P. 67 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	halfnq 6601*	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.)
|- ( A e. Q. -> E. x ( x +Q x ) = A )
Theorem	nsmallnqq 6602*	There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A e. Q. -> E. x e. Q. x <Q A )
Theorem	nsmallnq 6603*	There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.)
|- ( A e. Q. -> E. x x <Q A )
Theorem	subhalfnqq 6604*	There is a number which is less than half of any positive fraction. The case where A 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 6600). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 6605*	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.)
|- ( A <Q B <-> E. x e. Q. ( A <Q x /\ x <Q B ) )
Theorem	ltbtwnnq 6606*	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.)
|- ( A <Q B <-> E. x ( A <Q x /\ x <Q B ) )
Theorem	archnqq 6607*	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.)
|- ( A e. Q. -> E. x e. N. A <Q [ <. x , 1o >. ] ~Q )
Theorem	prarloclemarch 6608*	A version of the Archimedean property. This variation is "stronger" than archnqq 6607 in the sense that we provide an integer which is larger than a given rational A even after being multiplied by a second rational B. (Contributed by Jim Kingdon, 30-Nov-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> E. x e. N. A <Q ( [ <. x , 1o >. ] ~Q .Q B ) )
Theorem	prarloclemarch2 6609*	Like prarloclemarch 6608 but the integer must be at least two, and there is also B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6693. (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) )
Theorem	ltrnqg 6610	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6611. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) )
Theorem	ltrnqi 6611	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6610. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	nnnq 6612	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
|- ( A e. N. -> [ <. A , 1o >. ] ~Q e. Q. )
Theorem	ltnnnq 6613	Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> [ <. A , 1o >. ] ~Q <Q [ <. B , 1o >. ] ~Q ) )
Definition	df-enq0 6614*	Define equivalence relation for non-negative 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 = { <. x , y >. | ( ( x e. ( om X. N. ) /\ y e. ( om X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .o u ) = ( w .o v ) ) ) }
Definition	df-nq0 6615	Define class of non-negative 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 = ( ( om X. N. ) /. ~Q0 )
Definition	df-0nq0 6616	Define non-negative 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.)
|- 0Q0 = [ <. (/) , 1o >. ] ~Q0
Definition	df-plq0 6617*	Define addition on non-negative 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 = { <. <. x , y >. , z >. | ( ( x e. Q0 /\ y e. Q0 ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 ) ) }
Definition	df-mq0 6618*	Define multiplication on non-negative 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 = { <. <. x , y >. , z >. | ( ( x e. Q0 /\ y e. Q0 ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( w .o u ) , ( v .o f ) >. ] ~Q0 ) ) }
Theorem	dfmq0qs 6619*	Multiplication on non-negative fractions. This definition is similar to df-mq0 6618 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
|- ·Q0 = { <. <. x , y >. , z >. | ( ( x e. ( ( om X. N. ) /. ~Q0 ) /\ y e. ( ( om X. N. ) /. ~Q0 ) ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( w .o u ) , ( v .o f ) >. ] ~Q0 ) ) }
Theorem	dfplq0qs 6620*	Addition on non-negative fractions. This definition is similar to df-plq0 6617 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
|- +Q0 = { <. <. x , y >. , z >. | ( ( x e. ( ( om X. N. ) /. ~Q0 ) /\ y e. ( ( om X. N. ) /. ~Q0 ) ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 ) ) }
Theorem	enq0enq 6621	Equivalence on positive fractions in terms of equivalence on non-negative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q = ( ~Q0 i^i ( ( N. X. N. ) X. ( N. X. N. ) ) )
Theorem	enq0sym 6622	The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6625. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f ~Q0 g -> g ~Q0 f )
Theorem	enq0ref 6623	The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6625. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f e. ( om X. N. ) <-> f ~Q0 f )
Theorem	enq0tr 6624	The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6625. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h )
Theorem	enq0er 6625	The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q0 Er ( om X. N. )
Theorem	enq0breq 6626	Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
|- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( <. A , B >. ~Q0 <. C , D >. <-> ( A .o D ) = ( B .o C ) ) )
Theorem	enq0eceq 6627	Equivalence class equality of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
|- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 = [ <. C , D >. ] ~Q0 <-> ( A .o D ) = ( B .o C ) ) )
Theorem	nqnq0pi 6628	A non-negative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( A e. N. /\ B e. N. ) -> [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q )
Theorem	enq0ex 6629	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- ~Q0 e. _V
Theorem	nq0ex 6630	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q0 e. _V
Theorem	nqnq0 6631	A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q. C_ Q0
Theorem	nq0nn 6632*	Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
|- ( A e. Q0 -> E. w E. v ( ( w e. om /\ v e. N. ) /\ A = [ <. w , v >. ] ~Q0 ) )
Theorem	addcmpblnq0 6633	Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
|- ( ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) /\ ( ( F e. om /\ G e. N. ) /\ ( R e. om /\ S e. N. ) ) ) -> ( ( ( A .o D ) = ( B .o C ) /\ ( F .o S ) = ( G .o R ) ) -> <. ( ( A .o G ) +o ( B .o F ) ) , ( B .o G ) >. ~Q0 <. ( ( C .o S ) +o ( D .o R ) ) , ( D .o S ) >. ) )
Theorem	mulcmpblnq0 6634	Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
|- ( ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) /\ ( ( F e. om /\ G e. N. ) /\ ( R e. om /\ S e. N. ) ) ) -> ( ( ( A .o D ) = ( B .o C ) /\ ( F .o S ) = ( G .o R ) ) -> <. ( A .o F ) , ( B .o G ) >. ~Q0 <. ( C .o R ) , ( D .o S ) >. ) )
Theorem	mulcanenq0ec 6635	Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( ( A e. N. /\ B e. om /\ C e. N. ) -> [ <. ( A .o B ) , ( A .o C ) >. ] ~Q0 = [ <. B , C >. ] ~Q0 )
Theorem	nnnq0lem1 6636*	Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6639 and mulnnnq0 6640. (Contributed by Jim Kingdon, 23-Nov-2019.)
|- ( ( ( A e. ( ( om X. N. ) /. ~Q0 ) /\ B e. ( ( om X. N. ) /. ~Q0 ) ) /\ ( ( ( A = [ <. w , v >. ] ~Q0 /\ B = [ <. u , t >. ] ~Q0 ) /\ z = [ C ] ~Q0 ) /\ ( ( A = [ <. s , f >. ] ~Q0 /\ B = [ <. g , h >. ] ~Q0 ) /\ q = [ D ] ~Q0 ) ) ) -> ( ( ( ( w e. om /\ v e. N. ) /\ ( s e. om /\ f e. N. ) ) /\ ( ( u e. om /\ t e. N. ) /\ ( g e. om /\ h e. N. ) ) ) /\ ( ( w .o f ) = ( v .o s ) /\ ( u .o h ) = ( t .o g ) ) ) )
Theorem	addnq0mo 6637*	There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
|- ( ( A e. ( ( om X. N. ) /. ~Q0 ) /\ B e. ( ( om X. N. ) /. ~Q0 ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~Q0 /\ B = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) )
Theorem	mulnq0mo 6638*	There is at most one result from multiplying non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
|- ( ( A e. ( ( om X. N. ) /. ~Q0 ) /\ B e. ( ( om X. N. ) /. ~Q0 ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] ~Q0 /\ B = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) )
Theorem	addnnnq0 6639	Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
|- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 +Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 )
Theorem	mulnnnq0 6640	Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
|- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 ·Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 )
Theorem	addclnq0 6641	Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A +Q0 B ) e. Q0 )
Theorem	mulclnq0 6642	Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A ·Q0 B ) e. Q0 )
Theorem	nqpnq0nq 6643	A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
|- ( ( A e. Q. /\ B e. Q0 ) -> ( A +Q0 B ) e. Q. )
Theorem	nqnq0a 6644	Addition of positive fractions is equal with +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( A +Q0 B ) )
Theorem	nqnq0m 6645	Multiplication of positive fractions is equal with .Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( A ·Q0 B ) )
Theorem	nq0m0r 6646	Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> (0Q0 ·Q0 A ) = 0Q0 )
Theorem	nq0a0 6647	Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
Theorem	nnanq0 6648	Addition of non-negative 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.)
|- ( ( N e. om /\ M e. om /\ A e. N. ) -> [ <. ( N +o M ) , A >. ] ~Q0 = ( [ <. N , A >. ] ~Q0 +Q0 [ <. M , A >. ] ~Q0 ) )
Theorem	distrnq0 6649	Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 /\ C e. Q0 ) -> ( A ·Q0 ( B +Q0 C ) ) = ( ( A ·Q0 B ) +Q0 ( A ·Q0 C ) ) )
Theorem	mulcomnq0 6650	Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A ·Q0 B ) = ( B ·Q0 A ) )
Theorem	addassnq0lemcl 6651	A natural number closure law. Lemma for addassnq0 6652. (Contributed by Jim Kingdon, 3-Dec-2019.)
