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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addpipqqslem 7301 | Lemma for addpipqqs 7302. (Contributed by Jim Kingdon, 11-Sep-2019.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → 〈((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)〉 ∈ (N × N)) | ||
Theorem | addpipqqs 7302 | Addition 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 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)〉] ~Q ) | ||
Theorem | mulpipq2 7303 | 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) = 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) | ||
Theorem | mulpipq 7304 | 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 7305 | 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 7306 | 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 7307 | Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q) | ||
Theorem | mulclnq 7308 | Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) | ||
Theorem | dmaddpqlem 7309* | Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7311. (Contributed by Jim Kingdon, 15-Sep-2019.) |
⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [〈𝑤, 𝑣〉] ~Q ) | ||
Theorem | nqpi 7310* | Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7309 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 7311 | Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) |
⊢ dom +Q = (Q × Q) | ||
Theorem | dmmulpq 7312 | Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) |
⊢ dom ·Q = (Q × Q) | ||
Theorem | addcomnqg 7313 | 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 7314 | 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 7315 | 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 7316 | 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 7317 | 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 7318 | 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 7319 | 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 7320 | The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) |
⊢ (𝐴 ∈ N → 1Q = [〈𝐴, 𝐴〉] ~Q ) | ||
Theorem | mulidnq 7321 | Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) |
⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) | ||
Theorem | recexnq 7322* | Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)) | ||
Theorem | recmulnqg 7323 | 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) = 1Q)) | ||
Theorem | recclnq 7324 | 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 7325 | 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)) = 1Q) | ||
Theorem | recrecnq 7326 | 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 7327 | Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.) |
⊢ (*Q‘1Q) = 1Q | ||
Theorem | nqtri3or 7328 | 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 7329 | Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵) | ||
Theorem | ltsonq 7330 | '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 7331 | 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 7332 | 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 7333 | 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 7334 | Ordering property of addition for positive fractions. One direction of ltanqg 7332. (Contributed by Jim Kingdon, 9-Dec-2019.) |
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)) | ||
Theorem | ltmnqi 7335 | Ordering property of multiplication for positive fractions. One direction of ltmnqg 7333. (Contributed by Jim Kingdon, 9-Dec-2019.) |
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)) | ||
Theorem | lt2addnq 7336 | 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 7337 | 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 7338 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 1Q <Q (1Q +Q 1Q) | ||
Theorem | ltaddnq 7339 | 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 7340* | 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 7341* | 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 7342* | 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 7343* | 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 7344* | There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐴) | ||
Theorem | nsmallnq 7345* | 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 7346* | 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 7342). (Contributed by Jim Kingdon, 25-Nov-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴) | ||
Theorem | ltbtwnnqq 7347* | 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 7348* | 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 7349* | 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, 1o〉] ~Q ) | ||
Theorem | prarloclemarch 7350* | A version of the Archimedean property. This variation is "stronger" than archnqq 7349 in the sense that we provide an integer which is larger than a given rational 𝐴 even after being multiplied by a second rational 𝐵. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N 𝐴 <Q ([〈𝑥, 1o〉] ~Q ·Q 𝐵)) | ||
Theorem | prarloclemarch2 7351* | Like prarloclemarch 7350 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 7435. (Contributed by Jim Kingdon, 25-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ N (1o <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([〈𝑥, 1o〉] ~Q ·Q 𝐶)))) | ||
Theorem | ltrnqg 7352 | Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7353. (Contributed by Jim Kingdon, 29-Dec-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))) | ||
