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Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulclnq0 6301 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
((A Q0 B Q0) → (A ·Q0 B) Q0)

Theoremnqpnq0nq 6302 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 6303 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 6304 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 6305 Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(A Q0 → (0Q0 ·Q0 A) = 0Q0)

Theoremnq0a0 6306 Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(A Q0 → (A +Q0 0Q0) = A)

Theoremnnanq0 6307 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 6308 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 6309 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 6310 A natural number closure law. Lemma for addassnq0 6311. (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 6311 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 6312 Multiplication of non-negative fractions is distributive. Version of distrnq0 6308 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)))

Theoremnq02m 6313 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 6314* 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 6315* 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 6316* 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 6356.

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 6317* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6316 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 6318* 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 6319 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
P ⊆ (𝒫 Q × 𝒫 Q)

Theorempreqlu 6320 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 6321 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
P V

Theoremelinp 6322* 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 6323 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 6324* 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 6325* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈 Px Q x 𝐿)

Theoremprmu 6326* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈 Px Q x 𝑈)

Theoremprssnql 6327 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 6328 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 6329 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 6330 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 6331 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
¬ ∅ P

Theoremprcdnql 6332 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 6333 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 6334 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 6335 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 6336* 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 6337* 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 6338* 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 6339 A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
((⟨𝐿, 𝑈 P A <Q B) → (A 𝐿 B 𝑈))

Theoremprdisj 6340 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
((⟨𝐿, 𝑈 P A Q) → ¬ (A 𝐿 A 𝑈))

Theoremprarloclemlt 6341 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6342* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6343 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6344* Induction step for prarloclem3 6345. (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 6345* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6346* A slight rearrangement of prarloclem3 6345. Lemma for prarloc 6351. (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 6347* Subtracting two from a positive integer. Lemma for prarloc 6351. (Contributed by Jim Kingdon, 5-Nov-2019.)
((𝑁 N 1𝑜 <N 𝑁) → x 𝜔 (2𝑜 +𝑜 x) = 𝑁)

Theoremprarloclem5 6348* A substitution of zero for y and 𝑁 minus two for x. Lemma for prarloc 6351. (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 6349* 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 6350 Some calculations for prarloc 6351. (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 6351* 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. (Contributed by Jim Kingdon, 22-Oct-2019.)
((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))

Theoremprarloc2 6352* A Dedekind cut is arithmetically located. This is a variation of prarloc 6351 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 6353 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
<P ⊆ (P × P)

Theoremltdfpr 6354* More convenient form of df-iltp 6318. (Contributed by Jim Kingdon, 15-Dec-2019.)
((A P B P) → (A<P B𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))

Theoremgenpdflem 6355* Simplification of upper or lower cut expression. Lemma for genpdf 6356. (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 6356* 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 6357* 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 6358* 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 6359* 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 6360* 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 6361* 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 6362* 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 6363* 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 6364* 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)

Theoremgenpelpw 6365* Result of general operation on positive reals is an ordered pair of sets of positive fractions. (Contributed by Jim Kingdon, 4-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) (𝒫 Q × 𝒫 Q))

Theoremgenpml 6366* 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 6367* 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)))

Theoremgenpcdl 6368* Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-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 g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))       ((A P B P) → (f (1st ‘(A𝐹B)) → (x <Q fx (1st ‘(A𝐹B)))))

Theoremgenpcuu 6369* Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-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 g (2ndA)) (B P (2ndB))) x Q) → ((g𝐺) <Q xx (2nd ‘(A𝐹B))))       ((A P B P) → (f (2nd ‘(A𝐹B)) → (f <Q xx (2nd ‘(A𝐹B)))))

Theoremgenprndl 6370* The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-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)    &   ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))    &   ((x Q y Q) → (x𝐺y) = (y𝐺x))    &   ((((A P g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))       ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))

Theoremgenprndu 6371* The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-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)    &   ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))    &   ((x Q y Q) → (x𝐺y) = (y𝐺x))    &   ((((A P g (2ndA)) (B P (2ndB))) x Q) → ((g𝐺) <Q xx (2nd ‘(A𝐹B))))       ((A P B P) → 𝑟 Q (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))

