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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremenq0ex 6901 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
~Q0 ∈ V

Theoremnq0ex 6902 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
Q0 ∈ V

Theoremnqnq0 6903 A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
QQ0

Theoremnq0nn 6904* Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
(𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))

Theoremaddcmpblnq0 6905 Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) ∧ (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨((𝐴 ·𝑜 𝐺) +𝑜 (𝐵 ·𝑜 𝐹)), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩))

Theoremmulcmpblnq0 6906 Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) ∧ (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨(𝐴 ·𝑜 𝐹), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨(𝐶 ·𝑜 𝑅), (𝐷 ·𝑜 𝑆)⟩))

Theoremmulcanenq0ec 6907 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴N𝐵 ∈ ω ∧ 𝐶N) → [⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )

Theoremnnnq0lem1 6908* Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6911 and mulnnnq0 6912. (Contributed by Jim Kingdon, 23-Nov-2019.)
(((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔))))

Theoremaddnq0mo 6909* There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))

Theoremmulnq0mo 6910* There is at most one result from multiplying non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))

Theoremaddnnnq0 6911 Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Theoremmulnnnq0 6912 Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Theoremaddclnq0 6913 Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴Q0𝐵Q0) → (𝐴 +Q0 𝐵) ∈ Q0)

Theoremmulclnq0 6914 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
((𝐴Q0𝐵Q0) → (𝐴 ·Q0 𝐵) ∈ Q0)

Theoremnqpnq0nq 6915 A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
((𝐴Q𝐵Q0) → (𝐴 +Q0 𝐵) ∈ Q)

Theoremnqnq0a 6916 Addition of positive fractions is equal with +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))

Theoremnqnq0m 6917 Multiplication of positive fractions is equal with ·Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))

Theoremnq0m0r 6918 Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(𝐴Q0 → (0Q0 ·Q0 𝐴) = 0Q0)

Theoremnq0a0 6919 Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(𝐴Q0 → (𝐴 +Q0 0Q0) = 𝐴)

Theoremnnanq0 6920 Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)
((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Theoremdistrnq0 6921 Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
((𝐴Q0𝐵Q0𝐶Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))

Theoremmulcomnq0 6922 Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
((𝐴Q0𝐵Q0) → (𝐴 ·Q0 𝐵) = (𝐵 ·Q0 𝐴))

Theoremaddassnq0lemcl 6923 A natural number closure law. Lemma for addassnq0 6924. (Contributed by Jim Kingdon, 3-Dec-2019.)
(((𝐼 ∈ ω ∧ 𝐽N) ∧ (𝐾 ∈ ω ∧ 𝐿N)) → (((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω ∧ (𝐽 ·𝑜 𝐿) ∈ N))

Theoremaddassnq0 6924 Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴Q0𝐵Q0𝐶Q0) → ((𝐴 +Q0 𝐵) +Q0 𝐶) = (𝐴 +Q0 (𝐵 +Q0 𝐶)))

Theoremdistnq0r 6925 Multiplication of non-negative fractions is distributive. Version of distrnq0 6921 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴Q0𝐵Q0𝐶Q0) → ((𝐵 +Q0 𝐶) ·Q0 𝐴) = ((𝐵 ·Q0 𝐴) +Q0 (𝐶 ·Q0 𝐴)))

Theoremaddpinq1 6926 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴N → [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q))

Theoremnq02m 6927 Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
(𝐴Q0 → ([⟨2𝑜, 1𝑜⟩] ~Q0 ·Q0 𝐴) = (𝐴 +Q0 𝐴))

Definitiondf-inp 6928* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other.

Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers.

A Dedekind cut is an ordered pair of a lower set 𝑙 and an upper set 𝑢 which is inhabited (𝑞Q𝑞𝑙 ∧ ∃𝑟Q𝑟𝑢), rounded (𝑞Q(𝑞𝑙 ↔ ∃𝑟Q(𝑞 <Q 𝑟𝑟𝑙)) and likewise for 𝑢), disjoint (𝑞Q¬ (𝑞𝑙𝑞𝑢)) and located (𝑞Q𝑟Q(𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts.

(Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)

P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))}

Definitiondf-i1p 6929* Define the positive 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 Jim Kingdon, 25-Sep-2019.)
1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩

Definitiondf-iplp 6930* Define addition on positive reals. From Section 11.2.1 of [HoTT], p. (varies). We write this definition to closely resemble the definition in HoTT although some of the conditions are redundant (for example, 𝑟 ∈ (1st𝑥) implies 𝑟Q) and can be simplified as shown at genpdf 6970.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.)

+P = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)

Definitiondf-imp 6931* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6930 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

·P = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)

Definitiondf-iltp 6932* Define ordering on positive reals. We define 𝑥<P 𝑦 if there is a positive fraction 𝑞 which is an element of the upper cut of 𝑥 and the lower cut of 𝑦. From the definition of < in Section 11.2.1 of [HoTT], p. (varies).

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

<P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}

Theoremnpsspw 6933 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
P ⊆ (𝒫 Q × 𝒫 Q)

Theorempreqlu 6934 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))

Theoremnpex 6935 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
P ∈ V

Theoremelinp 6936* Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿Q𝑈Q) ∧ (∃𝑞Q 𝑞𝐿 ∧ ∃𝑟Q 𝑟𝑈)) ∧ ((∀𝑞Q (𝑞𝐿 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝐿)) ∧ ∀𝑟Q (𝑟𝑈 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑈))) ∧ ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝐿𝑟𝑈)))))

Theoremprop 6937 A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
(𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)

Theoremelnp1st2nd 6938* Membership in positive reals, using 1st and 2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
(𝐴P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))

Theoremprml 6939* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝐿)

Theoremprmu 6940* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝑈)

Theoremprssnql 6941 The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P𝐿Q)

Theoremprssnqu 6942 The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P𝑈Q)

Theoremelprnql 6943 An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → 𝐵Q)

Theoremelprnqu 6944 An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → 𝐵Q)

Theorem0npr 6945 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
¬ ∅ ∈ P

Theoremprcdnql 6946 A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → (𝐶 <Q 𝐵𝐶𝐿))

Theoremprcunqu 6947 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝐵𝐵𝑈))

Theoremprubl 6948 A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) ∧ 𝐶Q) → (¬ 𝐶𝐿𝐵 <Q 𝐶))

Theoremprltlu 6949 An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐵 <Q 𝐶)

Theoremprnmaxl 6950* A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → ∃𝑥𝐿 𝐵 <Q 𝑥)

Theoremprnminu 6951* An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥𝑈 𝑥 <Q 𝐵)

Theoremprnmaddl 6952* A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → ∃𝑥Q (𝐵 +Q 𝑥) ∈ 𝐿)

Theoremprloc 6953 A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → (𝐴𝐿𝐵𝑈))

Theoremprdisj 6954 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐴Q) → ¬ (𝐴𝐿𝐴𝑈))

Theoremprarloclemlt 6955 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))

Theoremprarloclemlo 6956* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝐿 → (((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))))

Theoremprarloclemup 6957 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈 → (((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))))

Theoremprarloclem3step 6958* Induction step for prarloclem3 6959. (Contributed by Jim Kingdon, 9-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))

Theoremprarloclem3 6959* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 27-Oct-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑋 ∈ ω ∧ 𝑃Q) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([⟨𝑗, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨(𝑗 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))

Theoremprarloclem4 6960* A slight rearrangement of prarloclem3 6959. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 4-Nov-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ 𝑃Q) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([⟨𝑗, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨(𝑗 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)))

Theoremprarloclemn 6961* Subtracting two from a positive integer. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 5-Nov-2019.)
((𝑁N ∧ 1𝑜 <N 𝑁) → ∃𝑥 ∈ ω (2𝑜 +𝑜 𝑥) = 𝑁)

Theoremprarloclem5 6962* A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 6965. (Contributed by Jim Kingdon, 4-Nov-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑥), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))

Theoremprarloclem 6963* A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from 𝐴 to 𝐴 +Q (𝑁 ·Q 𝑃) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 1𝑜 <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([⟨𝑗, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨(𝑗 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))

Theoremprarloclemcalc 6964 Some calculations for prarloc 6965. (Contributed by Jim Kingdon, 26-Oct-2019.)
(((𝐴 = (𝑋 +Q0 ([⟨𝑀, 1𝑜⟩] ~Q0 ·Q0 𝑄)) ∧ 𝐵 = (𝑋 +Q ([⟨(𝑀 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑄))) ∧ ((𝑄Q ∧ (𝑄 +Q 𝑄) <Q 𝑃) ∧ (𝑋Q𝑀 ∈ ω))) → 𝐵 <Q (𝐴 +Q 𝑃))

Theoremprarloc 6965* A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are within that tolerance of each other.

Usually, proofs will be shorter if they use prarloc2 6966 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)

((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃))

Theoremprarloc2 6966* A Dedekind cut is arithmetically located. This is a variation of prarloc 6965 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)

Theoremltrelpr 6967 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
<P ⊆ (P × P)

Theoremltdfpr 6968* More convenient form of df-iltp 6932. (Contributed by Jim Kingdon, 15-Dec-2019.)
((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))

Theoremgenpdflem 6969* Simplification of upper or lower cut expression. Lemma for genpdf 6970. (Contributed by Jim Kingdon, 30-Sep-2019.)
((𝜑𝑟𝐴) → 𝑟Q)    &   ((𝜑𝑠𝐵) → 𝑠Q)       (𝜑 → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)})

Theoremgenpdf 6970* Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)       𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)

Theoremgenipv 6971* Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)

Theoremgenplt2i 6972* Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴𝐺𝐶) <Q (𝐵𝐺𝐷))

Theoremgenpelxp 6973* Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)       ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Theoremgenpelvl 6974* Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))

Theoremgenpelvu 6975* Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))

Theoremgenpprecll 6976* Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ((𝐶 ∈ (1st𝐴) ∧ 𝐷 ∈ (1st𝐵)) → (𝐶𝐺𝐷) ∈ (1st ‘(𝐴𝐹𝐵))))

Theoremgenppreclu 6977* Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ((𝐶 ∈ (2nd𝐴) ∧ 𝐷 ∈ (2nd𝐵)) → (𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵))))

Theoremgenipdm 6978* Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       dom 𝐹 = (P × P)

Theoremgenpml 6979* The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))

Theoremgenpmu 6980* The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))

Theoremgenpcdl 6981* Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))

Theoremgenpcuu 6982* Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Theoremgenprndl 6983* The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))

Theoremgenprndu 6984* The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Theoremgenpdisj 6985* The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))

Theoremgenpassl 6986* Associativity of lower cuts. Lemma for genpassg 6988. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))       ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))

Theoremgenpassu 6987* Associativity of upper cuts. Lemma for genpassg 6988. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))       ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))

Theoremgenpassg 6988* Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))       ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Theoremaddnqprllem 6989 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐺𝐿) ∧ 𝑋Q) → (𝑋 <Q 𝑆 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝐿))

Theoremaddnqprulem 6990 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈))

Theoremaddnqprl 6991 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))

Theoremaddnqpru 6992 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocprlemlt 6993 Lemma for addlocpr 6998. The 𝑄 <Q (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))

Theoremaddlocprlemeqgt 6994 Lemma for addlocpr 6998. This is a step used in both the 𝑄 = (𝐷 +Q 𝐸) and (𝐷 +Q 𝐸) <Q 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Theoremaddlocprlemeq 6995 Lemma for addlocpr 6998. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocprlemgt 6996 Lemma for addlocpr 6998. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocprlem 6997 Lemma for addlocpr 6998. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocpr 6998* Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 6965 to both 𝐴 and 𝐵, and uses nqtri3or 6858 rather than prloc 6953 to decide whether 𝑞 is too big to be in the lower cut of 𝐴 +P 𝐵 (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))

Theoremaddclpr 6999 Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)

Theoremplpvlu 7000* Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)

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