|- ( ( ( I e. om /\ J e. N. ) /\ ( K e. om /\ L e. N. ) ) -> ( ( ( I .o L ) +o ( J .o K ) ) e. om /\ ( J .o L ) e. N. ) )
Theorem	addassnq0 6652	Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 /\ C e. Q0 ) -> ( ( A +Q0 B ) +Q0 C ) = ( A +Q0 ( B +Q0 C ) ) )
Theorem	distnq0r 6653	Multiplication of non-negative fractions is distributive. Version of distrnq0 6649 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 /\ C e. Q0 ) -> ( ( B +Q0 C ) ·Q0 A ) = ( ( B ·Q0 A ) +Q0 ( C ·Q0 A ) ) )
Theorem	addpinq1 6654	Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
|- ( A e. N. -> [ <. ( A +N 1o ) , 1o >. ] ~Q = ( [ <. A , 1o >. ] ~Q +Q 1Q ) )
Theorem	nq02m 6655	Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( A e. Q0 -> ( [ <. 2o , 1o >. ] ~Q0 ·Q0 A ) = ( A +Q0 A ) )
Definition	df-inp 6656*	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 l and an upper set u which is inhabited ( E. q e. Q. q e. l /\ E. r e. Q. r e. u), rounded ( A. q e. Q. ( q e. l <-> E. r e. Q. ( q <Q r /\ r e. l ) ) and likewise for u), disjoint ( A. q e. Q. -. ( q e. l /\ q e. u )) and located ( A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. l \/ r e. u ) )). 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.)
|- P. = { <. l , u >. | ( ( ( l C_ Q. /\ u C_ Q. ) /\ ( E. q e. Q. q e. l /\ E. r e. Q. r e. u ) ) /\ ( ( A. q e. Q. ( q e. l <-> E. r e. Q. ( q <Q r /\ r e. l ) ) /\ A. r e. Q. ( r e. u <-> E. q e. Q. ( q <Q r /\ q e. u ) ) ) /\ A. q e. Q. -. ( q e. l /\ q e. u ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. l \/ r e. u ) ) ) ) }
Definition	df-i1p 6657*	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.)
|- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
Definition	df-iplp 6658*	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, r e. ( 1st ` x ) implies r e. Q.) and can be simplified as shown at genpdf 6698. 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.)
|- +P. = ( x e. P. , y e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r +Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r +Q s ) ) } >. )
Definition	df-imp 6659*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6658 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.)
|- .P. = ( x e. P. , y e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >. )
Definition	df-iltp 6660*	Define ordering on positive reals. We define x <P y if there is a positive fraction q which is an element of the upper cut of x and the lower cut of y. 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.)
|- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
Theorem	npsspw 6661	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- P. C_ ( ~P Q. X. ~P Q. )
Theorem	preqlu 6662	Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
Theorem	npex 6663	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
|- P. e. _V
Theorem	elinp 6664*	Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- ( <. L , U >. e. P. <-> ( ( ( L C_ Q. /\ U C_ Q. ) /\ ( E. q e. Q. q e. L /\ E. r e. Q. r e. U ) ) /\ ( ( A. q e. Q. ( q e. L <-> E. r e. Q. ( q <Q r /\ r e. L ) ) /\ A. r e. Q. ( r e. U <-> E. q e. Q. ( q <Q r /\ q e. U ) ) ) /\ A. q e. Q. -. ( q e. L /\ q e. U ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. L \/ r e. U ) ) ) ) )
Theorem	prop 6665	A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
Theorem	elnp1st2nd 6666*	Membership in positive reals, using 1st and 2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
|- ( A e. P. <-> ( ( A e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` A ) /\ E. r e. Q. r e. ( 2nd ` A ) ) ) /\ ( ( A. q e. Q. ( q e. ( 1st ` A ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` A ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` A ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` A ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` A ) \/ r e. ( 2nd ` A ) ) ) ) ) )
Theorem	prml 6667*	A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- ( <. L , U >. e. P. -> E. x e. Q. x e. L )
Theorem	prmu 6668*	A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- ( <. L , U >. e. P. -> E. x e. Q. x e. U )
Theorem	prssnql 6669	The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
|- ( <. L , U >. e. P. -> L C_ Q. )
Theorem	prssnqu 6670	The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
|- ( <. L , U >. e. P. -> U C_ Q. )
Theorem	elprnql 6671	An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
|- ( ( <. L , U >. e. P. /\ B e. L ) -> B e. Q. )
Theorem	elprnqu 6672	An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
|- ( ( <. L , U >. e. P. /\ B e. U ) -> B e. Q. )
Theorem	0npr 6673	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
|- -. (/) e. P.
