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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrelpr 6601 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
<P ⊆ (P × P)

Theoremltdfpr 6602* More convenient form of df-iltp 6566. (Contributed by Jim Kingdon, 15-Dec-2019.)
((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))

Theoremgenpdflem 6603* Simplification of upper or lower cut expression. Lemma for genpdf 6604. (Contributed by Jim Kingdon, 30-Sep-2019.)
((𝜑𝑟𝐴) → 𝑟Q)    &   ((𝜑𝑠𝐵) → 𝑠Q)       (𝜑 → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟𝐴𝑠𝐵𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟𝐴𝑠𝐵 𝑞 = (𝑟𝐺𝑠)})

Theoremgenpdf 6604* Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)       𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)

Theoremgenipv 6605* Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝐴)∃𝑠 ∈ (1st𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝐴)∃𝑠 ∈ (2nd𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)

Theoremgenplt2i 6606* Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴𝐺𝐶) <Q (𝐵𝐺𝐷))

Theoremgenpelxp 6607* Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)       ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))

Theoremgenpelvl 6608* Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝐶 = (𝑔𝐺)))

Theoremgenpelvu 6609* Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))

Theoremgenpprecll 6610* Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ((𝐶 ∈ (1st𝐴) ∧ 𝐷 ∈ (1st𝐵)) → (𝐶𝐺𝐷) ∈ (1st ‘(𝐴𝐹𝐵))))

Theoremgenppreclu 6611* Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ((𝐶 ∈ (2nd𝐴) ∧ 𝐷 ∈ (2nd𝐵)) → (𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵))))

Theoremgenipdm 6612* Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       dom 𝐹 = (P × P)

Theoremgenpml 6613* The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))

Theoremgenpmu 6614* The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))

Theoremgenpcdl 6615* Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))

Theoremgenpcuu 6616* Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Theoremgenprndl 6617* The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))

Theoremgenprndu 6618* The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))       ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Theoremgenpdisj 6619* The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))

Theoremgenpassl 6620* Associativity of lower cuts. Lemma for genpassg 6622. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))       ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))

Theoremgenpassu 6621* Associativity of upper cuts. Lemma for genpassg 6622. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))       ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))

Theoremgenpassg 6622* Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))       ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Theoremaddnqprllem 6623 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐺𝐿) ∧ 𝑋Q) → (𝑋 <Q 𝑆 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝐿))

Theoremaddnqprulem 6624 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((⟨𝐿, 𝑈⟩ ∈ P𝐺𝑈) ∧ 𝑋Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q𝑆)) ·Q 𝐺) ∈ 𝑈))

Theoremaddnqprl 6625 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))

Theoremaddnqpru 6626 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocprlemlt 6627 Lemma for addlocpr 6632. The 𝑄 <Q (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))

Theoremaddlocprlemeqgt 6628 Lemma for addlocpr 6632. 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.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Theoremaddlocprlemeq 6629 Lemma for addlocpr 6632. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocprlemgt 6630 Lemma for addlocpr 6632. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocprlem 6631 Lemma for addlocpr 6632. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑𝑄 <Q 𝑅)    &   (𝜑𝑃Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑𝐷 ∈ (1st𝐴))    &   (𝜑𝑈 ∈ (2nd𝐴))    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑𝐸 ∈ (1st𝐵))    &   (𝜑𝑇 ∈ (2nd𝐵))    &   (𝜑𝑇 <Q (𝐸 +Q 𝑃))       (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Theoremaddlocpr 6632* 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 6599 to both 𝐴 and 𝐵, and uses nqtri3or 6492 rather than prloc 6587 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.)
((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))

Theoremaddclpr 6633 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.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)

Theoremplpvlu 6634* Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)

Theoremmpvlu 6635* Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)

Theoremdmplp 6636 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
dom +P = (P × P)

Theoremdmmp 6637 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
dom ·P = (P × P)

Theoremnqprm 6638* A cut produced from a rational is inhabited. Lemma for nqprlu 6643. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))

Theoremnqprrnd 6639* A cut produced from a rational is rounded. Lemma for nqprlu 6643. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → (∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))))

Theoremnqprdisj 6640* A cut produced from a rational is disjoint. Lemma for nqprlu 6643. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → ∀𝑞Q ¬ (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥𝐴 <Q 𝑥}))

Theoremnqprloc 6641* A cut produced from a rational is located. Lemma for nqprlu 6643. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥𝐴 <Q 𝑥})))

Theoremnqprxx 6642* The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → ⟨{𝑥𝑥 <Q 𝐴}, {𝑥𝐴 <Q 𝑥}⟩ ∈ P)

Theoremnqprlu 6643* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
(𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)

Theoremrecnnpr 6644* The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
(𝐴N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)

Theoremltnqex 6645 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{𝑥𝑥 <Q 𝐴} ∈ V

Theoremgtnqex 6646 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{𝑥𝐴 <Q 𝑥} ∈ V

Theoremnqprl 6647* Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)
((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))

Theoremnqpru 6648* Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <P. (Contributed by Jim Kingdon, 29-Nov-2020.)
((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) ↔ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))

Theoremnnprlu 6649* The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
(𝐴N → ⟨{𝑙𝑙 <Q [⟨𝐴, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐴, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)

Theorem1pr 6650 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
1PP

Theorem1prl 6651 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(1st ‘1P) = {𝑥𝑥 <Q 1Q}

Theorem1pru 6652 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}

Theoremaddnqprlemrl 6653* Lemma for addnqpr 6657. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

Theoremaddnqprlemru 6654* Lemma for addnqpr 6657. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

Theoremaddnqprlemfl 6655* Lemma for addnqpr 6657. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremaddnqprlemfu 6656* Lemma for addnqpr 6657. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremaddnqpr 6657* Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → ⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

Theoremaddnqpr1 6658* Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 6657. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴Q → ⟨{𝑙𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P 1P))

Theoremappdivnq 6659* Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where 𝐴 and 𝐵 are positive, as well as 𝐶). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴 <Q 𝐵𝐶Q) → ∃𝑚Q (𝐴 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))

Theoremappdiv0nq 6660* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6659 in which 𝐴 is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
((𝐵Q𝐶Q) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)

Theoremprmuloclemcalc 6661 Calculations for prmuloc 6662. (Contributed by Jim Kingdon, 9-Dec-2019.)
(𝜑𝑅 <Q 𝑈)    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → (𝐴 +Q 𝑋) = 𝐵)    &   (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))    &   (𝜑𝐴Q)    &   (𝜑𝐵Q)    &   (𝜑𝐷Q)    &   (𝜑𝑃Q)    &   (𝜑𝑋Q)       (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))

Theoremprmuloc 6662* Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))

Theoremprmuloc2 6663* Positive reals are multiplicatively located. This is a variation of prmuloc 6662 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio 𝐵, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)

Theoremmulnqprl 6664 Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))

Theoremmulnqpru 6665 Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Theoremmullocprlem 6666 Calculations for mullocpr 6667. (Contributed by Jim Kingdon, 10-Dec-2019.)
(𝜑 → (𝐴P𝐵P))    &   (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))    &   (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))    &   (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))    &   (𝜑 → (𝑄Q𝑅Q))    &   (𝜑 → (𝐷Q𝑈Q))    &   (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))    &   (𝜑 → (𝐸Q𝑇Q))       (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Theoremmullocpr 6667* Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both 𝐴 and 𝐵 are positive, not just 𝐴). (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))

Theoremmulclpr 6668 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)

Theoremmulnqprlemrl 6669* Lemma for mulnqpr 6673. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

Theoremmulnqprlemru 6670* Lemma for mulnqpr 6673. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

Theoremmulnqprlemfl 6671* Lemma for mulnqpr 6673. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremmulnqprlemfu 6672* Lemma for mulnqpr 6673. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremmulnqpr 6673* Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → ⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

Theoremaddcomprg 6674 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))

Theoremaddassprg 6675 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)))

Theoremmulcomprg 6676 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))

Theoremmulassprg 6677 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P𝐶P) → ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶)))

Theoremdistrlem1prl 6678 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Theoremdistrlem1pru 6679 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Theoremdistrlem4prl 6680* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
(((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrlem4pru 6681* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
(((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrlem5prl 6682 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrlem5pru 6683 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrprg 6684 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Theoremltprordil 6685 If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
(𝐴<P 𝐵 → (1st𝐴) ⊆ (1st𝐵))

Theorem1idprl 6686 Lemma for 1idpr 6688. (Contributed by Jim Kingdon, 13-Dec-2019.)
(𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))

Theorem1idpru 6687 Lemma for 1idpr 6688. (Contributed by Jim Kingdon, 13-Dec-2019.)
(𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))

Theorem1idpr 6688 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.)
(𝐴P → (𝐴 ·P 1P) = 𝐴)

Theoremltnqpr 6689* We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

Theoremltnqpri 6690* We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
(𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)

Theoremltpopr 6691 Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6692. (Contributed by Jim Kingdon, 15-Dec-2019.)
<P Po P

Theoremltsopr 6692 Positive real 'less than' is a weak linear order (in the sense of df-iso 4034). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
<P Or P

Theoremltaddpr 6693 The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))

Theoremltexprlemell 6694* Element in lower cut of the constructed difference. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))

Theoremltexprlemelu 6695* Element in upper cut of the constructed difference. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))

Theoremltexprlemm 6696* Our constructed difference is inhabited. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶)))

Theoremltexprlemopl 6697* The lower cut of our constructed difference is open. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))

Theoremltexprlemlol 6698* The lower cut of our constructed difference is lower. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))

Theoremltexprlemopu 6699* The upper cut of our constructed difference is open. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))

Theoremltexprlemupu 6700* The upper cut of our constructed difference is upper. Lemma for ltexpri 6709. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))

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