P. 77 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	genpml 7601*	The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )
Theorem	genpmu 7602*	The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   =>    |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )
Theorem	genpcdl 7603*	Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 1st ` ( A F B ) ) -> ( x <Q f -> x e. ( 1st ` ( A F B ) ) ) ) )
Theorem	genpcuu 7604*	Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
Theorem	genprndl 7605*	The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )   &    |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )   &    |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. ( q e. ( 1st ` ( A F B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
Theorem	genprndu 7606*	The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )   &    |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )   &    |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> A. r e. Q. ( r e. ( 2nd ` ( A F B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
Theorem	genpdisj 7607*	The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )   &    |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )   =>    |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )
Theorem	genpassl 7608*	Associativity of lower cuts. Lemma for genpassg 7610. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- dom F = ( P. X. P. )   &    |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )   &    |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )   =>    |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( ( A F B ) F C ) ) = ( 1st ` ( A F ( B F C ) ) ) )
Theorem	genpassu 7609*	Associativity of upper cuts. Lemma for genpassg 7610. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- dom F = ( P. X. P. )   &    |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )   &    |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )   =>    |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) )
Theorem	genpassg 7610*	Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )   &    |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )   &    |- dom F = ( P. X. P. )   &    |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )   &    |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )   =>    |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
Theorem	addnqprllem 7611	Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
|- ( ( ( <. L , U >. e. P. /\ G e. L ) /\ X e. Q. ) -> ( X <Q S -> ( ( X .Q ( *Q ` S ) ) .Q G ) e. L ) )
Theorem	addnqprulem 7612	Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
|- ( ( ( <. L , U >. e. P. /\ G e. U ) /\ X e. Q. ) -> ( S <Q X -> ( ( X .Q ( *Q ` S ) ) .Q G ) e. U ) )
Theorem	addnqprl 7613	Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-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	addnqpru 7614	Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-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	addlocprlemlt 7615	Lemma for addlocpr 7620. The Q <Q ( D +Q E ) 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 -> ( Q <Q ( D +Q E ) -> Q e. ( 1st ` ( A +P. B ) ) ) )
Theorem	addlocprlemeqgt 7616	Lemma for addlocpr 7620. This is a step used in both the Q = ( D +Q E ) and ( D +Q E ) <Q Q cases. (Contributed by Jim Kingdon, 7-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 -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
Theorem	addlocprlemeq 7617	Lemma for addlocpr 7620. The Q = ( D +Q E ) 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 -> ( Q = ( D +Q E ) -> R e. ( 2nd ` ( A +P. B ) ) ) )
Theorem	addlocprlemgt 7618	Lemma for addlocpr 7620. 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 7619	Lemma for addlocpr 7620. 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 7620*	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 7587 to both A and B, and uses nqtri3or 7480 rather than prloc 7575 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 7621	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 7622*	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 7623*	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 7624	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom +P. = ( P. X. P. )
Theorem	dmmp 7625	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom .P. = ( P. X. P. )
Theorem	nqprm 7626*	A cut produced from a rational is inhabited. Lemma for nqprlu 7631. (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 7627*	A cut produced from a rational is rounded. Lemma for nqprlu 7631. (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 7628*	A cut produced from a rational is disjoint. Lemma for nqprlu 7631. (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 7629*	A cut produced from a rational is located. Lemma for nqprlu 7631. (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 7630*	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 7631*	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 7632*	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 7633	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 7634	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 7635*	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 7636*	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 7637*	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 7638	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- 1P e. P.
Theorem	1prl 7639	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 1st ` 1P ) = { x | x <Q 1Q }
Theorem	1pru 7640	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 2nd ` 1P ) = { x | 1Q <Q x }
Theorem	addnqprlemrl 7641*	Lemma for addnqpr 7645. 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 7642*	Lemma for addnqpr 7645. 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 7643*	Lemma for addnqpr 7645. 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 7644*	Lemma for addnqpr 7645. 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 7645*	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 7646*	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 7645. (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 7647*	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 7648*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7647 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 7649	Calculations for prmuloc 7650. (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 7650*	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 7651*	Positive reals are multiplicatively located. This is a variation of prmuloc 7650 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 7652	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 7653	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 7654	Calculations for mullocpr 7655. (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 7655*	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 7656	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 7657*	Lemma for mulnqpr 7661. 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 7658*	Lemma for mulnqpr 7661. 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 7659*	Lemma for mulnqpr 7661. 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 7660*	Lemma for mulnqpr 7661. 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 7661*	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 7662	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 7663	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 7664	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 7665	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 7666	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 7667	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 7668*	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 7669*	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 7670	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 7671	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 7672	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 7673	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 7674	Lemma for 1idpr 7676. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )
Theorem	1idpru 7675	Lemma for 1idpr 7676. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 2nd ` ( A .P. 1P ) ) = ( 2nd ` A ) )
Theorem	1idpr 7676	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 7677*	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 7678*	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 7679	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 7680. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- <P Po P.
Theorem	ltsopr 7680	Positive real 'less than' is a weak linear order (in the sense of df-iso 4333). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
|- <P Or P.
Theorem	ltaddpr 7681	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 7682*	Element in lower cut of the constructed difference. Lemma for ltexpri 7697. (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 7683*	Element in upper cut of the constructed difference. Lemma for ltexpri 7697. (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 7684*	Our constructed difference is inhabited. Lemma for ltexpri 7697. (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 7685*	The lower cut of our constructed difference is open. Lemma for ltexpri 7697. (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 7686*	The lower cut of our constructed difference is lower. Lemma for ltexpri 7697. (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 7687*	The upper cut of our constructed difference is open. Lemma for ltexpri 7697. (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 7688*	The upper cut of our constructed difference is upper. Lemma for ltexpri 7697. (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 7689*	Our constructed difference is rounded. Lemma for ltexpri 7697. (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 7690*	Our constructed difference is disjoint. Lemma for ltexpri 7697. (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 7691*	Our constructed difference is located. Lemma for ltexpri 7697. (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 7692*	Our constructed difference is a positive real. Lemma for ltexpri 7697. (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 7693*	Lemma for ltexpri 7697. 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 7694*	Lemma for ltexpri 7697. 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 7695*	Lemma for ltexpri 7697. 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 7696*	Lemma for ltexpri 7697. 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 7697*	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 7698	Lemma for addcanprg 7700. (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 7699	Lemma for addcanprg 7700. (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 7700	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 ) )