P. 75 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	prarloclemcalc 7401	Some calculations for prarloc 7402. (Contributed by Jim Kingdon, 26-Oct-2019.)
|- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B <Q ( A +Q P ) )
Theorem	prarloc 7402*	A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance P, there are elements of the lower and upper cut which are within that tolerance of each other. Usually, proofs will be shorter if they use prarloc2 7403 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)
|- ( ( <. L , U >. e. P. /\ P e. Q. ) -> E. a e. L E. b e. U b <Q ( a +Q P ) )
Theorem	prarloc2 7403*	A Dedekind cut is arithmetically located. This is a variation of prarloc 7402 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance P, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
|- ( ( <. L , U >. e. P. /\ P e. Q. ) -> E. a e. L ( a +Q P ) e. U )
Theorem	ltrelpr 7404	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
|- <P C_ ( P. X. P. )
Theorem	ltdfpr 7405*	More convenient form of df-iltp 7369. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) )
Theorem	genpdflem 7406*	Simplification of upper or lower cut expression. Lemma for genpdf 7407. (Contributed by Jim Kingdon, 30-Sep-2019.)
|- ( ( ph /\ r e. A ) -> r e. Q. )   &    |- ( ( ph /\ s e. B ) -> s e. Q. )   =>    |- ( ph -> { q e. Q. | E. r e. Q. E. s e. Q. ( r e. A /\ s e. B /\ q = ( r G s ) ) } = { q e. Q. | E. r e. A E. s e. B q = ( r G s ) } )
Theorem	genpdf 7407*	Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
|- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. )   =>    |- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } >. )
Theorem	genipv 7408*	Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-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. ) -> ( A F B ) = <. { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) } >. )
Theorem	genplt2i 7409*	Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
|- ( ( 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 <Q B /\ C <Q D ) -> ( A G C ) <Q ( B G D ) )
Theorem	genpelxp 7410*	Set containing the result of adding or multiplying positive reals. (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 ) ) } >. )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. ( ~P Q. X. ~P Q. ) )
Theorem	genpelvl 7411*	Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-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. ) -> ( C e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) )
Theorem	genpelvu 7412*	Membership in upper cut of general operation (addition or multiplication) on positive reals. (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. )   =>    |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) )
Theorem	genpprecll 7413*	Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-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. ) -> ( ( C e. ( 1st ` A ) /\ D e. ( 1st ` B ) ) -> ( C G D ) e. ( 1st ` ( A F B ) ) ) )
Theorem	genppreclu 7414*	Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-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. /\ B e. P. ) -> ( ( C e. ( 2nd ` A ) /\ D e. ( 2nd ` B ) ) -> ( C G D ) e. ( 2nd ` ( A F B ) ) ) )
Theorem	genipdm 7415*	Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-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. )   =>    |- dom F = ( P. X. P. )
Theorem	genpml 7416*	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 7417*	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 7418*	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 7419*	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 7420*	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 7421*	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 7422*	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 7423*	Associativity of lower cuts. Lemma for genpassg 7425. (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 7424*	Associativity of upper cuts. Lemma for genpassg 7425. (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 7425*	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 7426	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 7427	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 7428	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 7429	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 7430	Lemma for addlocpr 7435. 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 7431	Lemma for addlocpr 7435. 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 7432	Lemma for addlocpr 7435. 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 7433	Lemma for addlocpr 7435. 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 7434	Lemma for addlocpr 7435. 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 7435*	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 7402 to both A and B, and uses nqtri3or 7295 rather than prloc 7390 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 7436	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 7437*	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 7438*	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 7439	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom +P. = ( P. X. P. )
Theorem	dmmp 7440	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom .P. = ( P. X. P. )
Theorem	nqprm 7441*	A cut produced from a rational is inhabited. Lemma for nqprlu 7446. (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 7442*	A cut produced from a rational is rounded. Lemma for nqprlu 7446. (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 7443*	A cut produced from a rational is disjoint. Lemma for nqprlu 7446. (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 7444*	A cut produced from a rational is located. Lemma for nqprlu 7446. (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 7445*	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 7446*	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 7447*	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 7448	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 7449	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 7450*	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 7451*	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 7452*	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 7453	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- 1P e. P.
Theorem	1prl 7454	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 1st ` 1P ) = { x | x <Q 1Q }
Theorem	1pru 7455	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 2nd ` 1P ) = { x | 1Q <Q x }
Theorem	addnqprlemrl 7456*	Lemma for addnqpr 7460. 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 7457*	Lemma for addnqpr 7460. 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 7458*	Lemma for addnqpr 7460. 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 7459*	Lemma for addnqpr 7460. 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 7460*	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 7461*	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 7460. (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 7462*	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 7463*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7462 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 7464	Calculations for prmuloc 7465. (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 7465*	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 7466*	Positive reals are multiplicatively located. This is a variation of prmuloc 7465 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 7467	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 7468	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 7469	Calculations for mullocpr 7470. (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 7470*	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 7471	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 7472*	Lemma for mulnqpr 7476. 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 7473*	Lemma for mulnqpr 7476. 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 7474*	Lemma for mulnqpr 7476. 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 7475*	Lemma for mulnqpr 7476. 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 7476*	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 7477	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 7478	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 7479	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 7480	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 7481	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 7482	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 7483*	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 7484*	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 7485	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 7486	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 7487	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 7488	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 7489	Lemma for 1idpr 7491. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )
Theorem	1idpru 7490	Lemma for 1idpr 7491. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 2nd ` ( A .P. 1P ) ) = ( 2nd ` A ) )
Theorem	1idpr 7491	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 7492*	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 7493*	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 7494	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 7495. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- <P Po P.
Theorem	ltsopr 7495	Positive real 'less than' is a weak linear order (in the sense of df-iso 4252). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
|- <P Or P.
Theorem	ltaddpr 7496	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 7497*	Element in lower cut of the constructed difference. Lemma for ltexpri 7512. (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 7498*	Element in upper cut of the constructed difference. Lemma for ltexpri 7512. (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 7499*	Our constructed difference is inhabited. Lemma for ltexpri 7512. (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 7500*	The lower cut of our constructed difference is open. Lemma for ltexpri 7512. (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 ) ) )