P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	addlocprlemgt 7901	Lemma for addlocpr 7903. The ( D +Q E ) <Q Q case. (Contributed by Jim Kingdon, 6-Dec-2019.)
|- ( ph -> A e. P. )   &    |- ( ph -> B e. P. )   &    |- ( ph -> Q <Q R )   &    |- ( ph -> P e. Q. )   &    |- ( ph -> ( Q +Q ( P +Q P ) ) = R )   &    |- ( ph -> D e. ( 1st ` A ) )   &    |- ( ph -> U e. ( 2nd ` A ) )   &    |- ( ph -> U <Q ( D +Q P ) )   &    |- ( ph -> E e. ( 1st ` B ) )   &    |- ( ph -> T e. ( 2nd ` B ) )   &    |- ( ph -> T <Q ( E +Q P ) )   =>    |- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )
Theorem	addlocprlem 7902	Lemma for addlocpr 7903. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
|- ( ph -> A e. P. )   &    |- ( ph -> B e. P. )   &    |- ( ph -> Q <Q R )   &    |- ( ph -> P e. Q. )   &    |- ( ph -> ( Q +Q ( P +Q P ) ) = R )   &    |- ( ph -> D e. ( 1st ` A ) )   &    |- ( ph -> U e. ( 2nd ` A ) )   &    |- ( ph -> U <Q ( D +Q P ) )   &    |- ( ph -> E e. ( 1st ` B ) )   &    |- ( ph -> T e. ( 2nd ` B ) )   &    |- ( ph -> T <Q ( E +Q P ) )   =>    |- ( ph -> ( Q e. ( 1st ` ( A +P. B ) ) \/ R e. ( 2nd ` ( A +P. B ) ) ) )
Theorem	addlocpr 7903*	Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 7870 to both A and B, and uses nqtri3or 7763 rather than prloc 7858 to decide whether q is too big to be in the lower cut of A +P. B (and deduce that if it is, then r must be in the upper cut). What the two proofs have in common is that they take the difference between q and r to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
Theorem	addclpr 7904	Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.)
|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
Theorem	plpvlu 7905*	Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y +Q z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y +Q z ) } >. )
Theorem	mpvlu 7906*	Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) } >. )
Theorem	dmplp 7907	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom +P. = ( P. X. P. )
Theorem	dmmp 7908	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom .P. = ( P. X. P. )
Theorem	nqprm 7909*	A cut produced from a rational is inhabited. Lemma for nqprlu 7914. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( A e. Q. -> ( E. q e. Q. q e. { x | x <Q A } /\ E. r e. Q. r e. { x | A <Q x } ) )
Theorem	nqprrnd 7910*	A cut produced from a rational is rounded. Lemma for nqprlu 7914. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( A e. Q. -> ( A. q e. Q. ( q e. { x | x <Q A } <-> E. r e. Q. ( q <Q r /\ r e. { x | x <Q A } ) ) /\ A. r e. Q. ( r e. { x | A <Q x } <-> E. q e. Q. ( q <Q r /\ q e. { x | A <Q x } ) ) ) )
Theorem	nqprdisj 7911*	A cut produced from a rational is disjoint. Lemma for nqprlu 7914. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( A e. Q. -> A. q e. Q. -. ( q e. { x | x <Q A } /\ q e. { x | A <Q x } ) )
Theorem	nqprloc 7912*	A cut produced from a rational is located. Lemma for nqprlu 7914. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( A e. Q. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
Theorem	nqprxx 7913*	The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( A e. Q. -> <. { x | x <Q A } , { x | A <Q x } >. e. P. )
Theorem	nqprlu 7914*	The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
|- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
Theorem	recnnpr 7915*	The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
|- ( A e. N. -> <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. e. P. )
Theorem	ltnqex 7916	The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- { x | x <Q A } e. _V
Theorem	gtnqex 7917	The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- { x | A <Q x } e. _V
Theorem	nqprl 7918*	Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)
|- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 1st ` B ) <-> <. { l | l <Q A } , { u | A <Q u } >. <P B ) )
Theorem	nqpru 7919*	Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <P. (Contributed by Jim Kingdon, 29-Nov-2020.)
|- ( ( A e. Q. /\ B e. P. ) -> ( A e. ( 2nd ` B ) <-> B <P <. { l | l <Q A } , { u | A <Q u } >. ) )
Theorem	nnprlu 7920*	The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
|- ( A e. N. -> <. { l | l <Q [ <. A , 1o >. ] ~Q } , { u | [ <. A , 1o >. ] ~Q <Q u } >. e. P. )
Theorem	1pr 7921	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- 1P e. P.
