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Type | Label | Description |
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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.) |
⊢ 0_{Q0} = [⟨∅, 1_{𝑜}⟩] ~_{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 ∈ Q_{0} ∧ y ∈ Q_{0}) ∧ ∃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 ∈ Q_{0} ∧ y ∈ Q_{0}) ∧ ∃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 Q_{0} (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 Q_{0} (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.) |
⊢ Q_{0} ∈ V | ||
Theorem | nqnq0 6416 | A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q_{0} | ||
Theorem | nq0nn 6417* | Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.) |
⊢ (A ∈ Q_{0} → ∃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 ∈ Q_{0} ∧ B ∈ Q_{0}) → (A +_{Q0} B) ∈ Q_{0}) | ||
Theorem | mulclnq0 6427 | Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0}) → (A ·_{Q0} B) ∈ Q_{0}) | ||
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 ∈ Q_{0}) → (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 ∈ Q_{0} → (0_{Q0} ·_{Q0} A) = 0_{Q0}) | ||
Theorem | nq0a0 6432 | Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (A ∈ Q_{0} → (A +_{Q0} 0_{Q0}) = 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 ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → (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 ∈ Q_{0} ∧ B ∈ Q_{0}) → (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 ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → ((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 ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → ((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} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} = ([⟨A, 1_{𝑜}⟩] ~_{Q} +_{Q} 1_{Q})) | ||
Theorem | nq02m 6440 | Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ (A ∈ Q_{0} → ([⟨2_{𝑜}, 1_{𝑜}⟩] ~_{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.) |
⊢ 1_{P} = ⟨{𝑙 ∣ 𝑙 <_{Q} 1_{Q}}, {u ∣ 1_{Q} <_{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
∈ (1^{st} ‘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 ∈ (1^{st} ‘x) ∧ 𝑠 ∈ (1^{st} ‘y) ∧ 𝑞 = (𝑟 +_{Q} 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2^{nd} ‘x) ∧ 𝑠 ∈ (2^{nd} ‘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 ∈ (1^{st} ‘x) ∧ 𝑠 ∈ (1^{st} ‘y) ∧ 𝑞 = (𝑟 ·_{Q} 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2^{nd} ‘x) ∧ 𝑠 ∈ (2^{nd} ‘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 ∈ (2^{nd} ‘x) ∧ 𝑞 ∈ (1^{st} ‘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 ↔ ((1^{st} ‘A) = (1^{st} ‘B) ∧ (2^{nd} ‘A) = (2^{nd} ‘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 → ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩ ∈ P) | ||
Theorem | elnp1st2nd 6451* | Membership in positive reals, using 1^{st} and 2^{nd} 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 ∈ (1^{st} ‘A) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^{nd} ‘A))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^{st} ‘A) ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘A))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^{nd} ‘A) ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘A)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^{st} ‘A) ∧ 𝑞 ∈ (2^{nd} ‘A)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ (1^{st} ‘A) ∨ 𝑟 ∈ (2^{nd} ‘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} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) <_{Q} (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{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} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝐿 → (((A +_{Q0} ([⟨y, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} suc 𝑋), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈) → ∃y ∈ 𝜔 ((A +_{Q0} ([⟨y, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{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} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈 → (((A +_{Q0} ([⟨y, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} suc 𝑋), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈) → ∃y ∈ 𝜔 ((A +_{Q0} ([⟨y, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{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, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} suc 