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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnnq 6501 | 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 6502 | 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 6503* | 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 6504 | 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 6505 | 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 6506* | 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 6507* | 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 6508* | Multiplication on non-negative fractions. This definition is similar to df-mq0 6507 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 6509* | Addition on non-negative fractions. This definition is similar to df-plq0 6506 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 6510 | 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 6511 | The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6514. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ~_{Q0} 𝑔 → 𝑔 ~_{Q0} 𝑓) | ||
Theorem | enq0ref 6512 | The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6514. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~_{Q0} 𝑓) | ||
Theorem | enq0tr 6513 | The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6514. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ ((𝑓 ~_{Q0} 𝑔 ∧ 𝑔 ~_{Q0} ℎ) → 𝑓 ~_{Q0} ℎ) | ||
Theorem | enq0er 6514 | The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~_{Q0} Er (ω × N) | ||
Theorem | enq0breq 6515 | 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 6516 | 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 6517 | 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 6518 | The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ ~_{Q0} ∈ V | ||
Theorem | nq0ex 6519 | The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q_{0} ∈ V | ||
Theorem | nqnq0 6520 | A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q_{0} | ||
Theorem | nq0nn 6521* | 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 6522 | 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 6523 | 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 6524 | 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 6525* | Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6528 and mulnnnq0 6529. (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 6526* | 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 6527* | 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 6528 | 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 6529 | 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 6530 | 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 6531 | 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 6532 | 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 6533 | 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 6534 | 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 6535 | 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 6536 | Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q_{0} → (𝐴 +_{Q0} 0_{Q0}) = 𝐴) | ||
Theorem | nnanq0 6537 | 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 6538 | 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 6539 | 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 6540 | A natural number closure law. Lemma for addassnq0 6541. (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 6541 | 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 6542 | Multiplication of non-negative fractions is distributive. Version of distrnq0 6538 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 6543 | 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 6544 | 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 6545* |
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 6546* | 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 6547* |
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 6587.
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 6548* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 6547 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 6549* |
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 6550 | Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ P ⊆ (𝒫 Q × 𝒫 Q) | ||
Theorem | preqlu 6551 | 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 6552 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
⊢ P ∈ V | ||
Theorem | elinp 6553* | 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 6554 | 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 6555* | 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 6556* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿) | ||
Theorem | prmu 6557* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝑈) | ||
Theorem | prssnql 6558 | 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 6559 | 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 6560 | 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 6561 | 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 6562 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
⊢ ¬ ∅ ∈ P | ||
Theorem | prcdnql 6563 | 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 6564 | 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 6565 | 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 6566 | 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 6567* | 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 6568* | 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 6569* | 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 6570 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <_{Q} 𝐵) → (𝐴 ∈ 𝐿 ∨ 𝐵 ∈ 𝑈)) | ||
Theorem | prdisj 6571 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ Q) → ¬ (𝐴 ∈ 𝐿 ∧ 𝐴 ∈ 𝑈)) | ||
Theorem | prarloclemlt 6572 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6582. (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 6573* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6582. (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 6574 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6582. (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 6575* | Induction step for prarloclem3 6576. (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 6576* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6582. (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 6577* | A slight rearrangement of prarloclem3 6576. Lemma for prarloc 6582. (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 6578* | Subtracting two from a positive integer. Lemma for prarloc 6582. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ ((𝑁 ∈ N ∧ 1_{𝑜} <_{N} 𝑁) → ∃𝑥 ∈ ω (2_{𝑜} +_{𝑜} 𝑥) = 𝑁) | ||
Theorem | prarloclem5 6579* | A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 6582. (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 6580* | 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 6581 | Some calculations for prarloc 6582. (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)) | ||
Theorem | prarloc 6582* |
A Dedekind cut is arithmetically located. Part of Proposition 11.15 of
[BauerTaylor], p. 52, slightly
modified. It states that given a
tolerance 𝑃, there are elements of the lower and
upper cut which
are within that tolerance of each other.