Theorem	1prl 7922	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 1st ` 1P ) = { x | x <Q 1Q }
Theorem	1pru 7923	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 2nd ` 1P ) = { x | 1Q <Q x }
Theorem	addnqprlemrl 7924*	Lemma for addnqpr 7928. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
Theorem	addnqprlemru 7925*	Lemma for addnqpr 7928. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
Theorem	addnqprlemfl 7926*	Lemma for addnqpr 7928. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
Theorem	addnqprlemfu 7927*	Lemma for addnqpr 7928. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
Theorem	addnqpr 7928*	Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
|- ( ( A e. Q. /\ B e. Q. ) -> <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) )
Theorem	addnqpr1 7929*	Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 7928. (Contributed by Jim Kingdon, 26-Apr-2020.)
|- ( A e. Q. -> <. { l | l <Q ( A +Q 1Q ) } , { u | ( A +Q 1Q ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. +P. 1P ) )
Theorem	appdivnq 7930*	Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where A and B are positive, as well as C). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( ( A <Q B /\ C e. Q. ) -> E. m e. Q. ( A <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) )
Theorem	appdiv0nq 7931*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7930 in which A is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ( B e. Q. /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )
Theorem	prmuloclemcalc 7932	Calculations for prmuloc 7933. (Contributed by Jim Kingdon, 9-Dec-2019.)
|- ( ph -> R <Q U )   &    |- ( ph -> U <Q ( D +Q P ) )   &    |- ( ph -> ( A +Q X ) = B )   &    |- ( ph -> ( P .Q B ) <Q ( R .Q X ) )   &    |- ( ph -> A e. Q. )   &    |- ( ph -> B e. Q. )   &    |- ( ph -> D e. Q. )   &    |- ( ph -> P e. Q. )   &    |- ( ph -> X e. Q. )   =>    |- ( ph -> ( U .Q A ) <Q ( D .Q B ) )
Theorem	prmuloc 7933*	Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( ( <. L , U >. e. P. /\ A <Q B ) -> E. d e. Q. E. u e. Q. ( d e. L /\ u e. U /\ ( u .Q A ) <Q ( d .Q B ) ) )
Theorem	prmuloc2 7934*	Positive reals are multiplicatively located. This is a variation of prmuloc 7933 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio B, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( ( <. L , U >. e. P. /\ 1Q <Q B ) -> E. x e. L ( x .Q B ) e. U )
Theorem	mulnqprl 7935	Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
|- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) )
Theorem	mulnqpru 7936	Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
|- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X -> X e. ( 2nd ` ( A .P. B ) ) ) )
Theorem	mullocprlem 7937	Calculations for mullocpr 7938. (Contributed by Jim Kingdon, 10-Dec-2019.)
|- ( ph -> ( A e. P. /\ B e. P. ) )   &    |- ( ph -> ( U .Q Q ) <Q ( E .Q ( D .Q U ) ) )   &    |- ( ph -> ( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) )   &    |- ( ph -> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) )   &    |- ( ph -> ( Q e. Q. /\ R e. Q. ) )   &    |- ( ph -> ( D e. Q. /\ U e. Q. ) )   &    |- ( ph -> ( D e. ( 1st ` A ) /\ U e. ( 2nd ` A ) ) )   &    |- ( ph -> ( E e. Q. /\ T e. Q. ) )   =>    |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )
Theorem	mullocpr 7938*	Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both A and B are positive, not just A). (Contributed by Jim Kingdon, 8-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A .P. B ) ) \/ r e. ( 2nd ` ( A .P. B ) ) ) ) )
Theorem	mulclpr 7939	Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
Theorem	mulnqprlemrl 7940*	Lemma for mulnqpr 7944. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
Theorem	mulnqprlemru 7941*	Lemma for mulnqpr 7944. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
Theorem	mulnqprlemfl 7942*	Lemma for mulnqpr 7944. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
Theorem	mulnqprlemfu 7943*	Lemma for mulnqpr 7944. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
Theorem	mulnqpr 7944*	Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) )
Theorem	addcomprg 7945	Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = ( B +P. A ) )
Theorem	addassprg 7946	Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A +P. B ) +P. C ) = ( A +P. ( B +P. C ) ) )
Theorem	mulcomprg 7947	Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )
Theorem	mulassprg 7948	Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) .P. C ) = ( A .P. ( B .P. C ) ) )
Theorem	distrlem1prl 7949	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( A .P. ( B +P. C ) ) ) C_ ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
Theorem	distrlem1pru 7950	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( A .P. ( B +P. C ) ) ) C_ ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
Theorem	distrlem4prl 7951*	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
Theorem	distrlem4pru 7952*	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
Theorem	distrlem5prl 7953	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_ ( 1st ` ( A .P. ( B +P. C ) ) ) )
Theorem	distrlem5pru 7954	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_ ( 2nd ` ( A .P. ( B +P. C ) ) ) )
Theorem	distrprg 7955	Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. ( B +P. C ) ) = ( ( A .P. B ) +P. ( A .P. C ) ) )
Theorem	ltprordil 7956	If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
|- ( A <P B -> ( 1st ` A ) C_ ( 1st ` B ) )
Theorem	1idprl 7957	Lemma for 1idpr 7959. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )
Theorem	1idpru 7958	Lemma for 1idpr 7959. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 2nd ` ( A .P. 1P ) ) = ( 2nd ` A ) )
Theorem	1idpr 7959	1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.)
