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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltmnqi 6501 Ordering property of multiplication for positive fractions. One direction of ltmnqg 6499. (Contributed by Jim Kingdon, 9-Dec-2019.)
((𝐴 <Q 𝐵𝐶Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))

Theoremlt2addnq 6502 Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 +Q 𝐶) <Q (𝐵 +Q 𝐷)))

Theoremlt2mulnq 6503 Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
(((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))

Theorem1lt2nq 6504 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
1Q <Q (1Q +Q 1Q)

Theoremltaddnq 6505 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))

Theoremltexnqq 6506* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵))

Theoremltexnqi 6507* Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
(𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)

Theoremhalfnqq 6508* One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
(𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝐴)

Theoremhalfnq 6509* One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
(𝐴Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)

Theoremnsmallnqq 6510* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
(𝐴Q → ∃𝑥Q 𝑥 <Q 𝐴)

Theoremnsmallnq 6511* There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.)
(𝐴Q → ∃𝑥 𝑥 <Q 𝐴)

Theoremsubhalfnqq 6512* 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 6508). (Contributed by Jim Kingdon, 25-Nov-2019.)
(𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴)

Theoremltbtwnnqq 6513* 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 ↔ ∃𝑥Q (𝐴 <Q 𝑥𝑥 <Q 𝐵))

Theoremltbtwnnq 6514* 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 𝑥𝑥 <Q 𝐵))

Theoremarchnqq 6515* 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.)
(𝐴Q → ∃𝑥N 𝐴 <Q [⟨𝑥, 1𝑜⟩] ~Q )

Theoremprarloclemarch 6516* A version of the Archimedean property. This variation is "stronger" than archnqq 6515 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.)
((𝐴Q𝐵Q) → ∃𝑥N 𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))

Theoremprarloclemarch2 6517* Like prarloclemarch 6516 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 6601. (Contributed by Jim Kingdon, 25-Nov-2019.)
((𝐴Q𝐵Q𝐶Q) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))

Theoremltrnqg 6518 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6519. (Contributed by Jim Kingdon, 29-Dec-2019.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴)))

Theoremltrnqi 6519 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6518. (Contributed by Jim Kingdon, 24-Sep-2019.)
(𝐴 <Q 𝐵 → (*Q𝐵) <Q (*Q𝐴))

Theoremnnnq 6520 The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴N → [⟨𝐴, 1𝑜⟩] ~QQ)

Theoremltnnnq 6521 Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ [⟨𝐴, 1𝑜⟩] ~Q <Q [⟨𝐵, 1𝑜⟩] ~Q ))

Definitiondf-enq0 6522* 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)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))}

Definitiondf-nq0 6523 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.)
Q0 = ((ω × N) / ~Q0 )

Definitiondf-0nq0 6524 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.)
0Q0 = [⟨∅, 1𝑜⟩] ~Q0

Definitiondf-plq0 6525* 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⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}

Definitiondf-mq0 6526* 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⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}

Theoremdfmq0qs 6527* Multiplication on non-negative fractions. This definition is similar to df-mq0 6526 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}

Theoremdfplq0qs 6528* Addition on non-negative fractions. This definition is similar to df-plq0 6525 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
+Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}

Theoremenq0enq 6529 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)))

Theoremenq0sym 6530 The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6533. (Contributed by Jim Kingdon, 14-Nov-2019.)
(𝑓 ~Q0 𝑔𝑔 ~Q0 𝑓)

Theoremenq0ref 6531 The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6533. (Contributed by Jim Kingdon, 14-Nov-2019.)
(𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)

Theoremenq0tr 6532 The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6533. (Contributed by Jim Kingdon, 14-Nov-2019.)
((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → 𝑓 ~Q0 )

Theoremenq0er 6533 The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
~Q0 Er (ω × N)

Theoremenq0breq 6534 Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (⟨𝐴, 𝐵⟩ ~Q0𝐶, 𝐷⟩ ↔ (𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶)))

Theoremenq0eceq 6535 Equivalence class equality of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ↔ (𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶)))

Theoremnqnq0pi 6536 A non-negative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴N𝐵N) → [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q )

