P. 75 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulpiord 7401	Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( A .o B ) )
Theorem	mulidpi 7402	1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( A e. N. -> ( A .N 1o ) = A )
Theorem	ltpiord 7403	Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )
Theorem	ltsopi 7404	Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
|- <N Or N.
Theorem	pitric 7405	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> -. ( A = B \/ B <N A ) ) )
Theorem	pitri3or 7406	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B \/ A = B \/ B <N A ) )
Theorem	ltdcpi 7407	Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. N. /\ B e. N. ) -> DECID A <N B )
Theorem	ltrelpi 7408	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
|- <N C_ ( N. X. N. )
Theorem	dmaddpi 7409	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom +N = ( N. X. N. )
Theorem	dmmulpi 7410	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom .N = ( N. X. N. )
Theorem	addclpi 7411	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) e. N. )
Theorem	mulclpi 7412	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) e. N. )
Theorem	addcompig 7413	Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) )
Theorem	addasspig 7414	Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +N B ) +N C ) = ( A +N ( B +N C ) ) )
Theorem	mulcompig 7415	Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( B .N A ) )
Theorem	mulasspig 7416	Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A .N B ) .N C ) = ( A .N ( B .N C ) ) )
Theorem	distrpig 7417	Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A .N ( B +N C ) ) = ( ( A .N B ) +N ( A .N C ) ) )
Theorem	addcanpig 7418	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 7419	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 7420	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 7421*	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 7422	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 7423	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 7424	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
|- 1o <N ( 1o +N 1o )
Theorem	nlt1pig 7425	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( A e. N. -> -. A <N 1o )
Theorem	indpi 7426*	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 7427	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 7428*	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 7433) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7431). (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 7429*	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 7430*	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 7431*	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 7432	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 7433*	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 7434*	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 7435	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 7436*	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 7437*	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 7438*	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 7439*	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 7440	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 7441	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 7442	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 7443	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 7444	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
|- ~Q e. _V
Theorem	enqdc 7445	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 7446	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 7447	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
|- Q. e. _V
Theorem	0nnq 7448	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 7449	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 7450	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
|- 1Q e. Q.
Theorem	addcmpblnq 7451	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 7452	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 7453	Lemma for addpipqqs 7454. (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 7454	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 7455	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 7456	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 7457	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 7458	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 7459	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 7460	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 7461*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7463. (Contributed by Jim Kingdon, 15-Sep-2019.)
|- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
Theorem	nqpi 7462*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7461 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 7463	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom +Q = ( Q. X. Q. )
Theorem	dmmulpq 7464	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom .Q = ( Q. X. Q. )
Theorem	addcomnqg 7465	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 7466	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 7467	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 7468	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 7469	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 7470	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 7471	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 7472	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
|- ( A e. N. -> 1Q = [ <. A , A >. ] ~Q )
Theorem	mulidnq 7473	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
|- ( A e. Q. -> ( A .Q 1Q ) = A )
Theorem	recexnq 7474*	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 7475	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 7476	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 7477	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 7478	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 7479	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( *Q ` 1Q ) = 1Q
Theorem	nqtri3or 7480	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 7481	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 7482	'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 7483	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 7484	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 7485	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 7486	Ordering property of addition for positive fractions. One direction of ltanqg 7484. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) )
Theorem	ltmnqi 7487	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7485. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) )
Theorem	lt2addnq 7488	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 7489	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 7490	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 7491	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 7492*	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 7493*	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 7494*	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 7495*	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 7496*	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 7497*	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 7498*	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 7494). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 7499*	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 7500*	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 ) )