![]() |
Intuitionistic Logic Explorer Theorem List (p. 75 of 145) | < 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 | subhalfnqq 7401* | There is a number which is less than half of any positive fraction. The case where 𝐴 is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 7397). (Contributed by Jim Kingdon, 25-Nov-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴) | ||
Theorem | ltbtwnnqq 7402* | There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) | ||
Theorem | ltbtwnnq 7403* | There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) | ||
Theorem | archnqq 7404* | For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.) |
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q [〈𝑥, 1o〉] ~Q ) | ||
Theorem | prarloclemarch 7405* | A version of the Archimedean property. This variation is "stronger" than archnqq 7404 in the sense that we provide an integer which is larger than a given rational 𝐴 even after being multiplied by a second rational 𝐵. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N 𝐴 <Q ([〈𝑥, 1o〉] ~Q ·Q 𝐵)) | ||
Theorem | prarloclemarch2 7406* | Like prarloclemarch 7405 but the integer must be at least two, and there is also 𝐵 added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 7490. (Contributed by Jim Kingdon, 25-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ N (1o <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([〈𝑥, 1o〉] ~Q ·Q 𝐶)))) | ||
Theorem | ltrnqg 7407 | Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7408. (Contributed by Jim Kingdon, 29-Dec-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))) | ||
Theorem | ltrnqi 7408 | Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7407. (Contributed by Jim Kingdon, 24-Sep-2019.) |
⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴)) | ||
Theorem | nnnq 7409 | The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.) |
⊢ (𝐴 ∈ N → [〈𝐴, 1o〉] ~Q ∈ Q) | ||
Theorem | ltnnnq 7410 | Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ [〈𝐴, 1o〉] ~Q <Q [〈𝐵, 1o〉] ~Q )) | ||
Definition | df-enq0 7411* | Define equivalence relation for nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
⊢ ~Q0 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))} | ||
Definition | df-nq0 7412 | Define class of nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
⊢ Q0 = ((ω × N) / ~Q0 ) | ||
Definition | df-0nq0 7413 | Define nonnegative fraction constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ 0Q0 = [〈∅, 1o〉] ~Q0 | ||
Definition | df-plq0 7414* | Define addition on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
⊢ +Q0 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Definition | df-mq0 7415* | Define multiplication on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.) |
⊢ ·Q0 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Theorem | dfmq0qs 7416* | Multiplication on nonnegative fractions. This definition is similar to df-mq0 7415 but expands Q0. (Contributed by Jim Kingdon, 22-Nov-2019.) |
⊢ ·Q0 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Theorem | dfplq0qs 7417* | Addition on nonnegative fractions. This definition is similar to df-plq0 7414 but expands Q0. (Contributed by Jim Kingdon, 24-Nov-2019.) |
⊢ +Q0 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝑦 = [〈𝑢, 𝑓〉] ~Q0 ) ∧ 𝑧 = [〈((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)〉] ~Q0 ))} | ||
Theorem | enq0enq 7418 | Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~Q = ( ~Q0 ∩ ((N × N) × (N × N))) | ||
Theorem | enq0sym 7419 | The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7422. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓) | ||
Theorem | enq0ref 7420 | The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7422. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓) | ||
Theorem | enq0tr 7421 | The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7422. (Contributed by Jim Kingdon, 14-Nov-2019.) |
⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ) | ||
Theorem | enq0er 7422 | The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
⊢ ~Q0 Er (ω × N) | ||
Theorem | enq0breq 7423 | Equivalence relation for nonnegative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q0 〈𝐶, 𝐷〉 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶))) | ||
Theorem | enq0eceq 7424 | Equivalence class equality of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([〈𝐴, 𝐵〉] ~Q0 = [〈𝐶, 𝐷〉] ~Q0 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶))) | ||
Theorem | nqnq0pi 7425 | A nonnegative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → [〈𝐴, 𝐵〉] ~Q0 = [〈𝐴, 𝐵〉] ~Q ) | ||
Theorem | enq0ex 7426 | The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ ~Q0 ∈ V | ||
Theorem | nq0ex 7427 | The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q0 ∈ V | ||
Theorem | nqnq0 7428 | A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
⊢ Q ⊆ Q0 | ||
Theorem | nq0nn 7429* | Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [〈𝑤, 𝑣〉] ~Q0 )) | ||
Theorem | addcmpblnq0 7430 | Lemma showing compatibility of addition on nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.) |
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → 〈((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)〉 ~Q0 〈((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)〉)) | ||
Theorem | mulcmpblnq0 7431 | Lemma showing compatibility of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) |
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → 〈(𝐴 ·o 𝐹), (𝐵 ·o 𝐺)〉 ~Q0 〈(𝐶 ·o 𝑅), (𝐷 ·o 𝑆)〉)) | ||
Theorem | mulcanenq0ec 7432 | 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 7433* | Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7436 and mulnnnq0 7437. (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 7434* | There is at most one result from adding nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.) |
⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝐵 = [〈𝑢, 𝑡〉] ~Q0 ) ∧ 𝑧 = [〈((𝑤 ·o 𝑡) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑡)〉] ~Q0 )) | ||
Theorem | mulnq0mo 7435* | There is at most one result from multiplying nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) |
⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [〈𝑤, 𝑣〉] ~Q0 ∧ 𝐵 = [〈𝑢, 𝑡〉] ~Q0 ) ∧ 𝑧 = [〈(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)〉] ~Q0 )) | ||
Theorem | addnnnq0 7436 | Addition of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([〈𝐴, 𝐵〉] ~Q0 +Q0 [〈𝐶, 𝐷〉] ~Q0 ) = [〈((𝐴 ·o 𝐷) +o (𝐵 ·o 𝐶)), (𝐵 ·o 𝐷)〉] ~Q0 ) | ||
Theorem | mulnnnq0 7437 | Multiplication of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([〈𝐴, 𝐵〉] ~Q0 ·Q0 [〈𝐶, 𝐷〉] ~Q0 ) = [〈(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)〉] ~Q0 ) | ||
Theorem | addclnq0 7438 | Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q0) | ||
Theorem | mulclnq0 7439 | Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0) | ||
Theorem | nqpnq0nq 7440 | A positive fraction plus a nonnegative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q) | ||
Theorem | nqnq0a 7441 | 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 7442 | 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 7443 | Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0) | ||
Theorem | nq0a0 7444 | Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → (𝐴 +Q0 0Q0) = 𝐴) | ||
Theorem | nnanq0 7445 | Addition of nonnegative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.) |
⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [〈(𝑁 +o 𝑀), 𝐴〉] ~Q0 = ([〈𝑁, 𝐴〉] ~Q0 +Q0 [〈𝑀, 𝐴〉] ~Q0 )) | ||
Theorem | distrnq0 7446 | Multiplication of nonnegative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))) | ||
Theorem | mulcomnq0 7447 | Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) = (𝐵 ·Q0 𝐴)) | ||
Theorem | addassnq0lemcl 7448 | A natural number closure law. Lemma for addassnq0 7449. (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 7449 | Addition of nonnegative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐴 +Q0 𝐵) +Q0 𝐶) = (𝐴 +Q0 (𝐵 +Q0 𝐶))) | ||
Theorem | distnq0r 7450 | Multiplication of nonnegative fractions is distributive. Version of distrnq0 7446 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐵 +Q0 𝐶) ·Q0 𝐴) = ((𝐵 ·Q0 𝐴) +Q0 (𝐶 ·Q0 𝐴))) | ||
Theorem | addpinq1 7451 | Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.) |
⊢ (𝐴 ∈ N → [〈(𝐴 +N 1o), 1o〉] ~Q = ([〈𝐴, 1o〉] ~Q +Q 1Q)) | ||
Theorem | nq02m 7452 | Multiply a nonnegative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.) |
⊢ (𝐴 ∈ Q0 → ([〈2o, 1o〉] ~Q0 ·Q0 𝐴) = (𝐴 +Q0 𝐴)) | ||
Definition | df-inp 7453* |
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 7454* | Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by Jim Kingdon, 25-Sep-2019.) |
⊢ 1P = 〈{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}〉 | ||
Definition | df-iplp 7455* |
Define addition on positive reals. From Section 11.2.1 of [HoTT], p.
(varies). We write this definition to closely resemble the definition
in HoTT although some of the conditions are redundant (for example,
𝑟
∈ (1st ‘𝑥) implies 𝑟 ∈ Q)
and can be simplified as
shown at genpdf 7495.
