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	lteupri 7701*	The difference from ltexpri 7697 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
|- ( A <P B -> E! x e. P. ( A +P. x ) = B )
Theorem	ltaprlem 7702	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 7703	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 7704*	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 7705	Strong extensionality of addition (ordering version). This is similar to addext 8654 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 7706*	Membership in the lower cut of B. Lemma for recexpr 7722. (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 7707*	Membership in the upper cut of B. Lemma for recexpr 7722. (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 7708*	B is inhabited. Lemma for recexpr 7722. (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 7709*	The lower cut of B is open. Lemma for recexpr 7722. (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 7710*	The lower cut of B is lower. Lemma for recexpr 7722. (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 7711*	The upper cut of B is open. Lemma for recexpr 7722. (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 7712*	The upper cut of B is upper. Lemma for recexpr 7722. (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 7713*	B is rounded. Lemma for recexpr 7722. (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 7714*	B is disjoint. Lemma for recexpr 7722. (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 7715*	B is located. Lemma for recexpr 7722. (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 7716*	B is a positive real. Lemma for recexpr 7722. (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 7717*	The lower cut of one is a subset of the lower cut of A .P. B. Lemma for recexpr 7722. (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 7718*	The upper cut of one is a subset of the upper cut of A .P. B. Lemma for recexpr 7722. (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 7719*	The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7722. (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 7720*	The upper cut of A .P. B is a subset of the upper cut of one. Lemma for recexpr 7722. (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 7721*	B is the reciprocal of A. Lemma for recexpr 7722. (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 7722*	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 7723	Lemma for aptipr 7725. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
Theorem	aptiprlemu 7724	Lemma for aptipr 7725. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
Theorem	aptipr 7725	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 7726	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 7727*	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 7637. (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 7728*	Lemma for cauappcvgprlemladdrl 7741. 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 7729*	Lemma for cauappcvgpr 7746. 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 7730*	Lemma for cauappcvgpr 7746. 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 7731*	Lemma for cauappcvgpr 7746. 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 7732*	Lemma for cauappcvgpr 7746. 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 7733*	Lemma for cauappcvgpr 7746. 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 7734*	Lemma for cauappcvgpr 7746. 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 7735*	Lemma for cauappcvgpr 7746. 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 7736*	Lemma for cauappcvgpr 7746. 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 7737*	Lemma for cauappcvgpr 7746. 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 7738*	Lemma for cauappcvgprlemladd 7742. 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 7739*	Lemma for cauappcvgprlemladd 7742. 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 7740*	Lemma for cauappcvgprlemladd 7742. 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 7741*	Lemma for cauappcvgprlemladd 7742. 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 7742*	Lemma for cauappcvgpr 7746. 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 7743*	Lemma for cauappcvgpr 7746. 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 7744*	Lemma for cauappcvgpr 7746. 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 7745*	Lemma for cauappcvgpr 7746. 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 7746*	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 7766 and caucvgprpr 7796 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 7747*	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 7748*	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 7749	Lemma for caucvgpr 7766. 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 7750*	Lemma for caucvgpr 7766. 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 7751*	Lemma for caucvgpr 7766. 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 7752*	Lemma for caucvgpr 7766. 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 7753*	Lemma for caucvgpr 7766. 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 7754*	Lemma for caucvgpr 7766. 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 7755*	Lemma for caucvgpr 7766. 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 7756*	Lemma for caucvgpr 7766. 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 7757*	Lemma for caucvgpr 7766. 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 7758*	Lemma for caucvgpr 7766. 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 7759*	Lemma for caucvgpr 7766. 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 7760*	Lemma for caucvgpr 7766. 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 7761*	Lemma for caucvgpr 7766. 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 7762*	Lemma for caucvgpr 7766. 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 7763*	Lemma for caucvgpr 7766. 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 7764*	Lemma for caucvgpr 7766. 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 7765*	Lemma for caucvgpr 7766. 