Home | Intuitionistic Logic Explorer Theorem List (p. 73 of 138) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cc4f 7201* | Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ Ⅎ𝑛𝐴 & ⊢ (𝜑 → 𝑁 ≈ ω) & ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒)) | ||
Theorem | cc4 7202* | Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ≈ ω) & ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒)) | ||
Theorem | cc4n 7203* | Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7202, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ≈ ω) & ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒)) | ||
This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6433 and similar theorems ), going from there to positive integers (df-ni 7236) and then positive rational numbers (df-nqqs 7280) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle or choice principles. With excluded middle, it is natural to define a cut as the lower set only (as Metamath Proof Explorer does), but here we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". When working constructively, there are several possible definitions of real numbers. Here we adopt the most common definition, as two-sided Dedekind cuts with the properties described at df-inp 7398. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7864 and the MacNeille reals fail to satisfy axltwlin 7957, and we do not develop them here. For more on differing definitions of the reals, see the introduction to Chapter 11 in [HoTT] or Section 1.2 of [BauerHanson]. | ||
Syntax | cnpi 7204 |
The set of positive integers, which is the set of natural numbers ω
with 0 removed.
Note: This is the start of the Dedekind-cut construction of real and complex numbers. |
class N | ||
Syntax | cpli 7205 | Positive integer addition. |
class +N | ||
Syntax | cmi 7206 | Positive integer multiplication. |
class ·N | ||
Syntax | clti 7207 | Positive integer ordering relation. |
class <N | ||
Syntax | cplpq 7208 | Positive pre-fraction addition. |
class +pQ | ||
Syntax | cmpq 7209 | Positive pre-fraction multiplication. |
class ·pQ | ||
Syntax | cltpq 7210 | Positive pre-fraction ordering relation. |
class <pQ | ||
Syntax | ceq 7211 | Equivalence class used to construct positive fractions. |
class ~Q | ||
Syntax | cnq 7212 | Set of positive fractions. |
class Q | ||
Syntax | c1q 7213 | The positive fraction constant 1. |
class 1Q | ||
Syntax | cplq 7214 | Positive fraction addition. |
class +Q | ||
Syntax | cmq 7215 | Positive fraction multiplication. |
class ·Q | ||
Syntax | crq 7216 | Positive fraction reciprocal operation. |
class *Q | ||
Syntax | cltq 7217 | Positive fraction ordering relation. |
class <Q | ||
Syntax | ceq0 7218 | Equivalence class used to construct nonnegative fractions. |
class ~Q0 | ||
Syntax | cnq0 7219 | Set of nonnegative fractions. |
class Q0 | ||
Syntax | c0q0 7220 | The nonnegative fraction constant 0. |
class 0Q0 | ||
Syntax | cplq0 7221 | Nonnegative fraction addition. |
class +Q0 | ||
Syntax | cmq0 7222 | Nonnegative fraction multiplication. |
class ·Q0 | ||
Syntax | cnp 7223 | Set of positive reals. |
class P | ||
Syntax | c1p 7224 | Positive real constant 1. |
class 1P | ||
Syntax | cpp 7225 | Positive real addition. |
class +P | ||
Syntax | cmp 7226 | Positive real multiplication. |
class ·P | ||
Syntax | cltp 7227 | Positive real ordering relation. |
class <P | ||
Syntax | cer 7228 | Equivalence class used to construct signed reals. |
class ~R | ||
Syntax | cnr 7229 | Set of signed reals. |
class R | ||
Syntax | c0r 7230 | The signed real constant 0. |
class 0R | ||
Syntax | c1r 7231 | The signed real constant 1. |
class 1R | ||
Syntax | cm1r 7232 | The signed real constant -1. |
class -1R | ||
Syntax | cplr 7233 | Signed real addition. |
class +R | ||
Syntax | cmr 7234 | Signed real multiplication. |
class ·R | ||
Syntax | cltr 7235 | Signed real ordering relation. |
class <R | ||
Definition | df-ni 7236 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) |
⊢ N = (ω ∖ {∅}) | ||
Definition | df-pli 7237 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
⊢ +N = ( +o ↾ (N × N)) | ||
Definition | df-mi 7238 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
⊢ ·N = ( ·o ↾ (N × N)) | ||
Definition | df-lti 7239 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) |
⊢ <N = ( E ∩ (N × N)) | ||
Theorem | elni 7240 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | ||
Theorem | pinn 7241 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | ||
Theorem | pion 7242 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ On) | ||
Theorem | piord 7243 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
⊢ (𝐴 ∈ N → Ord 𝐴) | ||
Theorem | niex 7244 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
⊢ N ∈ V | ||
Theorem | 0npi 7245 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
⊢ ¬ ∅ ∈ N | ||
Theorem | elni2 7246 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | ||
Theorem | 1pi 7247 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o ∈ N | ||
Theorem | addpiord 7248 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) | ||
Theorem | mulpiord 7249 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | ||
Theorem | mulidpi 7250 | 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | ||
Theorem | ltpiord 7251 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | ltsopi 7252 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) |
⊢ <N Or N | ||
Theorem | pitric 7253 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <N 𝐴))) | ||
Theorem | pitri3or 7254 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴)) | ||
Theorem | ltdcpi 7255 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵) | ||
Theorem | ltrelpi 7256 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) |
⊢ <N ⊆ (N × N) | ||
Theorem | dmaddpi 7257 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
