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	mulidnq 7501	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
|- ( A e. Q. -> ( A .Q 1Q ) = A )
Theorem	recexnq 7502*	Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
|- ( A e. Q. -> E. y ( y e. Q. /\ ( A .Q y ) = 1Q ) )
Theorem	recmulnqg 7503	Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( ( *Q ` A ) = B <-> ( A .Q B ) = 1Q ) )
Theorem	recclnq 7504	Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
|- ( A e. Q. -> ( *Q ` A ) e. Q. )
Theorem	recidnq 7505	A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
|- ( A e. Q. -> ( A .Q ( *Q ` A ) ) = 1Q )
Theorem	recrecnq 7506	Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.)
|- ( A e. Q. -> ( *Q ` ( *Q ` A ) ) = A )
Theorem	rec1nq 7507	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( *Q ` 1Q ) = 1Q
Theorem	nqtri3or 7508	Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B \/ A = B \/ B <Q A ) )
Theorem	ltdcnq 7509	Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> DECID A <Q B )
Theorem	ltsonq 7510	'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.)
|- <Q Or Q.
Theorem	nqtric 7511	Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> -. ( A = B \/ B <Q A ) ) )
Theorem	ltanqg 7512	Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) )
Theorem	ltmnqg 7513	Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
Theorem	ltanqi 7514	Ordering property of addition for positive fractions. One direction of ltanqg 7512. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) )
Theorem	ltmnqi 7515	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7513. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) )
Theorem	lt2addnq 7516	Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
|- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A +Q C ) <Q ( B +Q D ) ) )
Theorem	lt2mulnq 7517	Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A .Q C ) <Q ( B .Q D ) ) )
Theorem	1lt2nq 7518	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 7519	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 7520*	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 7521*	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 7522*	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 7523*	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 7524*	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 7525*	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 7526*	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 7522). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 7527*	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 7528*	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 7529*	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 7530*	A version of the Archimedean property. This variation is "stronger" than archnqq 7529 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 7531*	Like prarloclemarch 7530 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 7615. (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 7532	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7533. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) )
Theorem	ltrnqi 7533	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7532. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	nnnq 7534	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 7535	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 7536*	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 7537	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 7538	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 7539*	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 7540*	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 7541*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7540 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 7542*	Addition on nonnegative fractions. This definition is similar to df-plq0 7539 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 7543	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 7544	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7547. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f ~Q0 g -> g ~Q0 f )
Theorem	enq0ref 7545	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7547. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f e. ( om X. N. ) <-> f ~Q0 f )
Theorem	enq0tr 7546	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7547. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h )
Theorem	enq0er 7547	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q0 Er ( om X. N. )
Theorem	enq0breq 7548	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 7549	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 7550	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 7551	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- ~Q0 e. _V
Theorem	nq0ex 7552	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q0 e. _V
Theorem	nqnq0 7553	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q. C_ Q0
Theorem	nq0nn 7554*	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 7555	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 7556	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 7557	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 7558*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7561 and mulnnnq0 7562. (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 7559*	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 7560*	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 7561	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 7562	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 7563	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 7564	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 7565	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 7566	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 7567	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 7568	Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> (0Q0 ·Q0 A ) = 0Q0 )
Theorem	nq0a0 7569	Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
Theorem	nnanq0 7570	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 7571	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 7572	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 7573	A natural number closure law. Lemma for addassnq0 7574. (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 7574	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 7575	Multiplication of nonnegative fractions is distributive. Version of distrnq0 7571 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 7576	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 7577	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 7578*	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 7579*	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 7580*	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 7620. 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 7581*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7580 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 7582*	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 7583	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
|- P. C_ ( ~P Q. X. ~P Q. )
Theorem	preqlu 7584	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 7585	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
|- P. e. _V
Theorem	elinp 7586*	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 7587	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 7588*	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 7589*	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 7590*	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 7591	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 7592	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 7593	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 7594	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 7595	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
|- -. (/) e. P.
Theorem	prcdnql 7596	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 7597	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 7598	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 7599	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 7600*	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 )