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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mulpipq2 7201 | Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) |
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·_{pQ} 𝐵) = ⟨((1^{st} ‘𝐴) ·_{N} (1^{st} ‘𝐵)), ((2^{nd} ‘𝐴) ·_{N} (2^{nd} ‘𝐵))⟩) | ||
Theorem | mulpipq 7202 | Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ·_{pQ} ⟨𝐶, 𝐷⟩) = ⟨(𝐴 ·_{N} 𝐶), (𝐵 ·_{N} 𝐷)⟩) | ||
Theorem | mulpipqqs 7203 | Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~_{Q} ·_{Q} [⟨𝐶, 𝐷⟩] ~_{Q} ) = [⟨(𝐴 ·_{N} 𝐶), (𝐵 ·_{N} 𝐷)⟩] ~_{Q} ) | ||
Theorem | ordpipqqs 7204 | Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~_{Q} <_{Q} [⟨𝐶, 𝐷⟩] ~_{Q} ↔ (𝐴 ·_{N} 𝐷) <_{N} (𝐵 ·_{N} 𝐶))) | ||
Theorem | addclnq 7205 | Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +_{Q} 𝐵) ∈ Q) | ||
Theorem | mulclnq 7206 | Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·_{Q} 𝐵) ∈ Q) | ||
Theorem | dmaddpqlem 7207* | Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7209. (Contributed by Jim Kingdon, 15-Sep-2019.) |
⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~_{Q} ) | ||
Theorem | nqpi 7208* | Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7207 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~_{Q} )) | ||
Theorem | dmaddpq 7209 | Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) |
⊢ dom +_{Q} = (Q × Q) | ||
Theorem | dmmulpq 7210 | Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) |
⊢ dom ·_{Q} = (Q × Q) | ||
Theorem | addcomnqg 7211 | Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +_{Q} 𝐵) = (𝐵 +_{Q} 𝐴)) | ||
Theorem | addassnqg 7212 | Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +_{Q} 𝐵) +_{Q} 𝐶) = (𝐴 +_{Q} (𝐵 +_{Q} 𝐶))) | ||
Theorem | mulcomnqg 7213 | Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·_{Q} 𝐵) = (𝐵 ·_{Q} 𝐴)) | ||
Theorem | mulassnqg 7214 | Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_{Q} 𝐵) ·_{Q} 𝐶) = (𝐴 ·_{Q} (𝐵 ·_{Q} 𝐶))) | ||
Theorem | mulcanenq 7215 | Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·_{N} 𝐵), (𝐴 ·_{N} 𝐶)⟩ ~_{Q} ⟨𝐵, 𝐶⟩) | ||
Theorem | mulcanenqec 7216 | Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → [⟨(𝐴 ·_{N} 𝐵), (𝐴 ·_{N} 𝐶)⟩] ~_{Q} = [⟨𝐵, 𝐶⟩] ~_{Q} ) | ||
Theorem | distrnqg 7217 | Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_{Q} (𝐵 +_{Q} 𝐶)) = ((𝐴 ·_{Q} 𝐵) +_{Q} (𝐴 ·_{Q} 𝐶))) | ||
Theorem | 1qec 7218 | The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) |
⊢ (𝐴 ∈ N → 1_{Q} = [⟨𝐴, 𝐴⟩] ~_{Q} ) | ||
Theorem | mulidnq 7219 | Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) |
⊢ (𝐴 ∈ Q → (𝐴 ·_{Q} 1_{Q}) = 𝐴) | ||
Theorem | recexnq 7220* | Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·_{Q} 𝑦) = 1_{Q})) | ||
Theorem | recmulnqg 7221 | Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*_{Q}‘𝐴) = 𝐵 ↔ (𝐴 ·_{Q} 𝐵) = 1_{Q})) | ||
Theorem | recclnq 7222 | Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) |
⊢ (𝐴 ∈ Q → (*_{Q}‘𝐴) ∈ Q) | ||
Theorem | recidnq 7223 | A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) |
⊢ (𝐴 ∈ Q → (𝐴 ·_{Q} (*_{Q}‘𝐴)) = 1_{Q}) | ||
Theorem | recrecnq 7224 | Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) |
⊢ (𝐴 ∈ Q → (*_{Q}‘(*_{Q}‘𝐴)) = 𝐴) | ||
Theorem | rec1nq 7225 | Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.) |
⊢ (*_{Q}‘1_{Q}) = 1_{Q} | ||
Theorem | nqtri3or 7226 | Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_{Q} 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <_{Q} 𝐴)) | ||
Theorem | ltdcnq 7227 | Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <_{Q} 𝐵) | ||
Theorem | ltsonq 7228 | 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) |
⊢ <_{Q} Or Q | ||
Theorem | nqtric 7229 | Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_{Q} 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_{Q} 𝐴))) | ||
Theorem | ltanqg 7230 | Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <_{Q} 𝐵 ↔ (𝐶 +_{Q} 𝐴) <_{Q} (𝐶 +_{Q} 𝐵))) | ||
Theorem | ltmnqg 7231 | Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <_{Q} 𝐵 ↔ (𝐶 ·_{Q} 𝐴) <_{Q} (𝐶 ·_{Q} 𝐵))) | ||
Theorem | ltanqi 7232 | Ordering property of addition for positive fractions. One direction of ltanqg 7230. (Contributed by Jim Kingdon, 9-Dec-2019.) |
⊢ ((𝐴 <_{Q} 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +_{Q} 𝐴) <_{Q} (𝐶 +_{Q} 𝐵)) | ||
Theorem | ltmnqi 7233 | Ordering property of multiplication for positive fractions. One direction of ltmnqg 7231. (Contributed by Jim Kingdon, 9-Dec-2019.) |
⊢ ((𝐴 <_{Q} 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·_{Q} 𝐴) <_{Q} (𝐶 ·_{Q} 𝐵)) | ||
Theorem | lt2addnq 7234 | 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 7235 | 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 7236 | 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 7237 | 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 7238* | 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 7239* | 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 7240* | 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 7241* | 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 7242* | There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <_{Q} 𝐴) | ||
Theorem | nsmallnq 7243* | 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 7244* | 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 7240). (Contributed by Jim Kingdon, 25-Nov-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +_{Q} 𝑥) <_{Q} 𝐴) | ||
