P. 78 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prloc 7701	A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
|- ( ( <. L , U >. e. P. /\ A <Q B ) -> ( A e. L \/ B e. U ) )
Theorem	prdisj 7702	A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
|- ( ( <. L , U >. e. P. /\ A e. Q. ) -> -. ( A e. L /\ A e. U ) )
Theorem	prarloclemlt 7703	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ y e. om ) -> ( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) <Q ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) )
Theorem	prarloclemlo 7704*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ y e. om ) -> ( ( A +Q ( [ <. ( y +o 1o ) , 1o >. ] ~Q .Q P ) ) e. L -> ( ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) )
Theorem	prarloclemup 7705	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ y e. om ) -> ( ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U -> ( ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) ) )
Theorem	prarloclem3step 7706*	Induction step for prarloclem3 7707. (Contributed by Jim Kingdon, 9-Nov-2019.)
|- ( ( ( X e. om /\ ( <. L , U >. e. P. /\ A e. L /\ P e. Q. ) ) /\ E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o suc X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclem3 7707*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 27-Oct-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( X e. om /\ P e. Q. ) /\ E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o X ) , 1o >. ] ~Q .Q P ) ) e. U ) ) -> E. j e. om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclem4 7708*	A slight rearrangement of prarloclem3 7707. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 4-Nov-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ P e. Q. ) -> ( E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. j e. om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) ) )
Theorem	prarloclemn 7709*	Subtracting two from a positive integer. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
Theorem	prarloclem5 7710*	A substitution of zero for y and N minus two for x. Lemma for prarloc 7713. (Contributed by Jim Kingdon, 4-Nov-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclem 7711*	A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from A to A +Q ( N .Q P ) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.)
|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. j e. om ( ( A +Q0 ( [ <. j , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( j +o 2o ) , 1o >. ] ~Q .Q P ) ) e. U ) )
Theorem	prarloclemcalc 7712	Some calculations for prarloc 7713. (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 7713*	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 7714 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 7714*	A Dedekind cut is arithmetically located. This is a variation of prarloc 7713 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 7715	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
|- <P C_ ( P. X. P. )
Theorem	ltdfpr 7716*	More convenient form of df-iltp 7680. (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 7717*	Simplification of upper or lower cut expression. Lemma for genpdf 7718. (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 7718*	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 7719*	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 7720*	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 7721*	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 7722*	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 7723*	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 7724*	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 7725*	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 7726*	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 7727*	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 7728*	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 7729*	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 7730*	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 7731*	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 7732*	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 7733*	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 7734*	Associativity of lower cuts. Lemma for genpassg 7736. (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 7735*	Associativity of upper cuts. Lemma for genpassg 7736. (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 7736*	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 7737	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 7738	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 7739	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 7740	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 7741	Lemma for addlocpr 7746. 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 7742	Lemma for addlocpr 7746. 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 7743	Lemma for addlocpr 7746. 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 7744	Lemma for addlocpr 7746. 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 7745	Lemma for addlocpr 7746. 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 7746*	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 7713 to both A and B, and uses nqtri3or 7606 rather than prloc 7701 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 7747	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 7748*	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 7749*	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 7750	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom +P. = ( P. X. P. )
Theorem	dmmp 7751	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom .P. = ( P. X. P. )
Theorem	nqprm 7752*	A cut produced from a rational is inhabited. Lemma for nqprlu 7757. (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 7753*	A cut produced from a rational is rounded. Lemma for nqprlu 7757. (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 7754*	A cut produced from a rational is disjoint. Lemma for nqprlu 7757. (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 7755*	A cut produced from a rational is located. Lemma for nqprlu 7757. (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 7756*	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 7757*	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 7758*	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 7759	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 7760	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 7761*	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 7762*	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 7763*	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 7764	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- 1P e. P.
Theorem	1prl 7765	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 1st ` 1P ) = { x | x <Q 1Q }
Theorem	1pru 7766	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 2nd ` 1P ) = { x | 1Q <Q x }
Theorem	addnqprlemrl 7767*	Lemma for addnqpr 7771. 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 7768*	Lemma for addnqpr 7771. 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 7769*	Lemma for addnqpr 7771. 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 7770*	Lemma for addnqpr 7771. 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 7771*	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 7772*	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 7771. (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 7773*	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 7774*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7773 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 7775	Calculations for prmuloc 7776. (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 7776*	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 7777*	Positive reals are multiplicatively located. This is a variation of prmuloc 7776 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 7778	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 7779	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 7780	Calculations for mullocpr 7781. (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 7781*	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 7782	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 7783*	Lemma for mulnqpr 7787. 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 7784*	Lemma for mulnqpr 7787. 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 7785*	Lemma for mulnqpr 7787. 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 7786*	Lemma for mulnqpr 7787. 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 7787*	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 7788	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 7789	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 7790	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 7791	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 7792	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 7793	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 7794*	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 7795*	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 7796	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 7797	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 7798	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 7799	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 7800	Lemma for 1idpr 7802. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )