P. 81 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 8001-8100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	caucvgprlemcanl 8001*	Lemma for cauappcvgprlemladdrl 8014. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)


Theorem	cauappcvgprlemm 8002*	Lemma for cauappcvgpr 8019. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
$/\$ $/\$

Theorem	cauappcvgprlemopl 8003*	Lemma for cauappcvgpr 8019. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
$/\$ $/\$ $/\$

Theorem	cauappcvgprlemlol 8004*	Lemma for cauappcvgpr 8019. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
$/\$ $/\$ $/\$

Theorem	cauappcvgprlemopu 8005*	Lemma for cauappcvgpr 8019. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
$/\$ $/\$ $/\$

Theorem	cauappcvgprlemupu 8006*	Lemma for cauappcvgpr 8019. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
$/\$ $/\$ $/\$

Theorem	cauappcvgprlemrnd 8007*	Lemma for cauappcvgpr 8019. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
$/\$ $/\$ $/\$ $/\$

Theorem	cauappcvgprlemdisj 8008*	Lemma for cauappcvgpr 8019. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
$/\$ $/\$

Theorem	cauappcvgprlemloc 8009*	Lemma for cauappcvgpr 8019. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
$/\$ $\/$

Theorem	cauappcvgprlemcl 8010*	Lemma for cauappcvgpr 8019. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
$/\$

Theorem	cauappcvgprlemladdfu 8011*	Lemma for cauappcvgprlemladd 8015. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
$/\$

Theorem	cauappcvgprlemladdfl 8012*	Lemma for cauappcvgprlemladd 8015. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
$/\$

Theorem	cauappcvgprlemladdru 8013*	Lemma for cauappcvgprlemladd 8015. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
$/\$

Theorem	cauappcvgprlemladdrl 8014*	Lemma for cauappcvgprlemladd 8015. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
$/\$

Theorem	cauappcvgprlemladd 8015*	Lemma for cauappcvgpr 8019. This takes and offsets it by the positive fraction . (Contributed by Jim Kingdon, 23-Jun-2020.)
$/\$

Theorem	cauappcvgprlem1 8016*	Lemma for cauappcvgpr 8019. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
$/\$

Theorem	cauappcvgprlem2 8017*	Lemma for cauappcvgpr 8019. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
$/\$

Theorem	cauappcvgprlemlim 8018*	Lemma for cauappcvgpr 8019. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
$/\$ $/\$

Theorem	cauappcvgpr 8019*	A Cauchy approximation has a limit. A Cauchy approximation, here , 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 is rather than . We also specify that every term needs to be larger than a fraction , 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 8039 and caucvgprpr 8069 but is somewhat simpler, so reading this one first may help understanding the other two. (Contributed by Jim Kingdon, 19-Jun-2020.)
$/\$ $/\$

Theorem	archrecnq 8020*	Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)


Theorem	archrecpr 8021*	Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)


Theorem	caucvgprlemk 8022	Lemma for caucvgpr 8039. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)


Theorem	caucvgprlemnkj 8023*	Lemma for caucvgpr 8039. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
$/\$ $/\$

Theorem	caucvgprlemnbj 8024*	Lemma for caucvgpr 8039. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
$/\$

Theorem	caucvgprlemm 8025*	Lemma for caucvgpr 8039. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
$/\$ $/\$

Theorem	caucvgprlemopl 8026*	Lemma for caucvgpr 8039. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprlemlol 8027*	Lemma for caucvgpr 8039. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprlemopu 8028*	Lemma for caucvgpr 8039. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprlemupu 8029*	Lemma for caucvgpr 8039. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprlemrnd 8030*	Lemma for caucvgpr 8039. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
$/\$ $/\$ $/\$ $/\$

Theorem	caucvgprlemdisj 8031*	Lemma for caucvgpr 8039. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
$/\$ $/\$

Theorem	caucvgprlemloc 8032*	Lemma for caucvgpr 8039. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
$/\$ $\/$

Theorem	caucvgprlemcl 8033*	Lemma for caucvgpr 8039. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
$/\$

Theorem	caucvgprlemladdfu 8034*	Lemma for caucvgpr 8039. Adding after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
$/\$

Theorem	caucvgprlemladdrl 8035*	Lemma for caucvgpr 8039. Adding after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
$/\$

Theorem	caucvgprlem1 8036*	Lemma for caucvgpr 8039. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
$/\$

Theorem	caucvgprlem2 8037*	Lemma for caucvgpr 8039. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
$/\$

Theorem	caucvgprlemlim 8038*	Lemma for caucvgpr 8039. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
$/\$ $/\$

Theorem	caucvgpr 8039*	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 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 , 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 8019 and caucvgprpr 8069. Reading cauappcvgpr 8019 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.)
$/\$ $/\$

Theorem	caucvgprprlemk 8040*	Lemma for caucvgprpr 8069. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)


Theorem	caucvgprprlemloccalc 8041*	Lemma for caucvgprpr 8069. Rearranging some expressions for caucvgprprlemloc 8060. (Contributed by Jim Kingdon, 8-Feb-2021.)


