P. 76 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1lt2nq 7501	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
|- 1Q <Q ( 1Q +Q 1Q )
Theorem	ltaddnq 7502	The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
|- ( ( A e. Q. /\ B e. Q. ) -> A <Q ( A +Q B ) )
Theorem	ltexnqq 7503*	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> E. x e. Q. ( A +Q x ) = B ) )
Theorem	ltexnqi 7504*	Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
|- ( A <Q B -> E. x e. Q. ( A +Q x ) = B )
Theorem	halfnqq 7505*	One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) = A )
Theorem	halfnq 7506*	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 7507*	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 7508*	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 7509*	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 7505). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 7510*	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 7511*	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 7512*	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 7513*	A version of the Archimedean property. This variation is "stronger" than archnqq 7512 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 7514*	Like prarloclemarch 7513 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 7598. (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 7515	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7516. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) )
Theorem	ltrnqi 7516	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7515. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	nnnq 7517	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 7518	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 7519*	Define equivalence relation for nonnegative 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 7520	Define class of nonnegative 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 7521	Define nonnegative 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 7522*	Define addition on nonnegative 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 7523*	Define multiplication on nonnegative 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 7524*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7523 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 7525*	Addition on nonnegative fractions. This definition is similar to df-plq0 7522 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 7526	Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q = ( ~Q0 i^i ( ( N. X. N. ) X. ( N. X. N. ) ) )
Theorem	enq0sym 7527	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7530. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f ~Q0 g -> g ~Q0 f )
Theorem	enq0ref 7528	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7530. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f e. ( om X. N. ) <-> f ~Q0 f )
Theorem	enq0tr 7529	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7530. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h )
Theorem	enq0er 7530	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q0 Er ( om X. N. )
Theorem	enq0breq 7531	Equivalence relation for nonnegative 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 7532	Equivalence class equality of nonnegative 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 7533	A nonnegative 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 7534	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- ~Q0 e. _V
Theorem	nq0ex 7535	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q0 e. _V
Theorem	nqnq0 7536	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q. C_ Q0
Theorem	nq0nn 7537*	Decomposition of a nonnegative 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 7538	Lemma showing compatibility of addition on nonnegative 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 7539	Lemma showing compatibility of multiplication on nonnegative 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 7540	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 7541*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7544 and mulnnnq0 7545. (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 7542*	There is at most one result from adding nonnegative 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 7543*	There is at most one result from multiplying nonnegative 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 7544	Addition of nonnegative 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 7545	Multiplication of nonnegative 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 7546	Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A +Q0 B ) e. Q0 )
Theorem	mulclnq0 7547	Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A ·Q0 B ) e. Q0 )
Theorem	nqpnq0nq 7548	A positive fraction plus a nonnegative 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 7549	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 7550	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 7551	Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> (0Q0 ·Q0 A ) = 0Q0 )
Theorem	nq0a0 7552	Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
Theorem	nnanq0 7553	Addition of nonnegative 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 7554	Multiplication of nonnegative 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 7555	Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A ·Q0 B ) = ( B ·Q0 A ) )
Theorem	addassnq0lemcl 7556	A natural number closure law. Lemma for addassnq0 7557. (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 7557	Addition of nonnegative 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 7558	Multiplication of nonnegative fractions is distributive. Version of distrnq0 7554 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 7559	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 7560	Multiply a nonnegative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( A e. Q0 -> ( [ <. 2o , 1o >. ] ~Q0 ·Q0 A ) = ( A +Q0 A ) )
Definition	df-inp 7561*	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 7562*	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 7563*	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 7603. 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 7564*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7563 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 7565*	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 7566	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- P. C_ ( ~P Q. X. ~P Q. )
Theorem	preqlu 7567	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 7568	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
|- P. e. _V
Theorem	elinp 7569*	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 7570	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 7571*	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 7572*	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 7573*	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 7574	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 7575	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 7576	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 7577	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 7578	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
|- -. (/) e. P.
Theorem	prcdnql 7579	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 7580	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 7581	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 7582	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 7583*	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 7584*	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 7585*	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 7586	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 7587	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 7588	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7598. (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 7589*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7598. (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 7590	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7598. (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 7591*	Induction step for prarloclem3 7592. (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 7592*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7598. (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 7593*	A slight rearrangement of prarloclem3 7592. Lemma for prarloc 7598. (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 7594*	Subtracting two from a positive integer. Lemma for prarloc 7598. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
Theorem	prarloclem5 7595*	A substitution of zero for y and N minus two for x. Lemma for prarloc 7598. (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 7596*	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 7597	Some calculations for prarloc 7598. (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 7598*	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 7599 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 7599*	A Dedekind cut is arithmetically located. This is a variation of prarloc 7598 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 7600	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
|- <P C_ ( P. X. P. )