This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.) |
⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ 〈{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}〉) | ||
Definition | df-imp 7456* |
Define multiplication on positive reals. Here we use a simple
definition which is similar to df-iplp 7455 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 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}〉) | ||
Definition | df-iltp 7457* |
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 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} | ||
Theorem | npsspw 7458 | Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ P ⊆ (𝒫 Q × 𝒫 Q) | ||
Theorem | preqlu 7459 | 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 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) | ||
Theorem | npex 7460 | The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) |
⊢ P ∈ V | ||
Theorem | elinp 7461* | 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 7462 | A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (𝐴 ∈ P → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P) | ||
Theorem | elnp1st2nd 7463* | Membership in positive reals, using 1st and 2nd 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 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴)))))) | ||
Theorem | prml 7464* | A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (〈𝐿, 𝑈〉 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿) | ||
Theorem | prmu 7465* | A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.) |
⊢ (〈𝐿, 𝑈〉 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝑈) | ||
Theorem | prssnql 7466 | The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) | ||
Theorem | prssnqu 7467 | The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
⊢ (〈𝐿, 𝑈〉 ∈ P → 𝑈 ⊆ Q) | ||
Theorem | elprnql 7468 | 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 7469 | 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 7470 | The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
⊢ ¬ ∅ ∈ P | ||
Theorem | prcdnql 7471 | 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 7472 | 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 7473 | 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 7474 | 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 7475* | 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 7476* | 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 7477* | 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 7478 | A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 <Q 𝐵) → (𝐴 ∈ 𝐿 ∨ 𝐵 ∈ 𝑈)) | ||
Theorem | prdisj 7479 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ Q) → ¬ (𝐴 ∈ 𝐿 ∧ 𝐴 ∈ 𝑈)) | ||
Theorem | prarloclemlt 7480 | Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 10-Nov-2019.) |
⊢ (((𝑋 ∈ ω ∧ (〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([〈(𝑦 +o 1o), 1o〉] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑋), 1o〉] ~Q ·Q 𝑃))) | ||
Theorem | prarloclemlo 7481* | Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 10-Nov-2019.) |
⊢ (((𝑋 ∈ ω ∧ (〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([〈(𝑦 +o 1o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝐿 → (((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o suc 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)))) | ||
Theorem | prarloclemup 7482 | Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 10-Nov-2019.) |
⊢ (((𝑋 ∈ ω ∧ (〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈 → (((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o suc 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)))) | ||
Theorem | prarloclem3step 7483* | Induction step for prarloclem3 7484. (Contributed by Jim Kingdon, 9-Nov-2019.) |
⊢ (((𝑋 ∈ ω ∧ (〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o suc 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) | ||
Theorem | prarloclem3 7484* | Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 27-Oct-2019.) |
⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑋 ∈ ω ∧ 𝑃 ∈ Q) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑋), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) | ||
Theorem | prarloclem4 7485* | A slight rearrangement of prarloclem3 7484. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 4-Nov-2019.) |
⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ 𝑃 ∈ Q) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) | ||
Theorem | prarloclemn 7486* | Subtracting two from a positive integer. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 5-Nov-2019.) |
⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁) | ||
Theorem | prarloclem5 7487* | A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 7490. (Contributed by Jim Kingdon, 4-Nov-2019.) |
⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([〈𝑁, 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) | ||
Theorem | prarloclem 7488* | 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 ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([〈𝑁, 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) | ||
Theorem | prarloclemcalc 7489 | Some calculations for prarloc 7490. (Contributed by Jim Kingdon, 26-Oct-2019.) |
⊢ (((𝐴 = (𝑋 +Q0 ([〈𝑀, 1o〉] ~Q0 ·Q0 𝑄)) ∧ 𝐵 = (𝑋 +Q ([〈(𝑀 +o 2o), 1o〉] ~Q ·Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +Q 𝑄) <Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 <Q (𝐴 +Q 𝑃)) | ||
Theorem | prarloc 7490* |
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 7491 instead. (Contributed by Jim Kingdon, 22-Oct-2019.) |
⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃)) | ||
Theorem | prarloc2 7491* | A Dedekind cut is arithmetically located. This is a variation of prarloc 7490 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 7492 | Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) |
⊢ <P ⊆ (P × P) | ||
Theorem | ltdfpr 7493* | More convenient form of df-iltp 7457. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵)))) | ||
Theorem | genpdflem 7494* | Simplification of upper or lower cut expression. Lemma for genpdf 7495. (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 7495* | 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 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}〉) ⇒ ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ 〈{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)}〉) | ||
Theorem | genipv 7496* | 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 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}〉) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = 〈{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)}〉) | ||
Theorem | genplt2i 7497* | 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 7498* | 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 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}〉) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q)) | ||
Theorem | genpelvl 7499* | 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 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}〉) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝐶 = (𝑔𝐺ℎ))) | ||
Theorem | genpelvu 7500* | 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 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}〉) & ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) ⇒ ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |