P. 77 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	ltexprlemm 7601*	Our constructed difference is inhabited. Lemma for ltexpri 7614. (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 7602*	The lower cut of our constructed difference is open. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
Theorem	ltexprlemlol 7603*	The lower cut of our constructed difference is lower. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
Theorem	ltexprlemopu 7604*	The upper cut of our constructed difference is open. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )
Theorem	ltexprlemupu 7605*	The upper cut of our constructed difference is upper. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 21-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( ( A <P B /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
Theorem	ltexprlemrnd 7606*	Our constructed difference is rounded. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) )
Theorem	ltexprlemdisj 7607*	Our constructed difference is disjoint. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )
Theorem	ltexprlemloc 7608*	Our constructed difference is located. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
Theorem	ltexprlempr 7609*	Our constructed difference is a positive real. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> C e. P. )
Theorem	ltexprlemfl 7610*	Lemma for ltexpri 7614. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 1st ` ( A +P. C ) ) C_ ( 1st ` B ) )
Theorem	ltexprlemrl 7611*	Lemma for ltexpri 7614. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) )
Theorem	ltexprlemfu 7612*	Lemma for ltexpri 7614. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 2nd ` ( A +P. C ) ) C_ ( 2nd ` B ) )
Theorem	ltexprlemru 7613*	Lemma for ltexpri 7614. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.   =>    |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )
Theorem	ltexpri 7614*	Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
|- ( A <P B -> E. x e. P. ( A +P. x ) = B )
Theorem	addcanprleml 7615	Lemma for addcanprg 7617. (Contributed by Jim Kingdon, 25-Dec-2019.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
Theorem	addcanprlemu 7616	Lemma for addcanprg 7617. (Contributed by Jim Kingdon, 25-Dec-2019.)
|- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
Theorem	addcanprg 7617	Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A +P. B ) = ( A +P. C ) -> B = C ) )
Theorem	lteupri 7618*	The difference from ltexpri 7614 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
|- ( A <P B -> E! x e. P. ( A +P. x ) = B )
Theorem	ltaprlem 7619	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
|- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
Theorem	ltaprg 7620	Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )
Theorem	prplnqu 7621*	Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
|- ( ph -> X e. P. )   &    |- ( ph -> Q e. Q. )   &    |- ( ph -> A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )   =>    |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
Theorem	addextpr 7622	Strong extensionality of addition (ordering version). This is similar to addext 8569 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( A +P. B ) <P ( C +P. D ) -> ( A <P C \/ B <P D ) ) )
Theorem	recexprlemell 7623*	Membership in the lower cut of B. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
Theorem	recexprlemelu 7624*	Membership in the upper cut of B. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )
Theorem	recexprlemm 7625*	B is inhabited. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
Theorem	recexprlemopl 7626*	The lower cut of B is open. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ q e. Q. /\ q e. ( 1st ` B ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
Theorem	recexprlemlol 7627*	The lower cut of B is lower. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) -> q e. ( 1st ` B ) ) )
Theorem	recexprlemopu 7628*	The upper cut of B is open. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ r e. Q. /\ r e. ( 2nd ` B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
Theorem	recexprlemupu 7629*	The upper cut of B is upper. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 28-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( ( A e. P. /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) )
Theorem	recexprlemrnd 7630*	B is rounded. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( A. q e. Q. ( q e. ( 1st ` B ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) )
Theorem	recexprlemdisj 7631*	B is disjoint. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
Theorem	recexprlemloc 7632*	B is located. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
Theorem	recexprlempr 7633*	B is a positive real. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> B e. P. )
Theorem	recexprlem1ssl 7634*	The lower cut of one is a subset of the lower cut of A .P. B. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( 1st ` 1P ) C_ ( 1st ` ( A .P. B ) ) )
Theorem	recexprlem1ssu 7635*	The upper cut of one is a subset of the upper cut of A .P. B. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( 2nd ` 1P ) C_ ( 2nd ` ( A .P. B ) ) )
Theorem	recexprlemss1l 7636*	The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )
Theorem	recexprlemss1u 7637*	The upper cut of A .P. B is a subset of the upper cut of one. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )
Theorem	recexprlemex 7638*	B is the reciprocal of A. Lemma for recexpr 7639. (Contributed by Jim Kingdon, 27-Dec-2019.)
|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.   =>    |- ( A e. P. -> ( A .P. B ) = 1P )
Theorem	recexpr 7639*	The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
|- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P )
Theorem	aptiprleml 7640	Lemma for aptipr 7642. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
Theorem	aptiprlemu 7641	Lemma for aptipr 7642. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
Theorem	aptipr 7642	Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A = B )
Theorem	ltmprr 7643	Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( C .P. A ) <P ( C .P. B ) -> A <P B ) )
Theorem	archpr 7644*	For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer x is embedded into the reals as described at nnprlu 7554. (Contributed by Jim Kingdon, 22-Apr-2020.)