𝑋), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈)) → ∃y ∈ 𝜔 ((A +_{Q0} ([⟨y, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{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, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈)) → ∃𝑗 ∈ 𝜔 ((A +_{Q0} ([⟨𝑗, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨(𝑗 +_{𝑜} 2_{𝑜}), 1_{𝑜}⟩] ~_{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, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} x), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ 𝜔 ((A +_{Q0} ([⟨𝑗, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨(𝑗 +_{𝑜} 2_{𝑜}), 1_{𝑜}⟩] ~_{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 ∧ 1_{𝑜} <_{N} 𝑁) → ∃x ∈ 𝜔 (2_{𝑜} +_{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 ∧ 1_{𝑜} <_{N} 𝑁) ∧ (A +_{Q} ([⟨𝑁, 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈) → ∃x ∈ 𝜔 ∃y ∈ 𝜔 ((A +_{Q0} ([⟨y, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} x), 1_{𝑜}⟩] ~_{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 ∧ 1_{𝑜} <_{N} 𝑁) ∧ (A +_{Q} ([⟨𝑁, 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ 𝜔 ((A +_{Q0} ([⟨𝑗, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑃)) ∈ 𝐿 ∧ (A +_{Q} ([⟨(𝑗 +_{𝑜} 2_{𝑜}), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) ∈ 𝑈)) | ||
Theorem | prarloclemcalc 6477 | Some calculations for prarloc 6478. (Contributed by Jim Kingdon, 26-Oct-2019.) |
⊢ (((A = (𝑋 +_{Q0} ([⟨𝑀, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} 𝑄)) ∧ B = (𝑋 +_{Q} ([⟨(𝑀 +_{𝑜} 2_{𝑜}), 1_{𝑜}⟩] ~_{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 ∈ (2^{nd} ‘A) ∧ 𝑞 ∈ (1^{st} ‘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 ∈ (1^{st} ‘w) ∧ 𝑠 ∈ (1^{st} ‘v) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2^{nd} ‘w) ∧ 𝑠 ∈ (2^{nd} ‘v) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩) ⇒ ⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1^{st} ‘w)∃𝑠 ∈ (1^{st} ‘v)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2^{nd} ‘w)∃𝑠 ∈ (2^{nd} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → (A𝐹B) = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1^{st} ‘A)∃𝑠 ∈ (1^{st} ‘B)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2^{nd} ‘A)∃𝑠 ∈ (2^{nd} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (1^{st} ‘(A𝐹B)) ↔ ∃g ∈ (1^{st} ‘A)∃ℎ ∈ (1^{st} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (2^{nd} ‘(A𝐹B)) ↔ ∃g ∈ (2^{nd} ‘A)∃ℎ ∈ (2^{nd} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ((𝐶 ∈ (1^{st} ‘A) ∧ 𝐷 ∈ (1^{st} ‘B)) → (𝐶𝐺𝐷) ∈ (1^{st} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ((𝐶 ∈ (2^{nd} ‘A) ∧ 𝐷 ∈ (2^{nd} ‘B)) → (𝐶𝐺𝐷) ∈ (2^{nd} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^{st} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2^{nd} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) & ⊢ ((((A ∈ P ∧ g ∈ (1^{st} ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (1^{st} ‘B))) ∧ x ∈ Q) → (x <_{Q} (g𝐺ℎ) → x ∈ (1^{st} ‘(A𝐹B)))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → (f ∈ (1^{st} ‘(A𝐹B)) → (x <_{Q} f → x ∈ (1^{st} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) & ⊢ ((((A ∈ P ∧ g ∈ (2^{nd} ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (2^{nd} ‘B))) ∧ x ∈ Q) → ((g𝐺ℎ) <_{Q} x → x ∈ (2^{nd} ‘(A𝐹B)))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → (f ∈ (2^{nd} ‘(A𝐹B)) → (f <_{Q} x → x ∈ (2^{nd} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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 ∈ (1^{st} ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (1^{st} ‘B))) ∧ x ∈ Q) → (x <_{Q} (g𝐺ℎ) → x ∈ (1^{st} ‘(A𝐹B)))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^{st} ‘(A𝐹B)) ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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 ∈ (2^{nd} ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (2^{nd} ‘B))) ∧ x ∈ Q) → ((g𝐺ℎ) <_{Q} x → x ∈ (2^{nd} ‘(A𝐹B)))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2^{nd} ‘(A𝐹B)) ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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 ∈ (1^{st} ‘(A𝐹B)) ∧ 𝑞 ∈ (2^{nd} ‘(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 ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘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) → (1^{st} ‘((A𝐹B)𝐹𝐶)) = (1^{st} ‘(A𝐹(B𝐹𝐶)))) |
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