Theoremenq0ex 6537 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
~Q0 ∈ V

Theoremnq0ex 6538 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
Q0 ∈ V

Theoremnqnq0 6539 A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
QQ0

Theoremnq0nn 6540* Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
(𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))

Theoremaddcmpblnq0 6541 Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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)⟩))

Theoremmulcmpblnq0 6542 Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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)⟩))

Theoremmulcanenq0ec 6543 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴N𝐵 ∈ ω ∧ 𝐶N) → [⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )

Theoremnnnq0lem1 6544* Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6547 and mulnnnq0 6548. (Contributed by Jim Kingdon, 23-Nov-2019.)
(((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔))))

Theoremaddnq0mo 6545* 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 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))

Theoremmulnq0mo 6546* 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 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))

Theoremaddnnnq0 6547 Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Theoremmulnnnq0 6548 Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Theoremaddclnq0 6549 Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴Q0𝐵Q0) → (𝐴 +Q0 𝐵) ∈ Q0)

Theoremmulclnq0 6550 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
((𝐴Q0𝐵Q0) → (𝐴 ·Q0 𝐵) ∈ Q0)

Theoremnqpnq0nq 6551 A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
((𝐴Q𝐵Q0) → (𝐴 +Q0 𝐵) ∈ Q)

Theoremnqnq0a 6552 Addition of positive fractions is equal with +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))

Theoremnqnq0m 6553 Multiplication of positive fractions is equal with ·Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))

Theoremnq0m0r 6554 Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(𝐴Q0 → (0Q0 ·Q0 𝐴) = 0Q0)

Theoremnq0a0 6555 Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
(𝐴Q0 → (𝐴 +Q0 0Q0) = 𝐴)

Theoremnnanq0 6556 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 ∧ 𝐴N) → [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Theoremdistrnq0 6557 Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
((𝐴Q0𝐵Q0𝐶Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))

Theoremmulcomnq0 6558 Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
((𝐴Q0𝐵Q0) → (𝐴 ·Q0 𝐵) = (𝐵 ·Q0 𝐴))

Theoremaddassnq0lemcl 6559 A natural number closure law. Lemma for addassnq0 6560. (Contributed by Jim Kingdon, 3-Dec-2019.)
(((𝐼 ∈ ω ∧ 𝐽N) ∧ (𝐾 ∈ ω ∧ 𝐿N)) → (((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω ∧ (𝐽 ·𝑜 𝐿) ∈ N))

Theoremaddassnq0 6560 Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴Q0𝐵Q0𝐶Q0) → ((𝐴 +Q0 𝐵) +Q0 𝐶) = (𝐴 +Q0 (𝐵 +Q0 𝐶)))

Theoremdistnq0r 6561 Multiplication of non-negative fractions is distributive. Version of distrnq0 6557 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
((𝐴Q0𝐵Q0𝐶Q0) → ((𝐵 +Q0 𝐶) ·Q0 𝐴) = ((𝐵 ·Q0 𝐴) +Q0 (𝐶 ·Q0 𝐴)))

Theoremaddpinq1 6562 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴N → [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q))

Theoremnq02m 6563 Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
(𝐴Q0 → ([⟨2𝑜, 1𝑜⟩] ~Q0 ·Q0 𝐴) = (𝐴 +Q0 𝐴))

Definitiondf-inp 6564* 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 (𝑞Q𝑞𝑙 ∧ ∃𝑟Q𝑟𝑢), rounded (𝑞Q(𝑞𝑙 ↔ ∃𝑟Q(𝑞 <Q 𝑟𝑟𝑙)) and likewise for 𝑢), disjoint (𝑞Q¬ (𝑞𝑙𝑞𝑢)) and located (𝑞Q𝑟Q(𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))). 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⟩ ∣ (((𝑙Q𝑢Q) ∧ (∃𝑞Q 𝑞𝑙 ∧ ∃𝑟Q 𝑟𝑢)) ∧ ((∀𝑞Q (𝑞𝑙 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝑙)) ∧ ∀𝑟Q (𝑟𝑢 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑢))) ∧ ∀𝑞Q ¬ (𝑞𝑙𝑞𝑢) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝑙𝑟𝑢))))}

Definitiondf-i1p 6565* 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 = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩

Definitiondf-iplp 6566* 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 ∈ (1st𝑥) implies 𝑟Q) and can be simplified as shown at genpdf 6606.