Theorem	caucvgprprlemell 8042*	Lemma for caucvgprpr 8069. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
$/\$

Theorem	caucvgprprlemelu 8043*	Lemma for caucvgprpr 8069. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
$/\$

Theorem	caucvgprprlemcbv 8044*	Lemma for caucvgprpr 8069. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
$/\$ $/\$

Theorem	caucvgprprlemval 8045*	Lemma for caucvgprpr 8069. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemnkltj 8046*	Lemma for caucvgprpr 8069. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemnkeqj 8047*	Lemma for caucvgprpr 8069. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemnjltk 8048*	Lemma for caucvgprpr 8069. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemnkj 8049*	Lemma for caucvgprpr 8069. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
$/\$ $/\$

Theorem	caucvgprprlemnbj 8050*	Lemma for caucvgprpr 8069. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
$/\$

Theorem	caucvgprprlemml 8051*	Lemma for caucvgprpr 8069. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
$/\$

Theorem	caucvgprprlemmu 8052*	Lemma for caucvgprpr 8069. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
$/\$

Theorem	caucvgprprlemm 8053*	Lemma for caucvgprpr 8069. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$

Theorem	caucvgprprlemopl 8054*	Lemma for caucvgprpr 8069. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemlol 8055*	Lemma for caucvgprpr 8069. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemopu 8056*	Lemma for caucvgprpr 8069. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemupu 8057*	Lemma for caucvgprpr 8069. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$ $/\$

Theorem	caucvgprprlemrnd 8058*	Lemma for caucvgprpr 8069. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$ $/\$ $/\$

Theorem	caucvgprprlemdisj 8059*	Lemma for caucvgprpr 8069. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $/\$

Theorem	caucvgprprlemloc 8060*	Lemma for caucvgprpr 8069. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
$/\$ $\/$

Theorem	caucvgprprlemcl 8061*	Lemma for caucvgprpr 8069. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)
$/\$

Theorem	caucvgprprlemclphr 8062*	Lemma for caucvgprpr 8069. The putative limit is a positive real. Like caucvgprprlemcl 8061 but without a disjoint variable condition between and . (Contributed by Jim Kingdon, 19-Jun-2021.)
$/\$

Theorem	caucvgprprlemexbt 8063*	Lemma for caucvgprpr 8069. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
$/\$

Theorem	caucvgprprlemexb 8064*	Lemma for caucvgprpr 8069. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
$/\$

Theorem	caucvgprprlemaddq 8065*	Lemma for caucvgprpr 8069. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
$/\$

Theorem	caucvgprprlem1 8066*	Lemma for caucvgprpr 8069. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
$/\$

Theorem	caucvgprprlem2 8067*	Lemma for caucvgprpr 8069. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
$/\$

Theorem	caucvgprprlemlim 8068*	Lemma for caucvgprpr 8069. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
$/\$ $/\$

Theorem	caucvgprpr 8069*	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 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 , to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This is similar to caucvgpr 8039 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 8019) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)
$/\$ $/\$

Theorem	suplocexprlemell 8070*	Lemma for suplocexpr 8082. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)


Theorem	suplocexprlem2b 8071	Lemma for suplocexpr 8082. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)


Theorem	suplocexprlemss 8072*	Lemma for suplocexpr 8082. is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
$\/$

Theorem	suplocexprlemml 8073*	Lemma for suplocexpr 8082. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
$\/$

Theorem	suplocexprlemrl 8074*	Lemma for suplocexpr 8082. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
$\/$ $/\$

Theorem	suplocexprlemmu 8075*	Lemma for suplocexpr 8082. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
$\/$

Theorem	suplocexprlemru 8076*	Lemma for suplocexpr 8082. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
$\/$ $/\$

Theorem	suplocexprlemdisj 8077*	Lemma for suplocexpr 8082. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
$\/$ $/\$

Theorem	suplocexprlemloc 8078*	Lemma for suplocexpr 8082. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
$\/$ $\/$

Theorem	suplocexprlemex 8079*	Lemma for suplocexpr 8082. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
$\/$

Theorem	suplocexprlemub 8080*	Lemma for suplocexpr 8082. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
$\/$

Theorem	suplocexprlemlub 8081*	Lemma for suplocexpr 8082. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
$\/$

Theorem	suplocexpr 8082*	An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
$\/$ $/\$

Definition	df-enr 8083*	Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
$/\$ $/\$ $/\$ $/\$

Definition	df-nr 8084	Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)


Definition	df-plr 8085*	Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
$/\$ $/\$ $/\$ $/\$

Definition	df-mr 8086*	Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
$/\$ $/\$ $/\$ $/\$

Definition	df-ltr 8087*	Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.)
$/\$ $/\$ $/\$ $/\$

Definition	df-0r 8088	Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)


Definition	df-1r 8089	Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)


Definition	df-m1r 8090	Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.)


Theorem	enrbreq 8091	Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)
$/\$ $/\$ $/\$

Theorem	enrer 8092	The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)


Theorem	enreceq 8093	Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
$/\$ $/\$ $/\$

Theorem	enrex 8094	The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)


Theorem	ltrelsr 8095	Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)


Theorem	addcmpblnr 8096	Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	mulcmpblnrlemg 8097	Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	mulcmpblnr 8098	Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	prsrlem1 8099*	Decomposing signed reals into positive reals. Lemma for addsrpr 8102 and mulsrpr 8103. (Contributed by Jim Kingdon, 30-Dec-2019.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	addsrmo 8100*	There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
$/\$ $/\$ $/\$

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