Theoremgenpdisj 6372* The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (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)    &   ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))    &   ((x Q y Q) → (x𝐺y) = (y𝐺x))       ((A P B P) → 𝑞 Q ¬ (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))))

Theoremgenpassl 6373* Associativity of lower cuts. Lemma for genpassg 6375. (Contributed by Jim Kingdon, 11-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)    &   dom 𝐹 = (P × P)    &   ((f P g P) → (f𝐹g) P)    &   ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))       ((A P B P 𝐶 P) → (1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))))

Theoremgenpassu 6374* Associativity of upper cuts. Lemma for genpassg 6375. (Contributed by Jim Kingdon, 11-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)    &   dom 𝐹 = (P × P)    &   ((f P g P) → (f𝐹g) P)    &   ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))       ((A P B P 𝐶 P) → (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(A𝐹(B𝐹𝐶))))

Theoremgenpassg 6375* Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-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)    &   dom 𝐹 = (P × P)    &   ((f P g P) → (f𝐹g) P)    &   ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))       ((A P B P 𝐶 P) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))

Theoremaddnqprllem 6376 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((⟨𝐿, 𝑈 P 𝐺 𝐿) 𝑋 Q) → (𝑋 <Q 𝑆 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) 𝐿))

Theoremaddnqprulem 6377 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((⟨𝐿, 𝑈 P 𝐺 𝑈) 𝑋 Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) 𝑈))

Theoremaddnqprl 6378 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 (1st ‘(A +P B))))

Theoremaddnqpru 6379 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 (2nd ‘(A +P B))))

Theoremaddlocprlemlt 6380 Lemma for addlocpr 6385. The 𝑄 <Q (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6381 Lemma for addlocpr 6385. 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.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6382 Lemma for addlocpr 6385. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6383 Lemma for addlocpr 6385. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6384 Lemma for addlocpr 6385. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6385* 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 6351 to both A and B, and uses nqtri3or 6249 rather than prloc 6339 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.)
((A P B P) → 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))

Theoremaddclpr 6386 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.)
((A P B P) → (A +P B) P)

Theoremplpvlu 6387* Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((A P B P) → (A +P B) = ⟨{x Qy (1stA)z (1stB)x = (y +Q z)}, {x Qy (2ndA)z (2ndB)x = (y +Q z)}⟩)

Theoremmpvlu 6388* Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((A P B P) → (A ·P B) = ⟨{x Qy (1stA)z (1stB)x = (y ·Q z)}, {x Qy (2ndA)z (2ndB)x = (y ·Q z)}⟩)

Theoremdmplp 6389 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
dom +P = (P × P)

Theoremdmmp 6390 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
dom ·P = (P × P)

Theoremnqprm 6391* A cut produced from a rational is inhabited. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → (𝑞 Q 𝑞 {xx <Q A} 𝑟 Q 𝑟 {xA <Q x}))

Theoremnqprrnd 6392* A cut produced from a rational is rounded. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → (𝑞 Q (𝑞 {xx <Q A} ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 {xx <Q A})) 𝑟 Q (𝑟 {xA <Q x} ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 {xA <Q x}))))

Theoremnqprdisj 6393* A cut produced from a rational is disjoint. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q𝑞 Q ¬ (𝑞 {xx <Q A} 𝑞 {xA <Q x}))

Theoremnqprloc 6394* A cut produced from a rational is located. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 {xx <Q A} 𝑟 {xA <Q x})))

Theoremnqprlu 6395* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → ⟨{xx <Q A}, {xA <Q x}⟩ P)

Theoremltnqex 6396 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xx <Q A} V

Theoremgtnqex 6397 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xA <Q x} V

Theorem1pr 6398 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
1P P

Theorem1prl 6399 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(1st ‘1P) = {xx <Q 1Q}

Theorem1pru 6400 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(2nd ‘1P) = {x ∣ 1Q <Q x}

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