⊢ dom +N = (N × N) | ||
Theorem | dmmulpi 7258 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
⊢ dom ·N = (N × N) | ||
Theorem | addclpi 7259 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N) | ||
Theorem | mulclpi 7260 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) | ||
Theorem | addcompig 7261 | Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) | ||
Theorem | addasspig 7262 | Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = (𝐴 +N (𝐵 +N 𝐶))) | ||
Theorem | mulcompig 7263 | Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) | ||
Theorem | mulasspig 7264 | Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))) | ||
Theorem | distrpig 7265 | Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))) | ||
Theorem | addcanpig 7266 | Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | mulcanpig 7267 | Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | addnidpig 7268 | There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴) | ||
Theorem | ltexpi 7269* | Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ∃𝑥 ∈ N (𝐴 +N 𝑥) = 𝐵)) | ||
Theorem | ltapig 7270 | Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 <N 𝐵 ↔ (𝐶 +N 𝐴) <N (𝐶 +N 𝐵))) | ||
Theorem | ltmpig 7271 | Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 <N 𝐵 ↔ (𝐶 ·N 𝐴) <N (𝐶 ·N 𝐵))) | ||
Theorem | 1lt2pi 7272 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
⊢ 1o <N (1o +N 1o) | ||
Theorem | nlt1pig 7273 | No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) | ||
Theorem | indpi 7274* | Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) |
⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 +N 1o) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ N → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ N → 𝜏) | ||
Theorem | nnppipi 7275 | A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) ∈ N) | ||
Definition | df-plpq 7276* | Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +Q (df-plqqs 7281) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7279). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) |
⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | ||
Definition | df-mpq 7277* | Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) |
⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | ||
Definition | df-ltpq 7278* | Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) |
⊢ <pQ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))} | ||
Definition | df-enq 7279* | Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) |
⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} | ||
Definition | df-nqqs 7280 | Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.) |
⊢ Q = ((N × N) / ~Q ) | ||
Definition | df-plqqs 7281* | Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.) |
⊢ +Q = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 +pQ 〈𝑢, 𝑓〉)] ~Q ))} | ||
Definition | df-mqqs 7282* | Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.) |
⊢ ·Q = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 ·pQ 〈𝑢, 𝑓〉)] ~Q ))} | ||
Definition | df-1nqqs 7283 | Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.) |
⊢ 1Q = [〈1o, 1o〉] ~Q | ||
Definition | df-rq 7284* | Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by Jim Kingdon, 20-Sep-2019.) |
⊢ *Q = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q)} | ||
Definition | df-ltnqqs 7285* | Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) |
⊢ <Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ~Q ∧ 𝑦 = [〈𝑣, 𝑢〉] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))} | ||
Theorem | dfplpq2 7286* | Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.) |
⊢ +pQ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)〉))} | ||
Theorem | dfmpq2 7287* | Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.) |
⊢ ·pQ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)〉))} | ||
Theorem | enqbreq 7288 | Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q 〈𝐶, 𝐷〉 ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶))) | ||
Theorem | enqbreq2 7289 | Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) |
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))) | ||
Theorem | enqer 7290 | The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
⊢ ~Q Er (N × N) | ||
Theorem | enqeceq 7291 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([〈𝐴, 𝐵〉] ~Q = [〈𝐶, 𝐷〉] ~Q ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶))) | ||
Theorem | enqex 7292 | The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) |
⊢ ~Q ∈ V | ||
Theorem | enqdc 7293 | The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → DECID 〈𝐴, 𝐵〉 ~Q 〈𝐶, 𝐷〉) | ||
Theorem | enqdc1 7294 | The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID 〈𝐴, 𝐵〉 ~Q 𝐶) | ||
Theorem | nqex 7295 | The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) |
⊢ Q ∈ V | ||
Theorem | 0nnq 7296 | The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) |
⊢ ¬ ∅ ∈ Q | ||
Theorem | ltrelnq 7297 | Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) |
⊢ <Q ⊆ (Q × Q) | ||
Theorem | 1nq 7298 | The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) |
⊢ 1Q ∈ Q | ||
Theorem | addcmpblnq 7299 | Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.) |
⊢ ((((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ N ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ N ∧ 𝑆 ∈ N))) → (((𝐴 ·N 𝐷) = (𝐵 ·N 𝐶) ∧ (𝐹 ·N 𝑆) = (𝐺 ·N 𝑅)) → 〈((𝐴 ·N 𝐺) +N (𝐵 ·N 𝐹)), (𝐵 ·N 𝐺)〉 ~Q 〈((𝐶 ·N 𝑆) +N (𝐷 ·N 𝑅)), (𝐷 ·N 𝑆)〉)) | ||
Theorem | mulcmpblnq 7300 | Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.) |
⊢ ((((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ N ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ N ∧ 𝑆 ∈ N))) → (((𝐴 ·N 𝐷) = (𝐵 ·N 𝐶) ∧ (𝐹 ·N 𝑆) = (𝐺 ·N 𝑅)) → 〈(𝐴 ·N 𝐹), (𝐵 ·N 𝐺)〉 ~Q 〈(𝐶 ·N 𝑅), (𝐷 ·N 𝑆)〉)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |