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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprarloclemarch 6401* A version of the Archimedean property. This variation is "stronger" than archnqq 6400 in the sense that we provide an integer which is larger than a given rational A even after being multiplied by a second rational B. (Contributed by Jim Kingdon, 30-Nov-2019.)
((A Q B Q) → x N A <Q ([⟨x, 1𝑜⟩] ~Q ·Q B))

Theoremprarloclemarch2 6402* Like prarloclemarch 6401 but the integer must be at least two, and there is also B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6485. (Contributed by Jim Kingdon, 25-Nov-2019.)
((A Q B Q 𝐶 Q) → x N (1𝑜 <N x A <Q (B +Q ([⟨x, 1𝑜⟩] ~Q ·Q 𝐶))))

Theoremltrnqg 6403 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6404. (Contributed by Jim Kingdon, 29-Dec-2019.)
((A Q B Q) → (A <Q B ↔ (*QB) <Q (*QA)))

Theoremltrnqi 6404 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6403. (Contributed by Jim Kingdon, 24-Sep-2019.)
(A <Q B → (*QB) <Q (*QA))

Theoremnnnq 6405 The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
(A N → [⟨A, 1𝑜⟩] ~Q Q)

Definitiondf-enq0 6406* Define equivalence relation for non-negative fractions. 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, 2-Nov-2019.)
~Q0 = {⟨x, y⟩ ∣ ((x (𝜔 × N) y (𝜔 × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))}

Definitiondf-nq0 6407 Define class of non-negative fractions. 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, 2-Nov-2019.)
Q0 = ((𝜔 × N) / ~Q0 )

Definitiondf-0nq0 6408 Define non-negative fraction constant 0. 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, 5-Nov-2019.)
0Q0 = [⟨∅, 1𝑜⟩] ~Q0

Definitiondf-plq0 6409* Define addition on non-negative fractions. 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, 2-Nov-2019.)
+Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}

Definitiondf-mq0 6410* Define multiplication on non-negative fractions. 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, 2-Nov-2019.)
·Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}

Theoremdfmq0qs 6411* Multiplication on non-negative fractions. This definition is similar to df-mq0 6410 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
·Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}

Theoremdfplq0qs 6412* Addition on non-negative fractions. This definition is similar to df-plq0 6409 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
+Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}

Theoremenq0enq 6413 Equivalence on positive fractions in terms of equivalence on non-negative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
~Q = ( ~Q0 ∩ ((N × N) × (N × N)))

Theoremenq0sym 6414 The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6417. (Contributed by Jim Kingdon, 14-Nov-2019.)
(f ~Q0 gg ~Q0 f)

Theoremenq0ref 6415 The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6417. (Contributed by Jim Kingdon, 14-Nov-2019.)
(f (𝜔 × N) ↔ f ~Q0 f)

Theoremenq0tr 6416 The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6417. (Contributed by Jim Kingdon, 14-Nov-2019.)
((f ~Q0 g g ~Q0 ) → f ~Q0 )

Theoremenq0er 6417 The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
~Q0 Er (𝜔 × N)

Theoremenq0breq 6418 Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
(((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (⟨A, B⟩ ~Q0𝐶, 𝐷⟩ ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))

Theoremenq0eceq 6419 Equivalence class equality of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
(((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨A, B⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))

Theoremnqnq0pi 6420 A non-negative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.)
((A N B N) → [⟨A, B⟩] ~Q0 = [⟨A, B⟩] ~Q )

Theoremenq0ex 6421 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
~Q0 V

Theoremnq0ex 6422 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
Q0 V

Theoremnqnq0 6423 A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
QQ0

Theoremnq0nn 6424* Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
(A Q0wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))

Theoremaddcmpblnq0 6425 Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((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 6426 Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
((((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) ((𝐹 𝜔 𝐺 N) (𝑅 𝜔 𝑆 N))) → (((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 6427 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
((A N B 𝜔 𝐶 N) → [⟨(A ·𝑜 B), (A ·𝑜 𝐶)⟩] ~Q0 = [⟨B, 𝐶⟩] ~Q0 )

Theoremnnnq0lem1 6428* Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6431 and mulnnnq0 6432. (Contributed by Jim Kingdon, 23-Nov-2019.)
(((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ((((w 𝜔 v N) (𝑠 𝜔 f N)) ((u 𝜔 𝑡 N) (g 𝜔 N))) ((w ·𝑜 f) = (v ·𝑜 𝑠) (u ·𝑜 ) = (𝑡 ·𝑜 g))))

Theoremaddnq0mo 6429* 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⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))

Theoremmulnq0mo 6430* 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⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩] ~Q0 ))

Theoremaddnnnq0 6431 Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
(((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨A, B⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((A ·𝑜 𝐷) +𝑜 (B ·𝑜 𝐶)), (B ·𝑜 𝐷)⟩] ~Q0 )

Theoremmulnnnq0 6432 Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
(((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ([⟨A, B⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(A ·𝑜 𝐶), (B ·𝑜 𝐷)⟩] ~Q0 )

Theoremaddclnq0 6433 Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
((A Q0 B Q0) → (A +Q0 B) Q0)

Theoremmulclnq0 6434 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
((A Q0 B Q0) → (A ·Q0 B) Q0)

Theoremnqpnq0nq 6435 A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
((A Q B Q0) → (A +Q0 B) Q)

Theoremnqnq0a 6436 Addition of positive fractions is equal with +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
((A Q B Q) → (A +Q B) = (A +Q0 B))

Theoremnqnq0m 6437 Multiplication of positive fractions is equal with ·Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
((A Q B Q) → (A ·Q B) = (A ·Q0 B))

Theoremnq0m0r 6438 Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(A Q0 → (0Q0 ·Q0 A) = 0Q0)

Theoremnq0a0 6439 Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(A Q0 → (A +Q0 0Q0) = A)

Theoremnnanq0 6440 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 A N) → [⟨(𝑁 +𝑜 𝑀), A⟩] ~Q0 = ([⟨𝑁, A⟩] ~Q0 +Q0 [⟨𝑀, A⟩] ~Q0 ))

Theoremdistrnq0 6441 Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
((A Q0 B Q0 𝐶 Q0) → (A ·Q0 (B +Q0 𝐶)) = ((A ·Q0 B) +Q0 (A ·Q0 𝐶)))

Theoremmulcomnq0 6442 Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
((A Q0 B Q0) → (A ·Q0 B) = (B ·Q0 A))

Theoremaddassnq0lemcl 6443 A natural number closure law. Lemma for addassnq0 6444. (Contributed by Jim Kingdon, 3-Dec-2019.)
(((𝐼 𝜔 𝐽 N) (𝐾 𝜔 𝐿 N)) → (((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) 𝜔 (𝐽 ·𝑜 𝐿) N))

Theoremaddassnq0 6444 Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
((A Q0 B Q0 𝐶 Q0) → ((A +Q0 B) +Q0 𝐶) = (A +Q0 (B +Q0 𝐶)))

Theoremdistnq0r 6445 Multiplication of non-negative fractions is distributive. Version of distrnq0 6441 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
((A Q0 B Q0 𝐶 Q0) → ((B +Q0 𝐶) ·Q0 A) = ((B ·Q0 A) +Q0 (𝐶 ·Q0 A)))

Theoremaddpinq1 6446 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
(A N → [⟨(A +N 1𝑜), 1𝑜⟩] ~Q = ([⟨A, 1𝑜⟩] ~Q +Q 1Q))

Theoremnq02m 6447 Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
(A Q0 → ([⟨2𝑜, 1𝑜⟩] ~Q0 ·Q0 A) = (A +Q0 A))

Definitiondf-inp 6448* 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 (𝑞 Q𝑞 𝑙 𝑟 Q𝑟 u), rounded (𝑞 Q(𝑞 𝑙𝑟 Q(𝑞 <Q 𝑟 𝑟 𝑙)) and likewise for u), disjoint (𝑞 Q¬ (𝑞 𝑙 𝑞 u)) and located (𝑞 Q𝑟 Q(𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))). 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⟩ ∣ (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))}

Definitiondf-i1p 6449* 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 6450* 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 (for example, 𝑟 Q and 𝑟 (1stx)) conditions are redundant and can be simplified as shown at genpdf 6490.

