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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mulclnq0 6301 | 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 6302 | 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 6303 | 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 6304 | 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 6305 | 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 6306 | Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (A ∈ Q_{0} → (A +_{Q0} 0_{Q0}) = A) | ||
Theorem | nnanq0 6307 | 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 6308 | 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 6309 | 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 6310 | A natural number closure law. Lemma for addassnq0 6311. (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 6311 | 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 6312 | Multiplication of non-negative fractions is distributive. Version of distrnq0 6308 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 | nq02m 6313 | 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 6314* |
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 6315* | 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 6316* |
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 (for example, 𝑟 ∈ Q and
𝑟
∈ (1^{st} ‘x)) conditions are redundant and can be
simplified
as shown at genpdf 6356.
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 6317* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 6316 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 6318* |
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 6319 | Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ P ⊆ (𝒫 Q × 𝒫 Q) | ||
Theorem | preqlu 6320 | 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 6321 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
⊢ P ∈ V | ||
Theorem | elinp 6322* | 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 6323 | 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 6324* | 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 6325* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃x ∈ Q x ∈ 𝐿) | ||
Theorem | prmu 6326* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃x ∈ Q x ∈ 𝑈) | ||
Theorem | prssnql 6327 | 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 6328 | 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 6329 | 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 6330 | 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 6331 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
⊢ ¬ ∅ ∈ P | ||
Theorem | prcdnql 6332 | 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 6333 | 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 6334 | 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 6335 | 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 6336* | 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 6337* | 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 6338* | 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 6339 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A <_{Q} B) → (A ∈ 𝐿 ∨ B ∈ 𝑈)) | ||
Theorem | prdisj 6340 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A ∈ Q) → ¬ (A ∈ 𝐿 ∧ A ∈ 𝑈)) | ||
Theorem | prarloclemlt 6341 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6342* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6343 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6344* | Induction step for prarloclem3 6345. (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 6345* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6351. (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 6346* | A slight rearrangement of prarloclem3 6345. Lemma for prarloc 6351. (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 6347* | Subtracting two from a positive integer. Lemma for prarloc 6351. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ ((𝑁 ∈ N ∧ 1_{𝑜} <_{N} 𝑁) → ∃x ∈ 𝜔 (2_{𝑜} +_{𝑜} x) = 𝑁) | ||
Theorem | prarloclem5 6348* | A substitution of zero for y and 𝑁 minus two for x. Lemma for prarloc 6351. (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 6349* | 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 6350 | Some calculations for prarloc 6351. (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 6351* | 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. (Contributed by Jim Kingdon, 22-Oct-2019.) |
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <_{Q} (𝑎 +_{Q} 𝑃)) | ||
Theorem | prarloc2 6352* | A Dedekind cut is arithmetically located. This is a variation of prarloc 6351 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 6353 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
⊢ <_{P} ⊆ (P × P) | ||
Theorem | ltdfpr 6354* | More convenient form of df-iltp 6318. (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 6355* | Simplification of upper or lower cut expression. Lemma for genpdf 6356. (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 6356* | 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 6357* | 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 6358* | 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 6359* | 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 6360* | 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 6361* | 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 6362* | 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 6363* | 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 6364* | 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 | genpelpw 6365* | Result of general operation on positive reals is an ordered pair of sets of positive fractions. (Contributed by Jim Kingdon, 4-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) ∈ (𝒫 Q × 𝒫 Q)) | ||
Theorem | genpml 6366* | 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 6367* | 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 6368* | 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 6369* | 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 6370* | 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))))) | ||
Theorem | genprndu 6371* | The upper 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 ∈ (2^{nd} ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (2^{nd} ‘B))) ∧ x ∈ Q) → ((g𝐺ℎ) <_{Q} x → x ∈ (2^{nd} ‘(A𝐹B)))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2^{nd} ‘(A𝐹B)) ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘(A𝐹B))))) | ||
Theorem | genpdisj 6372* | The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (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) & ⊢ ((x ∈ Q ∧ y ∈ Q ∧ z ∈ Q) → (x <_{Q} y ↔ (z𝐺x) <_{Q} (z𝐺y))) & ⊢ ((x ∈ Q ∧ y ∈ Q) → (x𝐺y) = (y𝐺x)) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^{st} ‘(A𝐹B)) ∧ 𝑞 ∈ (2^{nd} ‘(A𝐹B)))) | ||
Theorem | genpassl 6373* | Associativity of lower cuts. Lemma for genpassg 6375. (Contributed by Jim Kingdon, 11-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) & ⊢ dom 𝐹 = (P × P) & ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P) & ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1^{st} ‘((A𝐹B)𝐹𝐶)) = (1^{st} ‘(A𝐹(B𝐹𝐶)))) | ||
Theorem | genpassu 6374* | Associativity of upper cuts. Lemma for genpassg 6375. (Contributed by Jim Kingdon, 11-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) & ⊢ dom 𝐹 = (P × P) & ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P) & ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (2^{nd} ‘((A𝐹B)𝐹𝐶)) = (2^{nd} ‘(A𝐹(B𝐹𝐶)))) | ||
Theorem | genpassg 6375* | Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-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) & ⊢ dom 𝐹 = (P × P) & ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P) & ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶))) | ||
Theorem | addnqprllem 6376 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝐿) ∧ 𝑋 ∈ Q) → (𝑋 <_{Q} 𝑆 → ((𝑋 ·_{Q} (*_{Q}‘𝑆)) ·_{Q} 𝐺) ∈ 𝐿)) | ||
Theorem | addnqprulem 6377 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) |
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝑈) ∧ 𝑋 ∈ Q) → (𝑆 <_{Q} 𝑋 → ((𝑋 ·_{Q} (*_{Q}‘𝑆)) ·_{Q} 𝐺) ∈ 𝑈)) | ||
Theorem | addnqprl 6378 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
⊢ ((((A ∈ P ∧ 𝐺 ∈ (1^{st} ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (1^{st} ‘B))) ∧ 𝑋 ∈ Q) → (𝑋 <_{Q} (𝐺 +_{Q} 𝐻) → 𝑋 ∈ (1^{st} ‘(A +_{P} B)))) | ||
Theorem | addnqpru 6379 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
⊢ ((((A ∈ P ∧ 𝐺 ∈ (2^{nd} ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2^{nd} ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 +_{Q} 𝐻) <_{Q} 𝑋 → 𝑋 ∈ (2^{nd} ‘(A +_{P} B)))) | ||
Theorem | addlocprlemlt 6380 | Lemma for addlocpr 6385. The 𝑄 <_{Q} (𝐷 +_{Q} 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑄 <_{Q} (𝐷 +_{Q} 𝐸) → 𝑄 ∈ (1^{st} ‘(A +_{P} B)))) | ||
Theorem | addlocprlemeqgt 6381 | Lemma for addlocpr 6385. This is a step used in both the 𝑄 = (𝐷 +_{Q} 𝐸) and (𝐷 +_{Q} 𝐸) <_{Q} 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.) |
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑈 +_{Q} 𝑇) <_{Q} ((𝐷 +_{Q} 𝐸) +_{Q} (𝑃 +_{Q} 𝑃))) | ||
Theorem | addlocprlemeq 6382 | Lemma for addlocpr 6385. The 𝑄 = (𝐷 +_{Q} 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑄 = (𝐷 +_{Q} 𝐸) → 𝑅 ∈ (2^{nd} ‘(A +_{P} B)))) | ||
Theorem | addlocprlemgt 6383 | Lemma for addlocpr 6385. The (𝐷 +_{Q} 𝐸) <_{Q} 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.) |
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → ((𝐷 +_{Q} 𝐸) <_{Q} 𝑄 → 𝑅 ∈ (2^{nd} ‘(A +_{P} B)))) | ||
Theorem | addlocprlem 6384 | Lemma for addlocpr 6385. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) |
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑄 ∈ (1^{st} ‘(A +_{P} B)) ∨ 𝑅 ∈ (2^{nd} ‘(A +_{P} B)))) | ||
Theorem | addlocpr 6385* | Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 6351 to both A and B, and uses nqtri3or 6249 rather than prloc 6339 to decide whether 𝑞 is too big to be in the lower cut of A +_{P} B (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.) |
⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ (1^{st} ‘(A +_{P} B)) ∨ 𝑟 ∈ (2^{nd} ‘(A +_{P} B))))) | ||
Theorem | addclpr 6386 | Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.) |
⊢ ((A ∈ P ∧ B ∈ P) → (A +_{P} B) ∈ P) | ||
Theorem | plpvlu 6387* | Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ ((A ∈ P ∧ B ∈ P) → (A +_{P} B) = ⟨{x ∈ Q ∣ ∃y ∈ (1^{st} ‘A)∃z ∈ (1^{st} ‘B)x = (y +_{Q} z)}, {x ∈ Q ∣ ∃y ∈ (2^{nd} ‘A)∃z ∈ (2^{nd} ‘B)x = (y +_{Q} z)}⟩) | ||
Theorem | mpvlu 6388* | Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ ((A ∈ P ∧ B ∈ P) → (A ·_{P} B) = ⟨{x ∈ Q ∣ ∃y ∈ (1^{st} ‘A)∃z ∈ (1^{st} ‘B)x = (y ·_{Q} z)}, {x ∈ Q ∣ ∃y ∈ (2^{nd} ‘A)∃z ∈ (2^{nd} ‘B)x = (y ·_{Q} z)}⟩) | ||
Theorem | dmplp 6389 | Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) |
⊢ dom +_{P} = (P × P) | ||
Theorem | dmmp 6390 | Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) |
⊢ dom ·_{P} = (P × P) | ||
Theorem | nqprm 6391* | A cut produced from a rational is inhabited. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ (A ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {x ∣ x <_{Q} A} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {x ∣ A <_{Q} x})) | ||
Theorem | nqprrnd 6392* | A cut produced from a rational is rounded. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ (A ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {x ∣ x <_{Q} A} ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ {x ∣ x <_{Q} A})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {x ∣ A <_{Q} x} ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ {x ∣ A <_{Q} x})))) | ||
Theorem | nqprdisj 6393* | A cut produced from a rational is disjoint. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ (A ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {x ∣ x <_{Q} A} ∧ 𝑞 ∈ {x ∣ A <_{Q} x})) | ||
Theorem | nqprloc 6394* | A cut produced from a rational is located. Lemma for nqprlu 6395. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ (A ∈ Q → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ {x ∣ x <_{Q} A} ∨ 𝑟 ∈ {x ∣ A <_{Q} x}))) | ||
Theorem | nqprlu 6395* | The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 8-Dec-2019.) |
⊢ (A ∈ Q → ⟨{x ∣ x <_{Q} A}, {x ∣ A <_{Q} x}⟩ ∈ P) | ||
Theorem | ltnqex 6396 | The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
⊢ {x ∣ x <_{Q} A} ∈ V | ||
Theorem | gtnqex 6397 | The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) |
⊢ {x ∣ A <_{Q} x} ∈ V | ||
Theorem | 1pr 6398 | The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
⊢ 1_{P} ∈ P | ||
Theorem | 1prl 6399 | The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
⊢ (1^{st} ‘1_{P}) = {x ∣ x <_{Q} 1_{Q}} | ||
Theorem | 1pru 6400 | The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
⊢ (2^{nd} ‘1_{P}) = {x ∣ 1_{Q} <_{Q} x} |
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