P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-rq 7301*	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 7302*	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 7303*	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 7304*	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 7305	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 7306	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 7307	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 7308	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 7309	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
|- ~Q e. _V
Theorem	enqdc 7310	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 7311	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 7312	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
|- Q. e. _V
Theorem	0nnq 7313	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 7314	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 7315	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
|- 1Q e. Q.
Theorem	addcmpblnq 7316	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 7317	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 7318	Lemma for addpipqqs 7319. (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 7319	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 7320	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 7321	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 7322	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 7323	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 7324	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 7325	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 7326*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7328. (Contributed by Jim Kingdon, 15-Sep-2019.)
|- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
Theorem	nqpi 7327*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7326 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 7328	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom +Q = ( Q. X. Q. )
Theorem	dmmulpq 7329	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom .Q = ( Q. X. Q. )
Theorem	addcomnqg 7330	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 7331	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 7332	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 7333	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 7334	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 7335	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 7336	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 7337	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
|- ( A e. N. -> 1Q = [ <. A , A >. ] ~Q )
Theorem	mulidnq 7338	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
|- ( A e. Q. -> ( A .Q 1Q ) = A )
Theorem	recexnq 7339*	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 7340	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 7341	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 7342	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 7343	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 7344	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( *Q ` 1Q ) = 1Q
Theorem	nqtri3or 7345	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 7346	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 7347	'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 7348	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 7349	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 7350	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 7351	Ordering property of addition for positive fractions. One direction of ltanqg 7349. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) )
Theorem	ltmnqi 7352	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7350. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) )
Theorem	lt2addnq 7353	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 7354	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 7355	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 7356	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 7357*	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 7358*	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 7359*	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 7360*	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 7361*	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 7362*	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 7363*	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 7359). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 7364*	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 7365*	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 7366*	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 7367*	A version of the Archimedean property. This variation is "stronger" than archnqq 7366 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 7368*	Like prarloclemarch 7367 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 7452. (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 7369	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7370. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) )
Theorem	ltrnqi 7370	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7369. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	nnnq 7371	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 7372	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 7373*	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 7374	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 7375	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 7376*	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 7377*	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 7378*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7377 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 7379*	Addition on nonnegative fractions. This definition is similar to df-plq0 7376 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 7380	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 7381	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7384. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f ~Q0 g -> g ~Q0 f )
Theorem	enq0ref 7382	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7384. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f e. ( om X. N. ) <-> f ~Q0 f )
Theorem	enq0tr 7383	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7384. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h )
Theorem	enq0er 7384	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q0 Er ( om X. N. )
Theorem	enq0breq 7385	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 7386	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 7387	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 7388	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- ~Q0 e. _V
Theorem	nq0ex 7389	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q0 e. _V
Theorem	nqnq0 7390	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q. C_ Q0
Theorem	nq0nn 7391*	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 7392	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 7393	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 7394	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 7395*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7398 and mulnnnq0 7399. (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 7396*	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 7397*	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 7398	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 7399	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 7400	Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
|- ( ( A e. Q0 /\ B e. Q0 ) -> ( A +Q0 B ) e. Q0 )