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Type | Label | Description |
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Statement | ||
Theorem | cc4n 7301* | Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7300, 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 6500 and similar theorems ), going from there to positive integers (df-ni 7334) and then positive rational numbers (df-nqqs 7378) 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 7496. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7962 and the MacNeille reals fail to satisfy axltwlin 8056, 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 7302 |
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 7303 | Positive integer addition. |
class +N | ||
Syntax | cmi 7304 | Positive integer multiplication. |
class ·N | ||
Syntax | clti 7305 | Positive integer ordering relation. |
class <N | ||
Syntax | cplpq 7306 | Positive pre-fraction addition. |
class +pQ | ||
Syntax | cmpq 7307 | Positive pre-fraction multiplication. |
class ·pQ | ||
Syntax | cltpq 7308 | Positive pre-fraction ordering relation. |
class <pQ | ||
Syntax | ceq 7309 | Equivalence class used to construct positive fractions. |
class ~Q | ||
Syntax | cnq 7310 | Set of positive fractions. |
class Q | ||
Syntax | c1q 7311 | The positive fraction constant 1. |
class 1Q | ||
Syntax | cplq 7312 | Positive fraction addition. |
class +Q | ||
Syntax | cmq 7313 | Positive fraction multiplication. |
class ·Q | ||
Syntax | crq 7314 | Positive fraction reciprocal operation. |
class *Q | ||
Syntax | cltq 7315 | Positive fraction ordering relation. |
class <Q | ||
Syntax | ceq0 7316 | Equivalence class used to construct nonnegative fractions. |
class ~Q0 | ||
Syntax | cnq0 7317 | Set of nonnegative fractions. |
class Q0 | ||
Syntax | c0q0 7318 | The nonnegative fraction constant 0. |
class 0Q0 | ||
Syntax | cplq0 7319 | Nonnegative fraction addition. |
class +Q0 | ||
Syntax | cmq0 7320 | Nonnegative fraction multiplication. |
class ·Q0 | ||
Syntax | cnp 7321 | Set of positive reals. |
class P | ||
Syntax | c1p 7322 | Positive real constant 1. |
class 1P | ||
Syntax | cpp 7323 | Positive real addition. |
class +P | ||
Syntax | cmp 7324 | Positive real multiplication. |
class ·P | ||
Syntax | cltp 7325 | Positive real ordering relation. |
class <P | ||
Syntax | cer 7326 | Equivalence class used to construct signed reals. |
class ~R | ||
Syntax | cnr 7327 | Set of signed reals. |
class R | ||
Syntax | c0r 7328 | The signed real constant 0. |
class 0R | ||
Syntax | c1r 7329 | The signed real constant 1. |
class 1R | ||
Syntax | cm1r 7330 | The signed real constant -1. |
class -1R | ||
Syntax | cplr 7331 | Signed real addition. |
class +R | ||
Syntax | cmr 7332 | Signed real multiplication. |
class ·R | ||
Syntax | cltr 7333 | Signed real ordering relation. |
class <R | ||
Definition | df-ni 7334 | 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 7335 | 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 7336 | 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 7337 | 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 7338 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | ||
Theorem | pinn 7339 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | ||
Theorem | pion 7340 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ On) | ||
Theorem | piord 7341 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
⊢ (𝐴 ∈ N → Ord 𝐴) | ||
Theorem | niex 7342 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
⊢ N ∈ V | ||
Theorem | 0npi 7343 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
⊢ ¬ ∅ ∈ N | ||
Theorem | elni2 7344 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | ||
Theorem | 1pi 7345 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o ∈ N | ||
Theorem | addpiord 7346 | 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 7347 | 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 7348 | 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 7349 | 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 7350 | 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 7351 | 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 7352 | 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 7353 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵) | ||
Theorem | ltrelpi 7354 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) |
⊢ <N ⊆ (N × N) | ||
Theorem | dmaddpi 7355 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
⊢ dom +N = (N × N) | ||
Theorem | dmmulpi 7356 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
⊢ dom ·N = (N × N) | ||
Theorem | addclpi 7357 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N) | ||
Theorem | mulclpi 7358 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) | ||
Theorem | addcompig 7359 | 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 7360 | 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 7361 | 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 7362 | 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 7363 | 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 7364 | 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 7365 | 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 7366 | 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 7367* | 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 7368 | 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 7369 | 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 7370 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
⊢ 1o <N (1o +N 1o) | ||
Theorem | nlt1pig 7371 | No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) | ||
Theorem | indpi 7372* | 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 7373 | 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 7374* | 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 7379) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7377). (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 7375* | 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 7376* | 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 7377* | 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 7378 | 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 7379* | 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 7380* | 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 7381 | 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 7382* | 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 7383* | 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 7384* | 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 7385* | 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 7386 | 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 7387 | 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 7388 | 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 7389 | 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 7390 | The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) |
⊢ ~Q ∈ V | ||
Theorem | enqdc 7391 | 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 7392 | 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 7393 | The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) |
⊢ Q ∈ V | ||
Theorem | 0nnq 7394 | The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) |
⊢ ¬ ∅ ∈ Q | ||
Theorem | ltrelnq 7395 | 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 7396 | The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) |
⊢ 1Q ∈ Q | ||
Theorem | addcmpblnq 7397 | 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 7398 | 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)〉)) | ||
Theorem | addpipqqslem 7399 | Lemma for addpipqqs 7400. (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 7400 | 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 ) |
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