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 = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)

Definitiondf-imp 6567* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6566 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 = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)

Definitiondf-iltp 6568* 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⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}

Theoremnpsspw 6569 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
P ⊆ (𝒫 Q × 𝒫 Q)

Theorempreqlu 6570 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))

Theoremnpex 6571 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
P ∈ V

Theoremelinp 6572* Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿Q𝑈Q) ∧ (∃𝑞Q 𝑞𝐿 ∧ ∃𝑟Q 𝑟𝑈)) ∧ ((∀𝑞Q (𝑞𝐿 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝐿)) ∧ ∀𝑟Q (𝑟𝑈 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑈))) ∧ ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝐿𝑟𝑈)))))

Theoremprop 6573 A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
(𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)

Theoremelnp1st2nd 6574* Membership in positive reals, using 1st and 2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
(𝐴P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐴))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐴) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐴)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐴) ∨ 𝑟 ∈ (2nd𝐴))))))

Theoremprml 6575* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝐿)

Theoremprmu 6576* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝑈)

Theoremprssnql 6577 A positive real's lower cut is a subset of the positive fractions. It would presumably be possible to also prove 𝐿Q, but we only need 𝐿Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P𝐿Q)

Theoremprssnqu 6578 A positive real's upper cut is a subset of the positive fractions. It would presumably be possible to also prove 𝑈Q, but we only need 𝑈Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
(⟨𝐿, 𝑈⟩ ∈ P𝑈Q)

Theoremelprnql 6579 An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → 𝐵Q)

Theoremelprnqu 6580 An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → 𝐵Q)

Theorem0npr 6581 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
¬ ∅ ∈ P

Theoremprcdnql 6582 A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → (𝐶 <Q 𝐵𝐶𝐿))

Theoremprcunqu 6583 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → (𝐶 <Q 𝐵𝐵𝑈))

Theoremprubl 6584 A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) ∧ 𝐶Q) → (¬ 𝐶𝐿𝐵 <Q 𝐶))

Theoremprltlu 6585 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⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐵 <Q 𝐶)

Theoremprnmaxl 6586* A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → ∃𝑥𝐿 𝐵 <Q 𝑥)

Theoremprnminu 6587* An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥𝑈 𝑥 <Q 𝐵)

Theoremprnmaddl 6588* A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) → ∃𝑥Q (𝐵 +Q 𝑥) ∈ 𝐿)

Theoremprloc 6589 A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → (𝐴𝐿𝐵𝑈))

Theoremprdisj 6590 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐴Q) → ¬ (𝐴𝐿𝐴𝑈))

Theoremprarloclemlt 6591 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))

Theoremprarloclemlo 6592* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 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))))

Theoremprarloclemup 6593 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 10-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 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))))

Theoremprarloclem3step 6594* Induction step for prarloclem3 6595. (Contributed by Jim Kingdon, 9-Nov-2019.)
(((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ ∃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))

Theoremprarloclem3 6595* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 27-Oct-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑋 ∈ ω ∧ 𝑃Q) ∧ ∃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))

Theoremprarloclem4 6596* A slight rearrangement of prarloclem3 6595. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 4-Nov-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ 𝑃Q) → (∃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)))

Theoremprarloclemn 6597* Subtracting two from a positive integer. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 5-Nov-2019.)
((𝑁N ∧ 1𝑜 <N 𝑁) → ∃𝑥 ∈ ω (2𝑜 +𝑜 𝑥) = 𝑁)

Theoremprarloclem5 6598* A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 4-Nov-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 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))

Theoremprarloclem 6599* 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⟩ ∈ P𝐴𝐿) ∧ (𝑁N𝑃Q ∧ 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))

Theoremprarloclemcalc 6600 Some calculations for prarloc 6601. (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))) ∧ ((𝑄Q ∧ (𝑄 +Q 𝑄) <Q 𝑃) ∧ (𝑋Q𝑀 ∈ ω))) → 𝐵 <Q (𝐴 +Q 𝑃))

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