P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	addpipqqslem 7201	Lemma for addpipqqs 7202. (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 7202	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 7203	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 7204	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 7205	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 7206	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 7207	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 7208	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 7209*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7211. (Contributed by Jim Kingdon, 15-Sep-2019.)
|- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )
Theorem	nqpi 7210*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7209 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 7211	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom +Q = ( Q. X. Q. )
Theorem	dmmulpq 7212	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
|- dom .Q = ( Q. X. Q. )
Theorem	addcomnqg 7213	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 7214	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 7215	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 7216	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 7217	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 7218	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 7219	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 7220	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
|- ( A e. N. -> 1Q = [ <. A , A >. ] ~Q )
Theorem	mulidnq 7221	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
|- ( A e. Q. -> ( A .Q 1Q ) = A )
Theorem	recexnq 7222*	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 7223	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 7224	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 7225	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 7226	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 7227	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( *Q ` 1Q ) = 1Q
Theorem	nqtri3or 7228	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 7229	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 7230	'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 7231	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 7232	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 7233	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 7234	Ordering property of addition for positive fractions. One direction of ltanqg 7232. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C +Q A ) <Q ( C +Q B ) )
Theorem	ltmnqi 7235	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7233. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> ( C .Q A ) <Q ( C .Q B ) )
Theorem	lt2addnq 7236	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 7237	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 7238	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 7239	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 7240*	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 7241*	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 7242*	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 7243*	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 7244*	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 7245*	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 7246*	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 7242). (Contributed by Jim Kingdon, 25-Nov-2019.)
|- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )
Theorem	ltbtwnnqq 7247*	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 7248*	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 7249*	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 7250*	A version of the Archimedean property. This variation is "stronger" than archnqq 7249 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 7251*	Like prarloclemarch 7250 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 7335. (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 7252	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7253. (Contributed by Jim Kingdon, 29-Dec-2019.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( *Q ` B ) <Q ( *Q ` A ) ) )
Theorem	ltrnqi 7253	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7252. (Contributed by Jim Kingdon, 24-Sep-2019.)
|- ( A <Q B -> ( *Q ` B ) <Q ( *Q ` A ) )
Theorem	nnnq 7254	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 7255	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 7256*	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 7257	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 7258	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 7259*	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 7260*	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 7261*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7260 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 7262*	Addition on nonnegative fractions. This definition is similar to df-plq0 7259 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 7263	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 7264	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7267. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f ~Q0 g -> g ~Q0 f )
Theorem	enq0ref 7265	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7267. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( f e. ( om X. N. ) <-> f ~Q0 f )
Theorem	enq0tr 7266	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7267. (Contributed by Jim Kingdon, 14-Nov-2019.)
|- ( ( f ~Q0 g /\ g ~Q0 h ) -> f ~Q0 h )
Theorem	enq0er 7267	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
|- ~Q0 Er ( om X. N. )
Theorem	enq0breq 7268	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 7269	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 7270	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 7271	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- ~Q0 e. _V
Theorem	nq0ex 7272	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q0 e. _V
Theorem	nqnq0 7273	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
|- Q. C_ Q0
Theorem	nq0nn 7274*	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 7275	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 7276	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 7277	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 7278*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7281 and mulnnnq0 7282. (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 7279*	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 7280*	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 7281	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 7282	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 7283	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 7284	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 7285	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 7286	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 7287	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 7288	Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> (0Q0 ·Q0 A ) = 0Q0 )
Theorem	nq0a0 7289	Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
Theorem	nnanq0 7290	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 7291	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 7292	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 7293	A natural number closure law. Lemma for addassnq0 7294. (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 7294	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 7295	Multiplication of nonnegative fractions is distributive. Version of distrnq0 7291 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 7296	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 7297	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 7298*	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 7299*	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 7300*	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 7340. 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 ) ) } >. )