P. 79 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltaprlem 7801	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 7802	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 7803*	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 7804	Strong extensionality of addition (ordering version). This is similar to addext 8753 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 7805*	Membership in the lower cut of B. Lemma for recexpr 7821. (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 7806*	Membership in the upper cut of B. Lemma for recexpr 7821. (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 7807*	B is inhabited. Lemma for recexpr 7821. (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 7808*	The lower cut of B is open. Lemma for recexpr 7821. (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 7809*	The lower cut of B is lower. Lemma for recexpr 7821. (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 7810*	The upper cut of B is open. Lemma for recexpr 7821. (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 7811*	The upper cut of B is upper. Lemma for recexpr 7821. (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 7812*	B is rounded. Lemma for recexpr 7821. (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 7813*	B is disjoint. Lemma for recexpr 7821. (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 7814*	B is located. Lemma for recexpr 7821. (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 7815*	B is a positive real. Lemma for recexpr 7821. (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 7816*	The lower cut of one is a subset of the lower cut of A .P. B. Lemma for recexpr 7821. (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 7817*	The upper cut of one is a subset of the upper cut of A .P. B. Lemma for recexpr 7821. (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 7818*	The lower cut of A .P. B is a subset of the lower cut of one. Lemma for recexpr 7821. (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 7819*	The upper cut of A .P. B is a subset of the upper cut of one. Lemma for recexpr 7821. (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 7820*	B is the reciprocal of A. Lemma for recexpr 7821. (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 7821*	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 7822	Lemma for aptipr 7824. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
Theorem	aptiprlemu 7823	Lemma for aptipr 7824. (Contributed by Jim Kingdon, 28-Jan-2020.)
|- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
Theorem	aptipr 7824	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 7825	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 7826*	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 7736. (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 7827*	Lemma for cauappcvgprlemladdrl 7840. 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 7828*	Lemma for cauappcvgpr 7845. 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 7829*	Lemma for cauappcvgpr 7845. 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 7830*	Lemma for cauappcvgpr 7845. 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 7831*	Lemma for cauappcvgpr 7845. 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 7832*	Lemma for cauappcvgpr 7845. 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 7833*	Lemma for cauappcvgpr 7845. 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 7834*	Lemma for cauappcvgpr 7845. 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 7835*	Lemma for cauappcvgpr 7845. 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 7836*	Lemma for cauappcvgpr 7845. 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 7837*	Lemma for cauappcvgprlemladd 7841. 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 7838*	Lemma for cauappcvgprlemladd 7841. 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 7839*	Lemma for cauappcvgprlemladd 7841. 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 7840*	Lemma for cauappcvgprlemladd 7841. 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 7841*	Lemma for cauappcvgpr 7845. 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 7842*	Lemma for cauappcvgpr 7845. 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 7843*	Lemma for cauappcvgpr 7845. 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 7844*	Lemma for cauappcvgpr 7845. 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 7845*	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 7865 and caucvgprpr 7895 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 7846*	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 7847*	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 7848	Lemma for caucvgpr 7865. 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 7849*	Lemma for caucvgpr 7865. 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 7850*	Lemma for caucvgpr 7865. 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 7851*	Lemma for caucvgpr 7865. 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 7852*	Lemma for caucvgpr 7865. 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 7853*	Lemma for caucvgpr 7865. 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 7854*	Lemma for caucvgpr 7865. 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 7855*	Lemma for caucvgpr 7865. 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 7856*	Lemma for caucvgpr 7865. 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 7857*	Lemma for caucvgpr 7865. 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 7858*	Lemma for caucvgpr 7865. 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 7859*	Lemma for caucvgpr 7865. 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 7860*	Lemma for caucvgpr 7865. 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 7861*	Lemma for caucvgpr 7865. 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 7862*	Lemma for caucvgpr 7865. 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 7863*	Lemma for caucvgpr 7865. 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 7864*	Lemma for caucvgpr 7865. 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 7865*	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 7845 and caucvgprpr 7895. Reading cauappcvgpr 7845 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 7866*	Lemma for caucvgprpr 7895. 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 7867*	Lemma for caucvgprpr 7895. Rearranging some expressions for caucvgprprlemloc 7886. (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 7868*	Lemma for caucvgprpr 7895. 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 7869*	Lemma for caucvgprpr 7895. 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 7870*	Lemma for caucvgprpr 7895. 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 7871*	Lemma for caucvgprpr 7895. 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 7872*	Lemma for caucvgprpr 7895. 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 7873*	Lemma for caucvgprpr 7895. 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 7874*	Lemma for caucvgprpr 7895. 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 7875*	Lemma for caucvgprpr 7895. 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 7876*	Lemma for caucvgprpr 7895. 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 7877*	Lemma for caucvgprpr 7895. 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 7878*	Lemma for caucvgprpr 7895. 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 7879*	Lemma for caucvgprpr 7895. 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 7880*	Lemma for caucvgprpr 7895. 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 7881*	Lemma for caucvgprpr 7895. 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 7882*	Lemma for caucvgprpr 7895. 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 7883*	Lemma for caucvgprpr 7895. 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 7884*	Lemma for caucvgprpr 7895. 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 7885*	Lemma for caucvgprpr 7895. 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 7886*	Lemma for caucvgprpr 7895. 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 7887*	Lemma for caucvgprpr 7895. 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 7888*	Lemma for caucvgprpr 7895. The putative limit is a positive real. Like caucvgprprlemcl 7887 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 7889*	Lemma for caucvgprpr 7895. 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 7890*	Lemma for caucvgprpr 7895. 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 7891*	Lemma for caucvgprpr 7895. 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 7892*	Lemma for caucvgprpr 7895. 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 7893*	Lemma for caucvgprpr 7895. 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 7894*	Lemma for caucvgprpr 7895. 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 7895*	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 7865 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7845) 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 7896*	Lemma for suplocexpr 7908. 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 7897	Lemma for suplocexpr 7908. 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 7898*	Lemma for suplocexpr 7908. 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 7899*	Lemma for suplocexpr 7908. 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 ) )
Theorem	suplocexprlemrl 7900*	Lemma for suplocexpr 7908. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-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. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )