P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prarloclemlt 7301	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7311. (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 7302*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7311. (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 7303	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7311. (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 7304*	Induction step for prarloclem3 7305. (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 7305*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7311. (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 7306*	A slight rearrangement of prarloclem3 7305. Lemma for prarloc 7311. (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 7307*	Subtracting two from a positive integer. Lemma for prarloc 7311. (Contributed by Jim Kingdon, 5-Nov-2019.)
|- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
Theorem	prarloclem5 7308*	A substitution of zero for y and N minus two for x. Lemma for prarloc 7311. (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 7309*	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 7310	Some calculations for prarloc 7311. (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 7311*	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 7312 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 7312*	A Dedekind cut is arithmetically located. This is a variation of prarloc 7311 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 7313	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
|- <P C_ ( P. X. P. )
Theorem	ltdfpr 7314*	More convenient form of df-iltp 7278. (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 7315*	Simplification of upper or lower cut expression. Lemma for genpdf 7316. (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 7316*	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 7317*	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 7318*	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 7319*	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 7320*	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 7321*	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 7322*	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 7323*	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 7324*	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 7325*	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 7326*	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 7327*	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 7328*	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 7329*	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 7330*	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 7331*	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 7332*	Associativity of lower cuts. Lemma for genpassg 7334. (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 7333*	Associativity of upper cuts. Lemma for genpassg 7334. (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 7334*	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 7335	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 7336	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 7337	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 7338	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 7339	Lemma for addlocpr 7344. 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 7340	Lemma for addlocpr 7344. 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 7341	Lemma for addlocpr 7344. 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 7342	Lemma for addlocpr 7344. 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 7343	Lemma for addlocpr 7344. 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 7344*	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 7311 to both A and B, and uses nqtri3or 7204 rather than prloc 7299 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 7345	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 7346*	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 7347*	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 7348	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom +P. = ( P. X. P. )
Theorem	dmmp 7349	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
|- dom .P. = ( P. X. P. )
Theorem	nqprm 7350*	A cut produced from a rational is inhabited. Lemma for nqprlu 7355. (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 7351*	A cut produced from a rational is rounded. Lemma for nqprlu 7355. (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 7352*	A cut produced from a rational is disjoint. Lemma for nqprlu 7355. (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 7353*	A cut produced from a rational is located. Lemma for nqprlu 7355. (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 7354*	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 7355*	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 7356*	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 7357	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 7358	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 7359*	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 7360*	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 7361*	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 7362	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- 1P e. P.
Theorem	1prl 7363	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 1st ` 1P ) = { x | x <Q 1Q }
Theorem	1pru 7364	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- ( 2nd ` 1P ) = { x | 1Q <Q x }
Theorem	addnqprlemrl 7365*	Lemma for addnqpr 7369. 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 7366*	Lemma for addnqpr 7369. 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 7367*	Lemma for addnqpr 7369. 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 7368*	Lemma for addnqpr 7369. 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 7369*	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 7370*	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 7369. (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 7371*	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 7372*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7371 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 7373	Calculations for prmuloc 7374. (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 7374*	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 7375*	Positive reals are multiplicatively located. This is a variation of prmuloc 7374 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 7376	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 7377	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 7378	Calculations for mullocpr 7379. (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 7379*	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 7380	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 7381*	Lemma for mulnqpr 7385. 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 7382*	Lemma for mulnqpr 7385. 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 7383*	Lemma for mulnqpr 7385. 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 7384*	Lemma for mulnqpr 7385. 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 7385*	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 7386	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 7387	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 7388	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 7389	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 7390	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 7391	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 7392*	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 7393*	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 7394	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 7395	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 7396	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 7397	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 7398	Lemma for 1idpr 7400. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )
Theorem	1idpru 7399	Lemma for 1idpr 7400. (Contributed by Jim Kingdon, 13-Dec-2019.)
|- ( A e. P. -> ( 2nd ` ( A .P. 1P ) ) = ( 2nd ` A ) )
Theorem	1idpr 7400	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 )