P. 78 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	addcanpig 7701	Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +N B ) = ( A +N C ) <-> B = C ) )
Theorem	mulcanpig 7702	Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A .N B ) = ( A .N C ) <-> B = C ) )
Theorem	addnidpig 7703	There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
|- ( ( A e. N. /\ B e. N. ) -> -. ( A +N B ) = A )
Theorem	ltexpi 7704*	Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> E. x e. N. ( A +N x ) = B ) )
Theorem	ltapig 7705	Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A <N B <-> ( C +N A ) <N ( C +N B ) ) )
Theorem	ltmpig 7706	Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A <N B <-> ( C .N A ) <N ( C .N B ) ) )
Theorem	1lt2pi 7707	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
|- 1o <N ( 1o +N 1o )
Theorem	nlt1pig 7708	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( A e. N. -> -. A <N 1o )
Theorem	indpi 7709*	Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
|- ( x = 1o -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y +N 1o ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. N. -> ( ch -> th ) )   =>    |- ( A e. N. -> ta )
Theorem	nnppipi 7710	A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( A e. om /\ B e. N. ) -> ( A +o B ) e. N. )
Definition	df-plpq 7711*	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +Q (df-plqqs 7716) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7714). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.)
|- +pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
Definition	df-mpq 7712*	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.)
|- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
Definition	df-ltpq 7713*	Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.)
|- <pQ = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) }
Definition	df-enq 7714*	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.)
|- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }
Definition	df-nqqs 7715	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.)
|- Q. = ( ( N. X. N. ) /. ~Q )
Definition	df-plqqs 7716*	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.)
|- +Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. +pQ <. u , f >. ) ] ~Q ) ) }
Definition	df-mqqs 7717*	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.)
|- .Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) ) }
Definition	df-1nqqs 7718	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.)
|- 1Q = [ <. 1o , 1o >. ] ~Q
Definition	df-rq 7719*	Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by Jim Kingdon, 20-Sep-2019.)
|- *Q = { <. x , y >. | ( x e. Q. /\ y e. Q. /\ ( x .Q y ) = 1Q ) }
Definition	df-ltnqqs 7720*	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.)
|- <Q = { <. x , y >. | ( ( x e. Q. /\ y e. Q. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~Q /\ y = [ <. v , u >. ] ~Q ) /\ ( z .N u ) <N ( w .N v ) ) ) }
Theorem	dfplpq2 7721*	Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
|- +pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) }
Theorem	dfmpq2 7722*	Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
|- .pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) }
Theorem	enqbreq 7723	Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. ~Q <. C , D >. <-> ( A .N D ) = ( B .N C ) ) )
Theorem	enqbreq2 7724	Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
|- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
Theorem	enqer 7725	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
|- ~Q Er ( N. X. N. )
Theorem	enqeceq 7726	Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q = [ <. C , D >. ] ~Q <-> ( A .N D ) = ( B .N C ) ) )
Theorem	enqex 7727	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
|- ~Q e. _V
Theorem	enqdc 7728	The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> DECID <. A , B >. ~Q <. C , D >. )
Theorem	enqdc1 7729	The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
|- ( ( ( A e. N. /\ B e. N. ) /\ C e. ( N. X. N. ) ) -> DECID <. A , B >. ~Q C )
Theorem	nqex 7730	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
|- Q. e. _V
Theorem	0nnq 7731	The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
|- -. (/) e. Q.
Theorem	ltrelnq 7732	Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.)
|- <Q C_ ( Q. X. Q. )
Theorem	1nq 7733	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
|- 1Q e. Q.