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 = (x P, y P ↦ ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stx) 𝑠 (1sty) 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndx) 𝑠 (2ndy) 𝑞 = (𝑟 +Q 𝑠))}⟩)

Definitiondf-imp 6451* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6450 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 = (x P, y P ↦ ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stx) 𝑠 (1sty) 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndx) 𝑠 (2ndy) 𝑞 = (𝑟 ·Q 𝑠))}⟩)

Definitiondf-iltp 6452* 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⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))}

Theoremnpsspw 6453 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
P ⊆ (𝒫 Q × 𝒫 Q)

Theorempreqlu 6454 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
((A P B P) → (A = B ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))

Theoremnpex 6455 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
P V

Theoremelinp 6456* Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈 P ↔ (((𝐿Q 𝑈Q) (𝑞 Q 𝑞 𝐿 𝑟 Q 𝑟 𝑈)) ((𝑞 Q (𝑞 𝐿𝑟 Q (𝑞 <Q 𝑟 𝑟 𝐿)) 𝑟 Q (𝑟 𝑈𝑞 Q (𝑞 <Q 𝑟 𝑞 𝑈))) 𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝐿 𝑟 𝑈)))))

Theoremprop 6457 A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
(A P → ⟨(1stA), (2ndA)⟩ P)

Theoremelnp1st2nd 6458* Membership in positive reals, using 1st and 2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
(A P ↔ ((A (𝒫 Q × 𝒫 Q) (𝑞 Q 𝑞 (1stA) 𝑟 Q 𝑟 (2ndA))) ((𝑞 Q (𝑞 (1stA) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stA))) 𝑟 Q (𝑟 (2ndA) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndA)))) 𝑞 Q ¬ (𝑞 (1stA) 𝑞 (2ndA)) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1stA) 𝑟 (2ndA))))))

Theoremprml 6459* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈 Px Q x 𝐿)

Theoremprmu 6460* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈 Px Q x 𝑈)

Theoremprssnql 6461 A positive real's lower cut is a subset of the positive fractions. It would presumably be possible to also prove 𝐿Q, but we only need 𝐿Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
(⟨𝐿, 𝑈 P𝐿Q)

Theoremprssnqu 6462 A positive real's upper cut is a subset of the positive fractions. It would presumably be possible to also prove 𝑈Q, but we only need 𝑈Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
(⟨𝐿, 𝑈 P𝑈Q)

Theoremelprnql 6463 An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈 P B 𝐿) → B Q)

Theoremelprnqu 6464 An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈 P B 𝑈) → B Q)

Theorem0npr 6465 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
¬ ∅ P

Theoremprcdnql 6466 A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈 P B 𝐿) → (𝐶 <Q B𝐶 𝐿))

Theoremprcunqu 6467 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
((⟨𝐿, 𝑈 P 𝐶 𝑈) → (𝐶 <Q BB 𝑈))

Theoremprubl 6468 A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
(((⟨𝐿, 𝑈 P B 𝐿) 𝐶 Q) → (¬ 𝐶 𝐿B <Q 𝐶))

Theoremprltlu 6469 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 P B 𝐿 𝐶 𝑈) → B <Q 𝐶)

Theoremprnmaxl 6470* A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
((⟨𝐿, 𝑈 P B 𝐿) → x 𝐿 B <Q x)

Theoremprnminu 6471* An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
((⟨𝐿, 𝑈 P B 𝑈) → x 𝑈 x <Q B)

Theoremprnmaddl 6472* A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
((⟨𝐿, 𝑈 P B 𝐿) → x Q (B +Q x) 𝐿)

Theoremprloc 6473 A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
((⟨𝐿, 𝑈 P A <Q B) → (A 𝐿 B 𝑈))

Theoremprdisj 6474 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
((⟨𝐿, 𝑈 P A Q) → ¬ (A 𝐿 A 𝑈))

Theoremprarloclemlt 6475 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 𝜔 (⟨𝐿, 𝑈 P A 𝐿 𝑃 Q)) y 𝜔) → (A +Q ([⟨(y +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (A +Q ([⟨((y +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))

