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
Theorem	prarloclemarch 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))
Theorem	prarloclemarch2 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 6486. (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))))
Theorem	ltrnqg 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 ↔ (*Q‘B) <Q (*Q‘A)))
Theorem	ltrnqi 6404	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6403. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (A <Q B → (*Q‘B) <Q (*Q‘A))
Theorem	nnnq 6405	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (A ∈ N → [⟨A, 1𝑜⟩] ~Q ∈ Q)
Theorem	ltnnnq 6406	Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
⊢ ((A ∈ N ∧ B ∈ N) → (A <N B ↔ [⟨A, 1𝑜⟩] ~Q <Q [⟨B, 1𝑜⟩] ~Q ))
Definition	df-enq0 6407*	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)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)))}
Definition	df-nq0 6408	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 )
Definition	df-0nq0 6409	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 6410*	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 6411*	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 6412*	Multiplication on non-negative fractions. This definition is similar to df-mq0 6411 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 6413*	Addition on non-negative fractions. This definition is similar to df-plq0 6410 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 6414	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 6415	The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6418. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (f ~Q0 g → g ~Q0 f)
Theorem	enq0ref 6416	The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6418. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (f ∈ (𝜔 × N) ↔ f ~Q0 f)
Theorem	enq0tr 6417	The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6418. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ ((f ~Q0 g ∧ g ~Q0 ℎ) → f ~Q0 ℎ)
Theorem	enq0er 6418	The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
⊢ ~Q0 Er (𝜔 × N)
Theorem	enq0breq 6419	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 6420	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 6421	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 6422	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ ~Q0 ∈ V
Theorem	nq0ex 6423	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q0 ∈ V
Theorem	nqnq0 6424	A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q ⊆ Q0
Theorem	nq0nn 6425*	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 6426	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 6427	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 6428	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 6429*	Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6432 and mulnnnq0 6433. (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 6430*	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 6431*	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 6432	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 6433	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 6434	Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((A ∈ Q0 ∧ B ∈ Q0) → (A +Q0 B) ∈ Q0)
Theorem	mulclnq0 6435	Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((A ∈ Q0 ∧ B ∈ Q0) → (A ·Q0 B) ∈ Q0)
Theorem	nqpnq0nq 6436	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 6437	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 6438	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 6439	Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (A ∈ Q0 → (0Q0 ·Q0 A) = 0Q0)
Theorem	nq0a0 6440	Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (A ∈ Q0 → (A +Q0 0Q0) = A)
Theorem	nnanq0 6441	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 6442	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 6443	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 6444	A natural number closure law. Lemma for addassnq0 6445. (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 6445	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 6446	Multiplication of non-negative fractions is distributive. Version of distrnq0 6442 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 6447	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 6448	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 6449*	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 6450*	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 6451*	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 ∈ (1st ‘x) implies 𝑟 ∈ Q) and can be simplified as shown at genpdf 6491. 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 6452*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6451 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 6453*	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 6454	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ P ⊆ (𝒫 Q × 𝒫 Q)
Theorem	preqlu 6455	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 6456	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
⊢ P ∈ V
Theorem	elinp 6457*	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 6458	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 6459*	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 6460*	A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃x ∈ Q x ∈ 𝐿)
Theorem	prmu 6461*	A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃x ∈ Q x ∈ 𝑈)
Theorem	prssnql 6462	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 6463	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 6464	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 6465	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 6466	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnql 6467	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 6468	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 6469	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 6470	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 6471*	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 6472*	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 6473*	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 6474	A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A <Q B) → (A ∈ 𝐿 ∨ B ∈ 𝑈))
Theorem	prdisj 6475	A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A ∈ Q) → ¬ (A ∈ 𝐿 ∧ A ∈ 𝑈))
Theorem	prarloclemlt 6476	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6486. (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 6477*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6486. (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 6478	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6486. (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 6479*	Induction step for prarloclem3 6480. (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 6480*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6486. (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 6481*	A slight rearrangement of prarloclem3 6480. Lemma for prarloc 6486. (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 6482*	Subtracting two from a positive integer. Lemma for prarloc 6486. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ ((𝑁 ∈ N ∧ 1𝑜 <N 𝑁) → ∃x ∈ 𝜔 (2𝑜 +𝑜 x) = 𝑁)
Theorem	prarloclem5 6483*	A substitution of zero for y and 𝑁 minus two for x. Lemma for prarloc 6486. (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 6484*	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 6485	Some calculations for prarloc 6486. (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 6486*	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 6487 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Theorem	prarloc2 6487*	A Dedekind cut is arithmetically located. This is a variation of prarloc 6486 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 6488	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
⊢ <P ⊆ (P × P)
Theorem	ltdfpr 6489*	More convenient form of df-iltp 6453. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((A ∈ P ∧ B ∈ P) → (A<P B ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘A) ∧ 𝑞 ∈ (1st ‘B))))
Theorem	genpdflem 6490*	Simplification of upper or lower cut expression. Lemma for genpdf 6491. (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 6491*	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 6492*	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 6493*	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 6494*	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 6495*	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 6496*	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 6497*	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 6498*	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 6499*	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	genpml 6500*	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)))