P. 65 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Definition	df-0nq0 6401	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
Definition	df-plq0 6402*	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) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q0 ∧ y = [⟨u, f⟩] ~Q0 ) ∧ z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}
Definition	df-mq0 6403*	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) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q0 ∧ y = [⟨u, f⟩] ~Q0 ) ∧ z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}
Theorem	dfmq0qs 6404*	Multiplication on non-negative fractions. This definition is similar to df-mq0 6403 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
⊢ ·Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ((𝜔 × N) / ~Q0 ) ∧ y ∈ ((𝜔 × N) / ~Q0 )) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q0 ∧ y = [⟨u, f⟩] ~Q0 ) ∧ z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}
Theorem	dfplq0qs 6405*	Addition on non-negative fractions. This definition is similar to df-plq0 6402 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
⊢ +Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ((𝜔 × N) / ~Q0 ) ∧ y ∈ ((𝜔 × N) / ~Q0 )) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q0 ∧ y = [⟨u, f⟩] ~Q0 ) ∧ z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}
Theorem	enq0enq 6406	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)))
Theorem	enq0sym 6407	The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6410. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (f ~Q0 g → g ~Q0 f)
Theorem	enq0ref 6408	The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6410. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (f ∈ (𝜔 × N) ↔ f ~Q0 f)
Theorem	enq0tr 6409	The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6410. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ ((f ~Q0 g ∧ g ~Q0 ℎ) → f ~Q0 ℎ)
Theorem	enq0er 6410	The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
⊢ ~Q0 Er (𝜔 × N)
Theorem	enq0breq 6411	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⟩ ~Q0 ⟨𝐶, 𝐷⟩ ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
Theorem	enq0eceq 6412	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)))
Theorem	nqnq0pi 6413	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 )
Theorem	enq0ex 6414	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ ~Q0 ∈ V
Theorem	nq0ex 6415	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q0 ∈ V
Theorem	nqnq0 6416	A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q ⊆ Q0
Theorem	nq0nn 6417*	Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
⊢ (A ∈ Q0 → ∃w∃v((w ∈ 𝜔 ∧ v ∈ N) ∧ A = [⟨w, v⟩] ~Q0 ))
Theorem	addcmpblnq0 6418	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)⟩))
Theorem	mulcmpblnq0 6419	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)⟩))
Theorem	mulcanenq0ec 6420	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 )
Theorem	nnnq0lem1 6421*	Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6424 and mulnnnq0 6425. (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, ℎ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((w ∈ 𝜔 ∧ v ∈ N) ∧ (𝑠 ∈ 𝜔 ∧ f ∈ N)) ∧ ((u ∈ 𝜔 ∧ 𝑡 ∈ N) ∧ (g ∈ 𝜔 ∧ ℎ ∈ N))) ∧ ((w ·𝑜 f) = (v ·𝑜 𝑠) ∧ (u ·𝑜 ℎ) = (𝑡 ·𝑜 g))))
Theorem	addnq0mo 6422*	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 )) → ∃*z∃w∃v∃u∃𝑡((A = [⟨w, v⟩] ~Q0 ∧ B = [⟨u, 𝑡⟩] ~Q0 ) ∧ z = [⟨((w ·𝑜 𝑡) +𝑜 (v ·𝑜 u)), (v ·𝑜 𝑡)⟩] ~Q0 ))
Theorem	mulnq0mo 6423*	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 )) → ∃*z∃w∃v∃u∃𝑡((A = [⟨w, v⟩] ~Q0 ∧ B = [⟨u, 𝑡⟩] ~Q0 ) ∧ z = [⟨(w ·𝑜 u), (v ·𝑜 𝑡)⟩] ~Q0 ))
Theorem	addnnnq0 6424	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 )
Theorem	mulnnnq0 6425	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 )
Theorem	addclnq0 6426	Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((A ∈ Q0 ∧ B ∈ Q0) → (A +Q0 B) ∈ Q0)
Theorem	mulclnq0 6427	Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((A ∈ Q0 ∧ B ∈ Q0) → (A ·Q0 B) ∈ Q0)
Theorem	nqpnq0nq 6428	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)
Theorem	nqnq0a 6429	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))
Theorem	nqnq0m 6430	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))
Theorem	nq0m0r 6431	Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (A ∈ Q0 → (0Q0 ·Q0 A) = 0Q0)
Theorem	nq0a0 6432	Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (A ∈ Q0 → (A +Q0 0Q0) = A)
Theorem	nnanq0 6433	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 ))
Theorem	distrnq0 6434	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)))
Theorem	mulcomnq0 6435	Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
⊢ ((A ∈ Q0 ∧ B ∈ Q0) → (A ·Q0 B) = (B ·Q0 A))
Theorem	addassnq0lemcl 6436	A natural number closure law. Lemma for addassnq0 6437. (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))
Theorem	addassnq0 6437	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)))
Theorem	distnq0r 6438	Multiplication of non-negative fractions is distributive. Version of distrnq0 6434 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)))
Theorem	addpinq1 6439	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))
Theorem	nq02m 6440	Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ (A ∈ Q0 → ([⟨2𝑜, 1𝑜⟩] ~Q0 ·Q0 A) = (A +Q0 A))
Definition	df-inp 6441*	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 ∈ 𝑙 ↔ ∃𝑟 ∈ Q(𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) and likewise for u), disjoint (∀𝑞 ∈ Q¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ u)) and located (∀𝑞 ∈ Q∀𝑟 ∈ 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⟩ ∣ (((𝑙 ⊆ Q ∧ u ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ u)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ u ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ u))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ u) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ u))))}
Definition	df-i1p 6442*	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 = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {u ∣ 1Q <Q u}⟩
Definition	df-iplp 6443*	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 ∈ (1st ‘x)) conditions are redundant and can be simplified as shown at genpdf 6483. 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 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘x) ∧ 𝑠 ∈ (1st ‘y) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘x) ∧ 𝑠 ∈ (2nd ‘y) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
Definition	df-imp 6444*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6443 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 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘x) ∧ 𝑠 ∈ (1st ‘y) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘x) ∧ 𝑠 ∈ (2nd ‘y) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)
Definition	df-iltp 6445*	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 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))}
Theorem	npsspw 6446	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ P ⊆ (𝒫 Q × 𝒫 Q)
Theorem	preqlu 6447	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 ↔ ((1st ‘A) = (1st ‘B) ∧ (2nd ‘A) = (2nd ‘B))))
Theorem	npex 6448	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
⊢ P ∈ V
Theorem	elinp 6449*	Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿 ⊆ Q ∧ 𝑈 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝐿 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑈)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝐿)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑈))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝐿 ∧ 𝑞 ∈ 𝑈) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))))
Theorem	prop 6450	A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
Theorem	elnp1st2nd 6451*	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 ∈ (1st ‘A) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘A))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘A) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘A))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘A) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘A)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘A) ∧ 𝑞 ∈ (2nd ‘A)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘A) ∨ 𝑟 ∈ (2nd ‘A))))))
Theorem	prml 6452*	A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃x ∈ Q x ∈ 𝐿)
Theorem	prmu 6453*	A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃x ∈ Q x ∈ 𝑈)
Theorem	prssnql 6454	A positive real's lower cut is a subset of the positive fractions. It would presumably be possible to also prove 𝐿 ⊊ Q, but we only need 𝐿 ⊆ Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → 𝐿 ⊆ Q)
Theorem	prssnqu 6455	A positive real's upper cut is a subset of the positive fractions. It would presumably be possible to also prove 𝑈 ⊊ Q, but we only need 𝑈 ⊆ Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → 𝑈 ⊆ Q)
Theorem	elprnql 6456	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)
Theorem	elprnqu 6457	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)
Theorem	0npr 6458	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnql 6459	A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ B ∈ 𝐿) → (𝐶 <Q B → 𝐶 ∈ 𝐿))
Theorem	prcunqu 6460	An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐶 ∈ 𝑈) → (𝐶 <Q B → B ∈ 𝑈))
Theorem	prubl 6461	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 ∈ 𝐿 → B <Q 𝐶))
Theorem	prltlu 6462	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)
Theorem	prnmaxl 6463*	A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ B ∈ 𝐿) → ∃x ∈ 𝐿 B <Q x)
Theorem	prnminu 6464*	An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ B ∈ 𝑈) → ∃x ∈ 𝑈 x <Q B)
Theorem	prnmaddl 6465*	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)
Theorem	prloc 6466	A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A <Q B) → (A ∈ 𝐿 ∨ B ∈ 𝑈))
Theorem	prdisj 6467	A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A ∈ Q) → ¬ (A ∈ 𝐿 ∧ A ∈ 𝑈))
Theorem	prarloclemlt 6468	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6478. (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)))
Theorem	prarloclemlo 6469*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6478. (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))))
Theorem	prarloclemup 6470	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6478. (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))))
Theorem	prarloclem3step 6471*	Induction step for prarloclem3 6472. (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))
Theorem	prarloclem3 6472*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6478. (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))
Theorem	prarloclem4 6473*	A slight rearrangement of prarloclem3 6472. Lemma for prarloc 6478. (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)))
Theorem	prarloclemn 6474*	Subtracting two from a positive integer. Lemma for prarloc 6478. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ ((𝑁 ∈ N ∧ 1𝑜 <N 𝑁) → ∃x ∈ 𝜔 (2𝑜 +𝑜 x) = 𝑁)
Theorem	prarloclem5 6475*	A substitution of zero for y and 𝑁 minus two for x. Lemma for prarloc 6478. (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))
Theorem	prarloclem 6476*	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))
Theorem	prarloclemcalc 6477	Some calculations for prarloc 6478. (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))
Theorem	prarloc 6478*	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 6479 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Theorem	prarloc2 6479*	A Dedekind cut is arithmetically located. This is a variation of prarloc 6478 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)
Theorem	ltrelpr 6480	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
⊢ <P ⊆ (P × P)
Theorem	ltdfpr 6481*	More convenient form of df-iltp 6445. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((A ∈ P ∧ B ∈ P) → (A<P B ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B))))
Theorem	genpdflem 6482*	Simplification of upper or lower cut expression. Lemma for genpdf 6483. (Contributed by Jim Kingdon, 30-Sep-2019.)