Theorem	addcmpblnq 7734	Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.)
|- ( ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) /\ ( ( F e. N. /\ G e. N. ) /\ ( R e. N. /\ S e. N. ) ) ) -> ( ( ( A .N D ) = ( B .N C ) /\ ( F .N S ) = ( G .N R ) ) -> <. ( ( A .N G ) +N ( B .N F ) ) , ( B .N G ) >. ~Q <. ( ( C .N S ) +N ( D .N R ) ) , ( D .N S ) >. ) )
Theorem	mulcmpblnq 7735	Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.)
|- ( ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) /\ ( ( F e. N. /\ G e. N. ) /\ ( R e. N. /\ S e. N. ) ) ) -> ( ( ( A .N D ) = ( B .N C ) /\ ( F .N S ) = ( G .N R ) ) -> <. ( A .N F ) , ( B .N G ) >. ~Q <. ( C .N R ) , ( D .N S ) >. ) )
Theorem	addpipqqslem 7736	Lemma for addpipqqs 7737. (Contributed by Jim Kingdon, 11-Sep-2019.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> <. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. e. ( N. X. N. ) )
Theorem	addpipqqs 7737	Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q +Q [ <. C , D >. ] ~Q ) = [ <. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. ] ~Q )
Theorem	mulpipq2 7738	Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
|- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
Theorem	mulpipq 7739	Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. .pQ <. C , D >. ) = <. ( A .N C ) , ( B .N D ) >. )
Theorem	mulpipqqs 7740	Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q .Q [ <. C , D >. ] ~Q ) = [ <. ( A .N C ) , ( B .N D ) >. ] ~Q )
Theorem	ordpipqqs 7741	Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q <Q [ <. C , D >. ] ~Q <-> ( A .N D ) <N ( B .N C ) ) )
Theorem	addclnq 7742	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
Theorem	mulclnq 7743	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. )
Theorem	dmaddpqlem 7744*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7746. (Contributed by Jim Kingdon, 15-Sep-2019.)
|- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
Theorem	nqpi 7745*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7744 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)
|- ( A e. Q. -> E. w E. v ( ( w e. N. /\ v e. N. ) /\ A = [ <. w , v >. ] ~Q ) )
Theorem	dmaddpq 7746	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom +Q = ( Q. X. Q. )
Theorem	dmmulpq 7747	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom .Q = ( Q. X. Q. )
Theorem	addcomnqg 7748	Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) )
Theorem	addassnqg 7749	Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +Q B ) +Q C ) = ( A +Q ( B +Q C ) ) )
Theorem	mulcomnqg 7750	Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( B .Q A ) )
Theorem	mulassnqg 7751	Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A .Q B ) .Q C ) = ( A .Q ( B .Q C ) ) )
Theorem	mulcanenq 7752	Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> <. ( A .N B ) , ( A .N C ) >. ~Q <. B , C >. )
Theorem	mulcanenqec 7753	Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> [ <. ( A .N B ) , ( A .N C ) >. ] ~Q = [ <. B , C >. ] ~Q )
Theorem	distrnqg 7754	Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.)
|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A .Q ( B +Q C ) ) = ( ( A .Q B ) +Q ( A .Q C ) ) )
Theorem	1qec 7755	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
|- ( A e. N. -> 1Q = [ <. A , A >. ] ~Q )
Theorem	mulidnq 7756	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
|- ( A e. Q. -> ( A .Q 1Q ) = A )
Theorem	recexnq 7757*	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 7758	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 7759	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 7760	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 7761	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 7762	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( *Q ` 1Q ) = 1Q
Theorem	nqtri3or 7763	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 7764	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 7765	'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 7766	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 7767	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 7768	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 7769	Ordering property of addition for positive fractions. One direction of ltanqg 7767. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) )
Theorem	ltmnqi 7770	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7768. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) )
Theorem	lt2addnq 7771	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 7772	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 7773	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 7774	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 7775*	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 7776*	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 7777*	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 7778*	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 7779*	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 7780*	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 7781*	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 7777). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 7782*	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 7783*	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 7784*	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 7785*	A version of the Archimedean property. This variation is "stronger" than archnqq 7784 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 7786*	Like prarloclemarch 7785 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 7870. (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 7787	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7788. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) )
Theorem	ltrnqi 7788	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7787. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	nnnq 7789	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 7790	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 7791*	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 7792	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 7793	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 7794*	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 7795*	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 7796*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7795 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 7797*	Addition on nonnegative fractions. This definition is similar to df-plq0 7794 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 7798	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 7799	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7802. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f ~Q0 g -> g ~Q0 f )
Theorem	enq0ref 7800	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7802. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f e. ( om X. N. ) <-> f ~Q0 f )