Theorem | ltbtwnnqq 7245* | 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 7246* | 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 7247* | 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 7248* | A version of the Archimedean property. This variation is "stronger" than archnqq 7247 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 7249* | Like prarloclemarch 7248 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 7333. (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 7250 | Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7251. (Contributed by Jim Kingdon, 29-Dec-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_{Q} 𝐵 ↔ (*_{Q}‘𝐵) <_{Q} (*_{Q}‘𝐴))) | ||
Theorem | ltrnqi 7251 | Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7250. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 <_{Q} 𝐵 → (*_{Q}‘𝐵) <_{Q} (*_{Q}‘𝐴)) | ||
Theorem | nnnq 7252 | 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 7253 | 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 7254* | Define equivalence relation for nonnegative 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 7255 | Define class of nonnegative 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 7256 | Define nonnegative 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 7257* | Define addition on nonnegative 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 7258* | Define multiplication on nonnegative 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 7259* | Multiplication on nonnegative fractions. This definition is similar to df-mq0 7258 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 7260* | Addition on nonnegative fractions. This definition is similar to df-plq0 7257 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 7261 | Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~_{Q} = ( ~_{Q0} ∩ ((N × N) × (N × N))) | ||
Theorem | enq0sym 7262 | The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7265. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ~_{Q0} 𝑔 → 𝑔 ~_{Q0} 𝑓) | ||
Theorem | enq0ref 7263 | The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7265. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~_{Q0} 𝑓) | ||
Theorem | enq0tr 7264 | The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7265. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ ((𝑓 ~_{Q0} 𝑔 ∧ 𝑔 ~_{Q0} ℎ) → 𝑓 ~_{Q0} ℎ) | ||
Theorem | enq0er 7265 | The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~_{Q0} Er (ω × N) | ||
Theorem | enq0breq 7266 | Equivalence relation for nonnegative 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 7267 | Equivalence class equality of nonnegative 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 7268 | A nonnegative 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 7269 | The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ ~_{Q0} ∈ V | ||
Theorem | nq0ex 7270 | The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q_{0} ∈ V | ||
Theorem | nqnq0 7271 | A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q_{0} | ||
Theorem | nq0nn 7272* | Decomposition of a nonnegative 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 7273 | Lemma showing compatibility of addition on nonnegative 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 7274 | Lemma showing compatibility of multiplication on nonnegative 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 7275 | 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 7276* | Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7279 and mulnnnq0 7280. (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 7277* | There is at most one result from adding nonnegative 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 7278* | There is at most one result from multiplying nonnegative 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 7279 | Addition of nonnegative 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 7280 | Multiplication of nonnegative 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 7281 | Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q_{0} ∧ 𝐵 ∈ Q_{0}) → (𝐴 +_{Q0} 𝐵) ∈ Q_{0}) | ||
Theorem | mulclnq0 7282 | Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q_{0} ∧ 𝐵 ∈ Q_{0}) → (𝐴 ·_{Q0} 𝐵) ∈ Q_{0}) | ||
Theorem | nqpnq0nq 7283 | A positive fraction plus a nonnegative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q_{0}) → (𝐴 +_{Q0} 𝐵) ∈ Q) | ||
Theorem | nqnq0a 7284 | 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 7285 | 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 7286 | Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q_{0} → (0_{Q0} ·_{Q0} 𝐴) = 0_{Q0}) | ||
Theorem | nq0a0 7287 | Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q_{0} → (𝐴 +_{Q0} 0_{Q0}) = 𝐴) | ||
Theorem | nnanq0 7288 | Addition of nonnegative 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 7289 | Multiplication of nonnegative 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 7290 | Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.) |
⊢ ((𝐴 ∈ Q_{0} ∧ 𝐵 ∈ Q_{0}) → (𝐴 ·_{Q0} 𝐵) = (𝐵 ·_{Q0} 𝐴)) | ||
Theorem | addassnq0lemcl 7291 | A natural number closure law. Lemma for addassnq0 7292. (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 7292 | Addition of nonnegative 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 7293 | Multiplication of nonnegative fractions is distributive. Version of distrnq0 7289 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 7294 | 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 7295 | Multiply a nonnegative 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 7296* |
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 7297* | 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 7298* |
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 7338.
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 7299* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 7298 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 7300* |
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)))} |
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