Theorem	prcdnql 6674	A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
|- ( ( <. L , U >. e. P. /\ B e. L ) -> ( C <Q B -> C e. L ) )
Theorem	prcunqu 6675	An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( ( <. L , U >. e. P. /\ C e. U ) -> ( C <Q B -> B e. U ) )
Theorem	prubl 6676	A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
|- ( ( ( <. L , U >. e. P. /\ B e. L ) /\ C e. Q. ) -> ( -. C e. L -> B <Q C ) )
Theorem	prltlu 6677	An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.)
|- ( ( <. L , U >. e. P. /\ B e. L /\ C e. U ) -> B <Q C )
Theorem	prnmaxl 6678*	A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
|- ( ( <. L , U >. e. P. /\ B e. L ) -> E. x e. L B <Q x )
Theorem	prnminu 6679*	An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
|- ( ( <. L , U >. e. P. /\ B e. U ) -> E. x e. U x <Q B )
Theorem	prnmaddl 6680*	A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
|- ( ( <. L , U >. e. P. /\ B e. L ) -> E. x e. Q. ( B +Q x ) e. L )
Theorem	prloc 6681	A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
|- ( ( <. L , U >. e. P. /\ A <Q B ) -> ( A e. L \/ B e. U ) )
Theorem	prdisj 6682	A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- ( ( <. L , U >. e. P. /\ A e. Q. ) -> -. ( A e. L /\ A e. U ) )
Theorem	prarloclemlt 6683	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ y e. om ) -> ( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) <Q ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) )
Theorem	prarloclemlo 6684*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ y e. om ) -> ( ( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) e. L -> ( ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) )
Theorem	prarloclemup 6685	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ y e. om ) -> ( ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U -> ( ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) )
Theorem	prarloclem3step 6686*	Induction step for prarloclem3 6687. (Contributed by Jim Kingdon, 9-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclem3 6687*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 27-Oct-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( X e. om /\ P e. Q. ) /\ E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) -> E. j e. om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclem4 6688*	A slight rearrangement of prarloclem3 6687. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 4-Nov-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ P e. Q. ) -> ( E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. j e. om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) ) )
Theorem	prarloclemn 6689*	Subtracting two from a positive integer. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
Theorem	prarloclem5 6690*	A substitution of zero for y and N minus two for x. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 4-Nov-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclem 6691*	A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from A to A +Q ( N .Q P ) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. j e. om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclemcalc 6692	Some calculations for prarloc 6693. (Contributed by Jim Kingdon, 26-Oct-2019.)
|- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B <Q ( A +Q P ) )
Theorem	prarloc 6693*	A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance P, there are elements of the lower and upper cut which are within that tolerance of each other. Usually, proofs will be shorter if they use prarloc2 6694 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)
|- ( ( <. L , U >. e. P. /\ P e. Q. ) -> E. a e. L E. b e. U b <Q ( a +Q P ) )
Theorem	prarloc2 6694*	A Dedekind cut is arithmetically located. This is a variation of prarloc 6693 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance P, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
|- ( ( <. L , U >. e. P. /\ P e. Q. ) -> E. a e. L ( a +Q P ) e. U )
Theorem	ltrelpr 6695	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
|- <P C_ ( P. X. P. )
Theorem	ltdfpr 6696*	More convenient form of df-iltp 6660. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) )
Theorem	genpdflem 6697*	Simplification of upper or lower cut expression. Lemma for genpdf 6698. (Contributed by Jim Kingdon, 30-Sep-2019.)
|- ( ( ph /\ r e. A ) -> r e. Q. )   &    |- ( ( ph /\ s e. B ) -> s e. Q. )   =>    |- ( ph -> { q e. Q. | E. r e. Q. E. s e. Q. ( r e. A /\ s e. B /\ q = ( r G s ) ) } = { q e. Q. | E. r e. A E. s e. B q = ( r G s ) } )
Theorem	genpdf 6698*	Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
|- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. )   =>    |- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } >. )
Theorem	genipv 6699*	Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` 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 ) = <. { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) } >. )
Theorem	genplt2i 6700*	Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
|- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )   &    |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )   =>    |- ( ( A <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G D ) )