⊢ ((φ ∧ 𝑟 ∈ A) → 𝑟 ∈ Q)    &   ⊢ ((φ ∧ 𝑠 ∈ B) → 𝑠 ∈ Q)    ⇒   ⊢ (φ → {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ A ∧ 𝑠 ∈ B ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ A ∃𝑠 ∈ B 𝑞 = (𝑟𝐺𝑠)})
Theorem	genpdf 6483*	Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘w) ∧ 𝑠 ∈ (1st ‘v) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘w) ∧ 𝑠 ∈ (2nd ‘v) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)    ⇒   ⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘w)∃𝑠 ∈ (1st ‘v)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘w)∃𝑠 ∈ (2nd ‘v)𝑞 = (𝑟𝐺𝑠)}⟩)
Theorem	genipv 6484*	Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (A𝐹B) = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘A)∃𝑠 ∈ (1st ‘B)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘A)∃𝑠 ∈ (2nd ‘B)𝑞 = (𝑟𝐺𝑠)}⟩)
Theorem	genplt2i 6485*	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))
Theorem	genpelxp 6486*	Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (A𝐹B) ∈ (𝒫 Q × 𝒫 Q))
Theorem	genpelvl 6487*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (1st ‘(A𝐹B)) ↔ ∃g ∈ (1st ‘A)∃ℎ ∈ (1st ‘B)𝐶 = (g𝐺ℎ)))
Theorem	genpelvu 6488*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (2nd ‘(A𝐹B)) ↔ ∃g ∈ (2nd ‘A)∃ℎ ∈ (2nd ‘B)𝐶 = (g𝐺ℎ)))
Theorem	genpprecll 6489*	Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ((𝐶 ∈ (1st ‘A) ∧ 𝐷 ∈ (1st ‘B)) → (𝐶𝐺𝐷) ∈ (1st ‘(A𝐹B))))
Theorem	genppreclu 6490*	Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ((𝐶 ∈ (2nd ‘A) ∧ 𝐷 ∈ (2nd ‘B)) → (𝐶𝐺𝐷) ∈ (2nd ‘(A𝐹B))))
Theorem	genipdm 6491*	Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ dom 𝐹 = (P × P)
Theorem	genpelpw 6492*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (A𝐹B) ∈ (𝒫 Q × 𝒫 Q))
Theorem	genpml 6493*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(A𝐹B)))
Theorem	genpmu 6494*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))
Theorem	genpcdl 6495*	Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    &   ⊢ ((((A ∈ P ∧ g ∈ (1st ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (1st ‘B))) ∧ x ∈ Q) → (x <Q (g𝐺ℎ) → x ∈ (1st ‘(A𝐹B))))    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (f ∈ (1st ‘(A𝐹B)) → (x <Q f → x ∈ (1st ‘(A𝐹B)))))
Theorem	genpcuu 6496*	Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    &   ⊢ ((((A ∈ P ∧ g ∈ (2nd ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (2nd ‘B))) ∧ x ∈ Q) → ((g𝐺ℎ) <Q x → x ∈ (2nd ‘(A𝐹B))))    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (f ∈ (2nd ‘(A𝐹B)) → (f <Q x → x ∈ (2nd ‘(A𝐹B)))))
Theorem	genprndl 6497*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ 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 ∈ (1st ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (1st ‘B))) ∧ x ∈ Q) → (x <Q (g𝐺ℎ) → x ∈ (1st ‘(A𝐹B))))    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(A𝐹B)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(A𝐹B)))))
Theorem	genprndu 6498*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ 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 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (2nd ‘B))) ∧ x ∈ Q) → ((g𝐺ℎ) <Q x → x ∈ (2nd ‘(A𝐹B))))    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(A𝐹B)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(A𝐹B)))))
Theorem	genpdisj 6499*	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 ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ 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))))
Theorem	genpassl 6500*	Associativity of lower cuts. Lemma for genpassg 6502. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y𝐺z))}⟩)    &   ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)    &   ⊢ dom 𝐹 = (P × P)    &   ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P)    &   ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ)))    ⇒   ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))))