Usually, proofs will be shorter if they use prarloc2 6583 instead. (Contributed by Jim Kingdon, 22-Oct-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <_{Q} (𝑎 +_{Q} 𝑃)) | ||
Theorem | prarloc2 6583* | A Dedekind cut is arithmetically located. This is a variation of prarloc 6582 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 (𝑎 +_{Q} 𝑃) ∈ 𝑈) | ||
Theorem | ltrelpr 6584 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
⊢ <_{P} ⊆ (P × P) | ||
Theorem | ltdfpr 6585* | More convenient form of df-iltp 6549. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_{P} 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^{nd} ‘𝐴) ∧ 𝑞 ∈ (1^{st} ‘𝐵)))) | ||
Theorem | genpdflem 6586* | Simplification of upper or lower cut expression. Lemma for genpdf 6587. (Contributed by Jim Kingdon, 30-Sep-2019.) |
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ Q) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ Q) ⇒ ⊢ (𝜑 → {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐵 ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐵 𝑞 = (𝑟𝐺𝑠)}) | ||
Theorem | genpdf 6587* | Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1^{st} ‘𝑤) ∧ 𝑠 ∈ (1^{st} ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2^{nd} ‘𝑤) ∧ 𝑠 ∈ (2^{nd} ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩) ⇒ ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1^{st} ‘𝑤)∃𝑠 ∈ (1^{st} ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2^{nd} ‘𝑤)∃𝑠 ∈ (2^{nd} ‘𝑣)𝑞 = (𝑟𝐺𝑠)}⟩) | ||
Theorem | genipv 6588* | Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1^{st} ‘𝐴)∃𝑠 ∈ (1^{st} ‘𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2^{nd} ‘𝐴)∃𝑠 ∈ (2^{nd} ‘𝐵)𝑞 = (𝑟𝐺𝑠)}⟩) | ||
Theorem | genplt2i 6589* | Operating on both sides of two inequalities, when the operation is consistent with <_{Q}. (Contributed by Jim Kingdon, 6-Oct-2019.) |
⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_{Q} 𝑦 ↔ (𝑧𝐺𝑥) <_{Q} (𝑧𝐺𝑦))) & ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) ⇒ ⊢ ((𝐴 <_{Q} 𝐵 ∧ 𝐶 <_{Q} 𝐷) → (𝐴𝐺𝐶) <_{Q} (𝐵𝐺𝐷)) | ||
Theorem | genpelxp 6590* | Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q)) | ||
Theorem | genpelvl 6591* | Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1^{st} ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^{st} ‘𝐴)∃ℎ ∈ (1^{st} ‘𝐵)𝐶 = (𝑔𝐺ℎ))) | ||
Theorem | genpelvu 6592* | Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2^{nd} ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2^{nd} ‘𝐴)∃ℎ ∈ (2^{nd} ‘𝐵)𝐶 = (𝑔𝐺ℎ))) | ||
Theorem | genpprecll 6593* | Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ (1^{st} ‘𝐴) ∧ 𝐷 ∈ (1^{st} ‘𝐵)) → (𝐶𝐺𝐷) ∈ (1^{st} ‘(𝐴𝐹𝐵)))) | ||
Theorem | genppreclu 6594* | Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ (2^{nd} ‘𝐴) ∧ 𝐷 ∈ (2^{nd} ‘𝐵)) → (𝐶𝐺𝐷) ∈ (2^{nd} ‘(𝐴𝐹𝐵)))) | ||
Theorem | genipdm 6595* | Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ dom 𝐹 = (P × P) | ||
Theorem | genpml 6596* | The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^{st} ‘(𝐴𝐹𝐵))) | ||
Theorem | genpmu 6597* | The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2^{nd} ‘(𝐴𝐹𝐵))) | ||
Theorem | genpcdl 6598* | Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) & ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^{st} ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^{st} ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_{Q} (𝑔𝐺ℎ) → 𝑥 ∈ (1^{st} ‘(𝐴𝐹𝐵)))) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1^{st} ‘(𝐴𝐹𝐵)) → (𝑥 <_{Q} 𝑓 → 𝑥 ∈ (1^{st} ‘(𝐴𝐹𝐵))))) | ||
Theorem | genpcuu 6599* | Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) & ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2^{nd} ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^{nd} ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <_{Q} 𝑥 → 𝑥 ∈ (2^{nd} ‘(𝐴𝐹𝐵)))) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2^{nd} ‘(𝐴𝐹𝐵)) → (𝑓 <_{Q} 𝑥 → 𝑥 ∈ (2^{nd} ‘(𝐴𝐹𝐵))))) | ||
Theorem | genprndl 6600* | The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.) |
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^{st} ‘𝑤) ∧ 𝑧 ∈ (1^{st} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^{nd} ‘𝑤) ∧ 𝑧 ∈ (2^{nd} ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) & ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_{Q} 𝑦 ↔ (𝑧𝐺𝑥) <_{Q} (𝑧𝐺𝑦))) & ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) & ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^{st} ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^{st} ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_{Q} (𝑔𝐺ℎ) → 𝑥 ∈ (1^{st} ‘(𝐴𝐹𝐵)))) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^{st} ‘(𝐴𝐹𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘(𝐴𝐹𝐵))))) |
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