w e. P. /\ v e. P. ) /\ ( s e. P. /\ f e. P. ) ) /\ ( ( u e. P. /\ t e. P. ) /\ ( g e. P. /\ h e. P. ) ) ) /\ ( ( w +P. f ) = ( v +P. s ) /\ ( u +P. h ) = ( t +P. g ) ) ) )
Theorem	addsrmo 9894*	There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
|- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
Theorem	mulsrmo 9895*	There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
|- ( ( A e. ( ( P. X. P. ) /. ~R ) /\ B e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~R /\ B = [ <. u , t >. ] ~R ) /\ z = [ <. ( ( w .P. u ) +P. ( v .P. t ) ) , ( ( w .P. t ) +P. ( v .P. u ) ) >. ] ~R ) )
Theorem	addsrpr 9896	Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
Theorem	mulsrpr 9897	Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )
Theorem	ltsrpr 9898	Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
|- ( [ <. A , B >. ] ~R <R [ <. C , D >. ] ~R <-> ( A +P. D ) <P ( B +P. C ) )
Theorem	gt0srpr 9899	Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
|- ( 0R <R [ <. A , B >. ] ~R <-> B <P A )
Theorem	0nsr 9900	The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
|- -. (/) e. R.