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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, 1_{𝑜}⟩] ~_{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 (1_{𝑜} <_{N} x ∧ A <_{Q} (B +_{Q} ([⟨x, 1_{𝑜}⟩] ~_{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, 1_{𝑜}⟩] ~_{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, 1_{𝑜}⟩] ~_{Q} <_{Q} [⟨B, 1_{𝑜}⟩] ~_{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.) |
⊢ Q_{0} = ((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.) |
⊢ 0_{Q0} = [⟨∅, 1_{𝑜}⟩] ~_{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 ∈ 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 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 ∈ 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 6412* | Multiplication on non-negative fractions. This definition is similar to df-mq0 6411 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 6413* | Addition on non-negative fractions. This definition is similar to df-plq0 6410 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 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.) |
⊢ Q_{0} ∈ V | ||
Theorem | nqnq0 6424 | A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q_{0} | ||
Theorem | nq0nn 6425* | 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 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 ∈ Q_{0} ∧ B ∈ Q_{0}) → (A +_{Q0} B) ∈ Q_{0}) | ||
Theorem | mulclnq0 6435 | 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 6436 | 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 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 ∈ Q_{0} → (0_{Q0} ·_{Q0} A) = 0_{Q0}) | ||
Theorem | nq0a0 6440 | Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (A ∈ Q_{0} → (A +_{Q0} 0_{Q0}) = 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 ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → (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 ∈ Q_{0} ∧ B ∈ Q_{0}) → (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 ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → ((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 ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → ((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} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} = ([⟨A, 1_{𝑜}⟩] ~_{Q} +_{Q} 1_{Q})) | ||
Theorem | nq02m 6448 | 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 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.) |
⊢ 1_{P} = ⟨{𝑙 ∣ 𝑙 <_{Q} 1_{Q}}, {u ∣ 1_{Q} <_{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
∈ (1^{st} ‘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 ∈ (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 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 ∈ (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 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 ∈ (2^{nd} ‘x) ∧ 𝑞 ∈ (1^{st} ‘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 ↔ ((1^{st} ‘A) = (1^{st} ‘B) ∧ (2^{nd} ‘A) = (2^{nd} ‘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 → ⟨(1^{st} ‘A), (2^{nd} ‘A)⟩ ∈ P) | ||
Theorem | elnp1st2nd 6459* | 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 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} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝑃)) <_{Q} (A +_{Q} ([⟨((y +_{𝑜} 2_{𝑜}) +_{𝑜} 𝑋), 1_{𝑜}⟩] ~_{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} 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 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} 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 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, 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 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, 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 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, 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 6482* | Subtracting two from a positive integer. Lemma for prarloc 6486. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ ((𝑁 ∈ N ∧ 1_{𝑜} <_{N} 𝑁) → ∃x ∈ 𝜔 (2_{𝑜} +_{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 ∧ 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 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 ∧ 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 6485 | Some calculations for prarloc 6486. (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 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 ∈ (2^{nd} ‘A) ∧ 𝑞 ∈ (1^{st} ‘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 ∈ (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 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 ∈ (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 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 ∈ (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 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 ∈ (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 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 ∈ (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 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 ∈ (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 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 ∈ (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 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 ∈ (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 | 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 ∈ (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))) |
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