|- ( A e. P. -> ( A .P. 1P ) = A )
Theorem	ltnqpr 7960*	We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> <. { l | l <Q A } , { u | A <Q u } >. <P <. { l | l <Q B } , { u | B <Q u } >. ) )
Theorem	ltnqpri 7961*	We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
|- ( A <Q B -> <. { l | l <Q A } , { u | A <Q u } >. <P <. { l | l <Q B } , { u | B <Q u } >. )
Theorem	ltpopr 7962	Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7963. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- <P Po P.
Theorem	ltsopr 7963	Positive real 'less than' is a weak linear order (in the sense of df-iso 4442). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
|- <P Or P.
Theorem	ltaddpr 7964	The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )
Theorem	ltexprlemell 7965*	Element in lower cut of the constructed difference. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
Theorem	ltexprlemelu 7966*	Element in upper cut of the constructed difference. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
Theorem	ltexprlemm 7967*	Our constructed difference is inhabited. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( E. q e. Q. q e. ( 1st ` C ) /\ E. r e. Q. r e. ( 2nd ` C ) ) )
Theorem	ltexprlemopl 7968*	The lower cut of our constructed difference is open. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
Theorem	ltexprlemlol 7969*	The lower cut of our constructed difference is lower. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
Theorem	ltexprlemopu 7970*	The upper cut of our constructed difference is open. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )
Theorem	ltexprlemupu 7971*	The upper cut of our constructed difference is upper. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
Theorem	ltexprlemrnd 7972*	Our constructed difference is rounded. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) )
Theorem	ltexprlemdisj 7973*	Our constructed difference is disjoint. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )
Theorem	ltexprlemloc 7974*	Our constructed difference is located. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
Theorem	ltexprlempr 7975*	Our constructed difference is a positive real. Lemma for ltexpri 7980. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> C e. P. )
Theorem	ltexprlemfl 7976*	Lemma for ltexpri 7980. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )
Theorem	ltexprlemrl 7977*	Lemma for ltexpri 7980. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) )
Theorem	ltexprlemfu 7978*	Lemma for ltexpri 7980. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )
Theorem	ltexprlemru 7979*	Lemma for ltexpri 7980. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )
Theorem	ltexpri 7980*	Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
|- ( A <P B -> E. x e. P. ( A +P. x ) = B )
Theorem	addcanprleml 7981	Lemma for addcanprg 7983. (Contributed by Jim Kingdon, 25-Dec-2019.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
Theorem	addcanprlemu 7982	Lemma for addcanprg 7983. (Contributed by Jim Kingdon, 25-Dec-2019.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
Theorem	addcanprg 7983	Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A +P. B ) = ( A +P. C ) -> B = C ) )
Theorem	lteupri 7984*	The difference from ltexpri 7980 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
|- ( A <P B -> E! x e. P. ( A +P. x ) = B )
Theorem	ltaprlem 7985	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
|- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
Theorem	ltaprg 7986	Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )
Theorem	prplnqu 7987*	Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
|- ( ph -> X e. P. )   &    |- ( ph -> Q e. Q. )   &    |- ( ph -> A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )   =>    |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
Theorem	addextpr 7988	Strong extensionality of addition (ordering version). This is similar to addext 8938 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( A +P. B ) <P ( C +P. D ) -> ( A <P C \/ B <P D ) ) )
Theorem	recexprlemell 7989*	Membership in the lower cut of B. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
Theorem	recexprlemelu 7990*	Membership in the upper cut of B. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )
Theorem	recexprlemm 7991*	B is inhabited. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
Theorem	recexprlemopl 7992*	The lower cut of B is open. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ q e. Q. /\ q e. ( 1st ` B ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
Theorem	recexprlemlol 7993*	The lower cut of B is lower. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) -> q e. ( 1st ` B ) ) )
Theorem	recexprlemopu 7994*	The upper cut of B is open. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ r e. Q. /\ r e. ( 2nd ` B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
Theorem	recexprlemupu 7995*	The upper cut of B is upper. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) )
Theorem	recexprlemrnd 7996*	B is rounded. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( A. q e. Q. ( q e. ( 1st ` B ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) )
Theorem	recexprlemdisj 7997*	B is disjoint. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
Theorem	recexprlemloc 7998*	B is located. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
Theorem	recexprlempr 7999*	B is a positive real. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> B e. P. )
Theorem	recexprlem1ssl 8000*	The lower cut of one is a subset of the lower cut of A .P. B. Lemma for recexpr 8005. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( 1st ` 1P ) C_ ( 1st ` ( A .P. B ) ) )