Theoremprarloclemlo 6476* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 𝜔 (⟨𝐿, 𝑈 P A 𝐿 𝑃 Q)) 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 6477 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 𝜔 (⟨𝐿, 𝑈 P A 𝐿 𝑃 Q)) 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 6478* Induction step for prarloclem3 6479. (Contributed by Jim Kingdon, 9-Nov-2019.)
(((𝑋 𝜔 (⟨𝐿, 𝑈 P A 𝐿 𝑃 Q)) 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 6479* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 27-Oct-2019.)
(((⟨𝐿, 𝑈 P A 𝐿) (𝑋 𝜔 𝑃 Q) 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 6480* A slight rearrangement of prarloclem3 6479. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 4-Nov-2019.)
(((⟨𝐿, 𝑈 P A 𝐿) 𝑃 Q) → (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 6481* Subtracting two from a positive integer. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 5-Nov-2019.)
((𝑁 N 1𝑜 <N 𝑁) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)

Theoremprarloclem5 6482* A substitution of zero for y and 𝑁 minus two for x. Lemma for prarloc 6485. (Contributed by Jim Kingdon, 4-Nov-2019.)
(((⟨𝐿, 𝑈 P A 𝐿) (𝑁 N 𝑃 Q 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 6483* 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 P A 𝐿) (𝑁 N 𝑃 Q 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 6484 Some calculations for prarloc 6485. (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))) ((𝑄 Q (𝑄 +Q 𝑄) <Q 𝑃) (𝑋 Q 𝑀 𝜔))) → B <Q (A +Q 𝑃))

Theoremprarloc 6485* 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 6486 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)

((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))

Theoremprarloc2 6486* A Dedekind cut is arithmetically located. This is a variation of prarloc 6485 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 P 𝑃 Q) → 𝑎 𝐿 (𝑎 +Q 𝑃) 𝑈)

Theoremltrelpr 6487 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
<P ⊆ (P × P)

Theoremltdfpr 6488* More convenient form of df-iltp 6452. (Contributed by Jim Kingdon, 15-Dec-2019.)
((A P B P) → (A<P B𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))

Theoremgenpdflem 6489* Simplification of upper or lower cut expression. Lemma for genpdf 6490. (Contributed by Jim Kingdon, 30-Sep-2019.)
((φ 𝑟 A) → 𝑟 Q)    &   ((φ 𝑠 B) → 𝑠 Q)       (φ → {𝑞 Q𝑟 Q 𝑠 Q (𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠))} = {𝑞 Q𝑟 A 𝑠 B 𝑞 = (𝑟𝐺𝑠)})

Theoremgenpdf 6490* Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
𝐹 = (w P, v P ↦ ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stw) 𝑠 (1stv) 𝑞 = (𝑟𝐺𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndw) 𝑠 (2ndv) 𝑞 = (𝑟𝐺𝑠))}⟩)       𝐹 = (w P, v P ↦ ⟨{𝑞 Q𝑟 (1stw)𝑠 (1stv)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndw)𝑠 (2ndv)𝑞 = (𝑟𝐺𝑠)}⟩)

Theoremgenipv 6491* Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → (A𝐹B) = ⟨{𝑞 Q𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠)}⟩)

Theoremgenplt2i 6492* Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))    &   ((x Q y Q) → (x𝐺y) = (y𝐺x))       ((A <Q B 𝐶 <Q 𝐷) → (A𝐺𝐶) <Q (B𝐺𝐷))

Theoremgenpelxp 6493* Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)       ((A P B P) → (A𝐹B) (𝒫 Q × 𝒫 Q))

Theoremgenpelvl 6494* Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → (𝐶 (1st ‘(A𝐹B)) ↔ g (1stA) (1stB)𝐶 = (g𝐺)))

Theoremgenpelvu 6495* Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → (𝐶 (2nd ‘(A𝐹B)) ↔ g (2ndA) (2ndB)𝐶 = (g𝐺)))

Theoremgenpprecll 6496* Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → ((𝐶 (1stA) 𝐷 (1stB)) → (𝐶𝐺𝐷) (1st ‘(A𝐹B))))

Theoremgenppreclu 6497* Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → ((𝐶 (2ndA) 𝐷 (2ndB)) → (𝐶𝐺𝐷) (2nd ‘(A𝐹B))))

Theoremgenipdm 6498* Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       dom 𝐹 = (P × P)

Theoremgenpml 6499* The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → 𝑞 Q 𝑞 (1st ‘(A𝐹B)))

Theoremgenpmu 6500* The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)       ((A P B P) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B)))

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