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 7766*	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 7746 and caucvgprpr 7796. Reading cauappcvgpr 7746 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 7767*	Lemma for caucvgprpr 7796. 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 7768*	Lemma for caucvgprpr 7796. Rearranging some expressions for caucvgprprlemloc 7787. (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 7769*	Lemma for caucvgprpr 7796. 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 7770*	Lemma for caucvgprpr 7796. 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 7771*	Lemma for caucvgprpr 7796. 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 7772*	Lemma for caucvgprpr 7796. 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 7773*	Lemma for caucvgprpr 7796. 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 7774*	Lemma for caucvgprpr 7796. 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 7775*	Lemma for caucvgprpr 7796. 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 7776*	Lemma for caucvgprpr 7796. 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 7777*	Lemma for caucvgprpr 7796. 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 7778*	Lemma for caucvgprpr 7796. 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 7779*	Lemma for caucvgprpr 7796. 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 7780*	Lemma for caucvgprpr 7796. 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 7781*	Lemma for caucvgprpr 7796. 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 7782*	Lemma for caucvgprpr 7796. 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 7783*	Lemma for caucvgprpr 7796. 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 ) ) )
Theorem	caucvgprprlemupu 7784*	Lemma for caucvgprpr 7796. The upper cut of the putative limit is upper. (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 /\ s e. ( 2nd ` L ) ) -> t e. ( 2nd ` L ) )
Theorem	caucvgprprlemrnd 7785*	Lemma for caucvgprpr 7796. The putative limit is rounded. (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 -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) /\ A. t e. Q. ( t e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) ) )
Theorem	caucvgprprlemdisj 7786*	Lemma for caucvgprpr 7796. The putative limit is disjoint. (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 -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
Theorem	caucvgprprlemloc 7787*	Lemma for caucvgprpr 7796. The putative limit is located. (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 -> A. s e. Q. A. t e. Q. ( s <Q t -> ( s e. ( 1st ` L ) \/ t e. ( 2nd ` L ) ) ) )
Theorem	caucvgprprlemcl 7788*	Lemma for caucvgprpr 7796. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-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 -> L e. P. )
Theorem	caucvgprprlemclphr 7789*	Lemma for caucvgprpr 7796. The putative limit is a positive real. Like caucvgprprlemcl 7788 but without a disjoint variable condition between ph and r. (Contributed by Jim Kingdon, 19-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 -> 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 -> L e. P. )
Theorem	caucvgprprlemexbt 7790*	Lemma for caucvgprpr 7796. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-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 -> 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 -> Q e. Q. )   &    |- ( ph -> T e. P. )   &    |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )   =>    |- ( ph -> 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 Q } , { q | Q <Q q } >. ) <P T )
Theorem	caucvgprprlemexb 7791*	Lemma for caucvgprpr 7796. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-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 -> 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 -> Q e. P. )   &    |- ( ph -> R e. N. )   =>    |- ( ph -> ( ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` R ) +P. Q ) -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )
Theorem	caucvgprprlemaddq 7792*	Lemma for caucvgprpr 7796. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-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 -> 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 -> X e. P. )   &    |- ( ph -> Q e. P. )   &    |- ( ph -> E. r e. N. ( X +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )   =>    |- ( ph -> X <P ( L +P. Q ) )
Theorem	caucvgprprlem1 7793*	Lemma for caucvgprpr 7796. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-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 -> Q e. P. )   &    |- ( ph -> J <N K )   &    |- ( ph -> <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. <P Q )   =>    |- ( ph -> ( F ` K ) <P ( L +P. Q ) )
Theorem	caucvgprprlem2 7794*	Lemma for caucvgprpr 7796. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-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 -> Q e. P. )   &    |- ( ph -> J <N K )   &    |- ( ph -> <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. <P Q )   =>    |- ( ph -> L <P ( ( F ` K ) +P. Q ) )
Theorem	caucvgprprlemlim 7795*	Lemma for caucvgprpr 7796. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-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 -> A. x e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) )
Theorem	caucvgprpr 7796*	A Cauchy sequence of positive reals 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 given value A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This is similar to caucvgpr 7766 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7746) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-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 ) )   =>    |- ( ph -> E. y e. P. A. x e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <P ( y +P. x ) /\ y <P ( ( F ` k ) +P. x ) ) ) )
Theorem	suplocexprlemell 7797*	Lemma for suplocexpr 7809. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- ( B e. U. ( 1st " A ) <-> E. x e. A B e. ( 1st ` x ) )
Theorem	suplocexprlem2b 7798	Lemma for suplocexpr 7809. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
Theorem	suplocexprlemss 7799*	Lemma for suplocexpr 7809. A is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   =>    |- ( ph -> A C_ P. )
Theorem	suplocexprlemml 7800*	Lemma for suplocexpr 7809. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   =>    |- ( ph -> E. s e. Q. s e. U. ( 1st " A ) )