Theorem | ltrnqi 7353 | Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7352. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴)) | ||
Theorem | nnnq 7354 | The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.) |
⊢ (𝐴 ∈ N → [〈𝐴, 1o〉] ~Q ∈ Q) | ||
Theorem | ltnnnq 7355 | Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ [〈𝐴, 1o〉] ~Q <Q [〈𝐵, 1o〉] ~Q )) | ||
Definition | df-enq0 7356* | 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 7357 | 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.) |
⊢ Q0 = ((ω × N) / ~Q0 ) | ||
Definition | df-0nq0 7358 | 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.) |
⊢ 0Q0 = [〈∅, 1o〉] ~Q0 | ||
Definition | df-plq0 7359* | 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 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Definition | df-mq0 7360* | 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 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Theorem | dfmq0qs 7361* | Multiplication on nonnegative fractions. This definition is similar to df-mq0 7360 but expands Q0. (Contributed by Jim Kingdon, 22-Nov-2019.) |
⊢ ·Q0 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Theorem | dfplq0qs 7362* | Addition on nonnegative fractions. This definition is similar to df-plq0 7359 but expands Q0. (Contributed by Jim Kingdon, 24-Nov-2019.) |
⊢ +Q0 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Theorem | enq0enq 7363 | 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 7364 | The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7367. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓) | ||
Theorem | enq0ref 7365 | The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7367. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓) | ||
Theorem | enq0tr 7366 | The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7367. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ) | ||
Theorem | enq0er 7367 | The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~Q0 Er (ω × N) | ||
Theorem | enq0breq 7368 | 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 7369 | 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 7370 | 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 7371 | The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ ~Q0 ∈ V | ||
Theorem | nq0ex 7372 | The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q0 ∈ V | ||
Theorem | nqnq0 7373 | A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q0 | ||
Theorem | nq0nn 7374* | Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [〈𝑤, 𝑣〉] ~Q0 )) | ||
Theorem | addcmpblnq0 7375 | 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 7376 | 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 7377 | 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 7378* | Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7381 and mulnnnq0 7382. (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 7379* | 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 7380* | 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 7381 | 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 7382 | 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 7383 | Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q0) | ||
Theorem | mulclnq0 7384 | Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0) | ||
Theorem | nqpnq0nq 7385 | A positive fraction plus a nonnegative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q) | ||
Theorem | nqnq0a 7386 | 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 7387 | 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 7388 | Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0) | ||
Theorem | nq0a0 7389 | Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → (𝐴 +Q0 0Q0) = 𝐴) | ||
Theorem | nnanq0 7390 | 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 7391 | Multiplication of nonnegative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))) | ||
Theorem | mulcomnq0 7392 | Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) = (𝐵 ·Q0 𝐴)) | ||
Theorem | addassnq0lemcl 7393 | A natural number closure law. Lemma for addassnq0 7394. (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 7394 | Addition of nonnegative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐴 +Q0 𝐵) +Q0 𝐶) = (𝐴 +Q0 (𝐵 +Q0 𝐶))) | ||
Theorem | distnq0r 7395 | Multiplication of nonnegative fractions is distributive. Version of distrnq0 7391 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐵 +Q0 𝐶) ·Q0 𝐴) = ((𝐵 ·Q0 𝐴) +Q0 (𝐶 ·Q0 𝐴))) | ||
Theorem | addpinq1 7396 | Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.) |
⊢ (𝐴 ∈ N → [〈(𝐴 +N 1o), 1o〉] ~Q = ([〈𝐴, 1o〉] ~Q +Q 1Q)) | ||
Theorem | nq02m 7397 | Multiply a nonnegative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → ([〈2o, 1o〉] ~Q0 ·Q0 𝐴) = (𝐴 +Q0 𝐴)) | ||
Definition | df-inp 7398* |
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 7399* | Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by Jim Kingdon, 25-Sep-2019.) |
⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | ||
Definition | df-iplp 7400* |
Define addition on positive reals. From Section 11.2.1 of [HoTT], p.
(varies). We write this definition to closely resemble the definition
in HoTT although some of the conditions are redundant (for example,
𝑟
∈ (1st ‘𝑥) implies 𝑟 ∈ Q)
and can be simplified as
shown at genpdf 7440.
This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.) |
⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ 〈{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}〉) |
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