|- ( A e. P. -> E. x e. N. A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )
Theorem	caucvgprlemcanl 7645*	Lemma for cauappcvgprlemladdrl 7658. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
|- ( ph -> L e. P. )   &    |- ( ph -> S e. Q. )   &    |- ( ph -> R e. Q. )   &    |- ( ph -> Q e. Q. )   =>    |- ( ph -> ( ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) <-> R e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )
Theorem	cauappcvgprlemm 7646*	Lemma for cauappcvgpr 7663. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
Theorem	cauappcvgprlemopl 7647*	Lemma for cauappcvgpr 7663. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
Theorem	cauappcvgprlemlol 7648*	Lemma for cauappcvgpr 7663. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
Theorem	cauappcvgprlemopu 7649*	Lemma for cauappcvgpr 7663. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )
Theorem	cauappcvgprlemupu 7650*	Lemma for cauappcvgpr 7663. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) )
Theorem	cauappcvgprlemrnd 7651*	Lemma for cauappcvgpr 7663. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) )
Theorem	cauappcvgprlemdisj 7652*	Lemma for cauappcvgpr 7663. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
Theorem	cauappcvgprlemloc 7653*	Lemma for cauappcvgpr 7663. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
Theorem	cauappcvgprlemcl 7654*	Lemma for cauappcvgpr 7663. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ph -> L e. P. )
Theorem	cauappcvgprlemladdfu 7655*	Lemma for cauappcvgprlemladd 7659. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
Theorem	cauappcvgprlemladdfl 7656*	Lemma for cauappcvgprlemladd 7659. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
Theorem	cauappcvgprlemladdru 7657*	Lemma for cauappcvgprlemladd 7659. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) C_ ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
Theorem	cauappcvgprlemladdrl 7658*	Lemma for cauappcvgprlemladd 7659. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) C_ ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
Theorem	cauappcvgprlemladd 7659*	Lemma for cauappcvgpr 7663. This takes L and offsets it by the positive fraction S. (Contributed by Jim Kingdon, 23-Jun-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. )
Theorem	cauappcvgprlem1 7660*	Lemma for cauappcvgpr 7663. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> Q e. Q. )   &    |- ( ph -> R e. Q. )   =>    |- ( ph -> <. { l | l <Q ( F ` Q ) } , { u | ( F ` Q ) <Q u } >. <P ( L +P. <. { l | l <Q ( Q +Q R ) } , { u | ( Q +Q R ) <Q u } >. ) )
Theorem	cauappcvgprlem2 7661*	Lemma for cauappcvgpr 7663. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   &    |- ( ph -> Q e. Q. )   &    |- ( ph -> R e. Q. )   =>    |- ( ph -> L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. )
Theorem	cauappcvgprlemlim 7662*	Lemma for cauappcvgpr 7663. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   &    |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.   =>    |- ( ph -> A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( L +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
Theorem	cauappcvgpr 7663*	A Cauchy approximation has a limit. A Cauchy approximation, here F, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of F is Q. rather than P.. We also specify that every term needs to be larger than a fraction A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This proof (including its lemmas) is similar to the proofs of caucvgpr 7683 and caucvgprpr 7713 but is somewhat simpler, so reading this one first may help understanding the other two. (Contributed by Jim Kingdon, 19-Jun-2020.)
|- ( ph -> F : Q. --> Q. )   &    |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )   &    |- ( ph -> A. p e. Q. A <Q ( F ` p ) )   =>    |- ( ph -> E. y e. P. A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
Theorem	archrecnq 7664*	Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
|- ( A e. Q. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q A )
Theorem	archrecpr 7665*	Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
|- ( A e. P. -> E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P A )
Theorem	caucvgprlemk 7666	Lemma for caucvgpr 7683. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
|- ( ph -> J <N K )   &    |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )   =>    |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
Theorem	caucvgprlemnkj 7667*	Lemma for caucvgpr 7683. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> K e. N. )   &    |- ( ph -> J e. N. )   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> -. ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )
Theorem	caucvgprlemnbj 7668*	Lemma for caucvgpr 7683. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> B e. N. )   &    |- ( ph -> J e. N. )   =>    |- ( ph -> -. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) )
Theorem	caucvgprlemm 7669*	Lemma for caucvgpr 7683. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
Theorem	caucvgprlemopl 7670*	Lemma for caucvgpr 7683. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
Theorem	caucvgprlemlol 7671*	Lemma for caucvgpr 7683. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
Theorem	caucvgprlemopu 7672*	Lemma for caucvgpr 7683. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )
Theorem	caucvgprlemupu 7673*	Lemma for caucvgpr 7683. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) )
Theorem	caucvgprlemrnd 7674*	Lemma for caucvgpr 7683. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) )
Theorem	caucvgprlemdisj 7675*	Lemma for caucvgpr 7683. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
Theorem	caucvgprlemloc 7676*	Lemma for caucvgpr 7683. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
Theorem	caucvgprlemcl 7677*	Lemma for caucvgpr 7683. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ph -> L e. P. )
Theorem	caucvgprlemladdfu 7678*	Lemma for caucvgpr 7683. Adding S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
Theorem	caucvgprlemladdrl 7679*	Lemma for caucvgpr 7683. Adding S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) } C_ ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
Theorem	caucvgprlem1 7680*	Lemma for caucvgpr 7683. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   &    |- ( ph -> Q e. Q. )   &    |- ( ph -> J <N K )   &    |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )   =>    |- ( ph -> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
Theorem	caucvgprlem2 7681*	Lemma for caucvgpr 7683. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   &    |- ( ph -> Q e. Q. )   &    |- ( ph -> J <N K )   &    |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )   =>    |- ( ph -> L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. )
Theorem	caucvgprlemlim 7682*	Lemma for caucvgpr 7683. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   &    |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.   =>    |- ( ph -> A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
Theorem	caucvgpr 7683*	A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This proof (including its lemmas) is similar to the proofs of cauappcvgpr 7663 and caucvgprpr 7713. Reading cauappcvgpr 7663 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.)
