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Type | Label | Description |
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Statement | ||
Theorem | ltmnqi 6501 | Ordering property of multiplication for positive fractions. One direction of ltmnqg 6499. (Contributed by Jim Kingdon, 9-Dec-2019.) |
⊢ ((𝐴 <_{Q} 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·_{Q} 𝐴) <_{Q} (𝐶 ·_{Q} 𝐵)) | ||
Theorem | lt2addnq 6502 | Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.) |
⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <_{Q} 𝐵 ∧ 𝐶 <_{Q} 𝐷) → (𝐴 +_{Q} 𝐶) <_{Q} (𝐵 +_{Q} 𝐷))) | ||
Theorem | lt2mulnq 6503 | Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.) |
⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <_{Q} 𝐵 ∧ 𝐶 <_{Q} 𝐷) → (𝐴 ·_{Q} 𝐶) <_{Q} (𝐵 ·_{Q} 𝐷))) | ||
Theorem | 1lt2nq 6504 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 1_{Q} <_{Q} (1_{Q} +_{Q} 1_{Q}) | ||
Theorem | ltaddnq 6505 | The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <_{Q} (𝐴 +_{Q} 𝐵)) | ||
Theorem | ltexnqq 6506* | Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_{Q} 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 +_{Q} 𝑥) = 𝐵)) | ||
Theorem | ltexnqi 6507* | Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.) |
⊢ (𝐴 <_{Q} 𝐵 → ∃𝑥 ∈ Q (𝐴 +_{Q} 𝑥) = 𝐵) | ||
Theorem | halfnqq 6508* | One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +_{Q} 𝑥) = 𝐴) | ||
Theorem | halfnq 6509* | One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +_{Q} 𝑥) = 𝐴) | ||
Theorem | nsmallnqq 6510* | There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <_{Q} 𝐴) | ||
Theorem | nsmallnq 6511* | There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <_{Q} 𝐴) | ||
Theorem | subhalfnqq 6512* | There is a number which is less than half of any positive fraction. The case where 𝐴 is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 6508). (Contributed by Jim Kingdon, 25-Nov-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +_{Q} 𝑥) <_{Q} 𝐴) | ||
Theorem | ltbtwnnqq 6513* | There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 <_{Q} 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <_{Q} 𝑥 ∧ 𝑥 <_{Q} 𝐵)) | ||
Theorem | ltbtwnnq 6514* | There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ (𝐴 <_{Q} 𝐵 ↔ ∃𝑥(𝐴 <_{Q} 𝑥 ∧ 𝑥 <_{Q} 𝐵)) | ||
Theorem | archnqq 6515* | For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <_{Q} [⟨𝑥, 1_{𝑜}⟩] ~_{Q} ) | ||
Theorem | prarloclemarch 6516* | A version of the Archimedean property. This variation is "stronger" than archnqq 6515 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 6517* | Like prarloclemarch 6516 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 6601. (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 6518 | Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6519. (Contributed by Jim Kingdon, 29-Dec-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_{Q} 𝐵 ↔ (*_{Q}‘𝐵) <_{Q} (*_{Q}‘𝐴))) | ||
Theorem | ltrnqi 6519 | Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6518. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 <_{Q} 𝐵 → (*_{Q}‘𝐵) <_{Q} (*_{Q}‘𝐴)) | ||
Theorem | nnnq 6520 | 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 6521 | 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 6522* | 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 6523 | 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 6524 | 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 6525* | 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 6526* | 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 6527* | Multiplication on non-negative fractions. This definition is similar to df-mq0 6526 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 6528* | Addition on non-negative fractions. This definition is similar to df-plq0 6525 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 6529 | 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 6530 | The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6533. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ~_{Q0} 𝑔 → 𝑔 ~_{Q0} 𝑓) | ||
Theorem | enq0ref 6531 | The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6533. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~_{Q0} 𝑓) | ||
Theorem | enq0tr 6532 | The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6533. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ ((𝑓 ~_{Q0} 𝑔 ∧ 𝑔 ~_{Q0} ℎ) → 𝑓 ~_{Q0} ℎ) | ||
Theorem | enq0er 6533 | The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~_{Q0} Er (ω × N) | ||
Theorem | enq0breq 6534 | 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 6535 | 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 6536 | 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 6537 | The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ ~_{Q0} ∈ V | ||
Theorem | nq0ex 6538 | The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q_{0} ∈ V | ||
Theorem | nqnq0 6539 | A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q_{0} | ||
Theorem | nq0nn 6540* | 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 6541 | 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 6542 | 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 6543 | 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 6544* | Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6547 and mulnnnq0 6548. (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 6545* | 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 6546* | 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 6547 | 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 6548 | 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 6549 | 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 6550 | 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 6551 | 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 6552 | 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 6553 | 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 6554 | 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 6555 | Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q_{0} → (𝐴 +_{Q0} 0_{Q0}) = 𝐴) | ||
Theorem | nnanq0 6556 | 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 6557 | 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 6558 | 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 6559 | A natural number closure law. Lemma for addassnq0 6560. (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 6560 | 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 6561 | Multiplication of non-negative fractions is distributive. Version of distrnq0 6557 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 6562 | 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 6563 | 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 6564* |
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 6565* | 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 6566* |
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 6606.
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 6567* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 6566 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 6568* |
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 6569 | Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ P ⊆ (𝒫 Q × 𝒫 Q) | ||
Theorem | preqlu 6570 | 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 6571 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
⊢ P ∈ V | ||
Theorem | elinp 6572* | 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 6573 | 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 6574* | 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 6575* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿) | ||
Theorem | prmu 6576* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝑈) | ||
Theorem | prssnql 6577 | 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 6578 | 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 6579 | 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 6580 | 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 6581 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
⊢ ¬ ∅ ∈ P | ||
Theorem | prcdnql 6582 | 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 6583 | 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 6584 | 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 6585 | 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 6586* | 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 6587* | 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 6588* | 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 6589 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <_{Q} 𝐵) → (𝐴 ∈ 𝐿 ∨ 𝐵 ∈ 𝑈)) | ||
Theorem | prdisj 6590 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ Q) → ¬ (𝐴 ∈ 𝐿 ∧ 𝐴 ∈ 𝑈)) | ||
Theorem | prarloclemlt 6591 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (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 6592* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (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 6593 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (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 6594* | Induction step for prarloclem3 6595. (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 6595* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (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 6596* | A slight rearrangement of prarloclem3 6595. Lemma for prarloc 6601. (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 6597* | Subtracting two from a positive integer. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ ((𝑁 ∈ N ∧ 1_{𝑜} <_{N} 𝑁) → ∃𝑥 ∈ ω (2_{𝑜} +_{𝑜} 𝑥) = 𝑁) | ||
Theorem | prarloclem5 6598* | A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 6601. (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 6599* | 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 6600 | Some calculations for prarloc 6601. (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)) |
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