|- ( ph -> F : N. --> Q. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )   &    |- ( ph -> A. j e. N. A <Q ( F ` j ) )   =>    |- ( ph -> E. y e. P. A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
Theorem	caucvgprprlemk 7684*	Lemma for caucvgprpr 7713. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
|- ( ph -> J <N K )   &    |- ( ph -> <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. <P Q )   =>    |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q )
Theorem	caucvgprprlemloccalc 7685*	Lemma for caucvgprpr 7713. Rearranging some expressions for caucvgprprlemloc 7704. (Contributed by Jim Kingdon, 8-Feb-2021.)
|- ( ph -> S <Q T )   &    |- ( ph -> Y e. Q. )   &    |- ( ph -> ( S +Q Y ) = T )   &    |- ( ph -> X e. Q. )   &    |- ( ph -> ( X +Q X ) <Q Y )   &    |- ( ph -> M e. N. )   &    |- ( ph -> ( *Q ` [ <. M , 1o >. ] ~Q ) <Q X )   =>    |- ( ph -> ( <. { l | l <Q ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. M , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q ( *Q ` [ <. M , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. M , 1o >. ] ~Q ) <Q u } >. ) <P <. { l | l <Q T } , { u | T <Q u } >. )
Theorem	caucvgprprlemell 7686*	Lemma for caucvgprpr 7713. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
|- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( X e. ( 1st ` L ) <-> ( X e. Q. /\ E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
Theorem	caucvgprprlemelu 7687*	Lemma for caucvgprpr 7713. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
|- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( X e. ( 2nd ` L ) <-> ( X e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )
Theorem	caucvgprprlemcbv 7688*	Lemma for caucvgprpr 7713. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   =>    |- ( ph -> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )
Theorem	caucvgprprlemval 7689*	Lemma for caucvgprpr 7713. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   =>    |- ( ( ph /\ A <N B ) -> ( ( F ` A ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) ) )
Theorem	caucvgprprlemnkltj 7690*	Lemma for caucvgprpr 7713. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> K e. N. )   &    |- ( ph -> J e. N. )   &    |- ( ph -> S e. Q. )   =>    |- ( ( ph /\ K <N J ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
Theorem	caucvgprprlemnkeqj 7691*	Lemma for caucvgprpr 7713. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> K e. N. )   &    |- ( ph -> J e. N. )   &    |- ( ph -> S e. Q. )   =>    |- ( ( ph /\ K = J ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
Theorem	caucvgprprlemnjltk 7692*	Lemma for caucvgprpr 7713. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> K e. N. )   &    |- ( ph -> J e. N. )   &    |- ( ph -> S e. Q. )   =>    |- ( ( ph /\ J <N K ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
Theorem	caucvgprprlemnkj 7693*	Lemma for caucvgprpr 7713. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> K e. N. )   &    |- ( ph -> J e. N. )   &    |- ( ph -> S e. Q. )   =>    |- ( ph -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
Theorem	caucvgprprlemnbj 7694*	Lemma for caucvgprpr 7713. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> B e. N. )   &    |- ( ph -> J e. N. )   =>    |- ( ph -> -. ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` J ) )
Theorem	caucvgprprlemml 7695*	Lemma for caucvgprpr 7713. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> A. m e. N. A <P ( F ` m ) )   &    |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( ph -> E. s e. Q. s e. ( 1st ` L ) )
Theorem	caucvgprprlemmu 7696*	Lemma for caucvgprpr 7713. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> A. m e. N. A <P ( F ` m ) )   &    |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) )
Theorem	caucvgprprlemm 7697*	Lemma for caucvgprpr 7713. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> A. m e. N. A <P ( F ` m ) )   &    |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. t e. Q. t e. ( 2nd ` L ) ) )
Theorem	caucvgprprlemopl 7698*	Lemma for caucvgprpr 7713. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> A. m e. N. A <P ( F ` m ) )   &    |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )
Theorem	caucvgprprlemlol 7699*	Lemma for caucvgprpr 7713. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> A. m e. N. A <P ( F ` m ) )   &    |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
Theorem	caucvgprprlemopu 7700*	Lemma for caucvgprpr 7713. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
|- ( ph -> F : N. --> P. )   &    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )   &    |- ( ph -> A. m e. N. A <P ( F ` m ) )   &    |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.   =>    |- ( ( ph /\ t e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) )