P. 77 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	genppreclu 7601*	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 ∈ (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 ‘𝐴) ∧ 𝐷 ∈ (2nd ‘𝐵)) → (𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵))))
Theorem	genipdm 7602*	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 ∈ (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)    ⇒   ⊢ dom 𝐹 = (P × P)
Theorem	genpml 7603*	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 ∈ (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) → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
Theorem	genpmu 7604*	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 ∈ (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) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
Theorem	genpcdl 7605*	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 ∈ (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 ∧ 𝑔 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓 → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
Theorem	genpcuu 7606*	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 ∈ (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 ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Theorem	genprndl 7607*	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 ∈ (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)    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
Theorem	genprndu 7608*	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 ∈ (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)    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Theorem	genpdisj 7609*	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 ∈ (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)    &   ⊢ ((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 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
Theorem	genpassl 7610*	Associativity of lower cuts. Lemma for genpassg 7612. (Contributed by Jim Kingdon, 11-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))}⟩)    &   ⊢ ((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) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
Theorem	genpassu 7611*	Associativity of upper cuts. Lemma for genpassg 7612. (Contributed by Jim Kingdon, 11-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))}⟩)    &   ⊢ ((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) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))
Theorem	genpassg 7612*	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 ∈ (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)    &   ⊢ 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 7613	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 7614	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 7615	Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))
Theorem	addnqpru 7616	Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocprlemlt 7617	Lemma for addlocpr 7622. 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 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))
Theorem	addlocprlemeqgt 7618	Lemma for addlocpr 7622. 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 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
Theorem	addlocprlemeq 7619	Lemma for addlocpr 7622. 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 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocprlemgt 7620	Lemma for addlocpr 7622. 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 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocprlem 7621	Lemma for addlocpr 7622. 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 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocpr 7622*	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 7589 to both 𝐴 and 𝐵, and uses nqtri3or 7482 rather than prloc 7577 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 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
Theorem	addclpr 7623	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 7624*	Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)
Theorem	mpvlu 7625*	Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)
Theorem	dmplp 7626	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
⊢ dom +P = (P × P)
Theorem	dmmp 7627	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
⊢ dom ·P = (P × P)
Theorem	nqprm 7628*	A cut produced from a rational is inhabited. Lemma for nqprlu 7633. (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 7629*	A cut produced from a rational is rounded. Lemma for nqprlu 7633. (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 7630*	A cut produced from a rational is disjoint. Lemma for nqprlu 7633. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ (𝐴 ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))
Theorem	nqprloc 7631*	A cut produced from a rational is located. Lemma for nqprlu 7633. (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	nqprxx 7632*	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.)
⊢ (𝐴 ∈ Q → ⟨{𝑥 ∣ 𝑥 <Q 𝐴}, {𝑥 ∣ 𝐴 <Q 𝑥}⟩ ∈ P)
Theorem	nqprlu 7633*	The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
Theorem	recnnpr 7634*	The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
⊢ (𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
Theorem	ltnqex 7635	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 7636	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	nqprl 7637*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (1st ‘𝐵) ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P 𝐵))
Theorem	nqpru 7638*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) ↔ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
Theorem	nnprlu 7639*	The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
⊢ (𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐴, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐴, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
Theorem	1pr 7640	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
⊢ 1P ∈ P
Theorem	1prl 7641	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q}
Theorem	1pru 7642	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Theorem	addnqprlemrl 7643*	Lemma for addnqpr 7647. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Theorem	addnqprlemru 7644*	Lemma for addnqpr 7647. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Theorem	addnqprlemfl 7645*	Lemma for addnqpr 7647. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	addnqprlemfu 7646*	Lemma for addnqpr 7647. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	addnqpr 7647*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
Theorem	addnqpr1 7648*	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 7647. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P 1P))
Theorem	appdivnq 7649*	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 𝐵 ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝐴 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))
Theorem	appdiv0nq 7650*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7649 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.)
⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
Theorem	prmuloclemcalc 7651	Calculations for prmuloc 7652. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ (𝜑 → 𝑅 <Q 𝑈)    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → (𝐴 +Q 𝑋) = 𝐵)    &   ⊢ (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Q)    &   ⊢ (𝜑 → 𝐵 ∈ Q)    &   ⊢ (𝜑 → 𝐷 ∈ Q)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → 𝑋 ∈ Q)    ⇒   ⊢ (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))
Theorem	prmuloc 7652*	Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <Q 𝐵) → ∃𝑑 ∈ Q ∃𝑢 ∈ Q (𝑑 ∈ 𝐿 ∧ 𝑢 ∈ 𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
Theorem	prmuloc2 7653*	Positive reals are multiplicatively located. This is a variation of prmuloc 7652 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) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
Theorem	mulnqprl 7654	Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
Theorem	mulnqpru 7655	Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Theorem	mullocprlem 7656	Calculations for mullocpr 7657. (Contributed by Jim Kingdon, 10-Dec-2019.)
⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐵 ∈ P))    &   ⊢ (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 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))    &   ⊢ (𝜑 → (𝐷 ∈ Q ∧ 𝑈 ∈ Q))    &   ⊢ (𝜑 → (𝐷 ∈ (1st ‘𝐴) ∧ 𝑈 ∈ (2nd ‘𝐴)))    &   ⊢ (𝜑 → (𝐸 ∈ Q ∧ 𝑇 ∈ Q))    ⇒   ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Theorem	mullocpr 7657*	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.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
Theorem	mulclpr 7658	Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
Theorem	mulnqprlemrl 7659*	Lemma for mulnqpr 7663. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
Theorem	mulnqprlemru 7660*	Lemma for mulnqpr 7663. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
Theorem	mulnqprlemfl 7661*	Lemma for mulnqpr 7663. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	mulnqprlemfu 7662*	Lemma for mulnqpr 7663. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	mulnqpr 7663*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
Theorem	addcomprg 7664	Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
Theorem	addassprg 7665	Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)))
Theorem	mulcomprg 7666	Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
Theorem	mulassprg 7667	Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶)))
Theorem	distrlem1prl 7668	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
Theorem	distrlem1pru 7669	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
Theorem	distrlem4prl 7670*	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)) ∧ (𝑓 ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (1st ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrlem4pru 7671*	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrlem5prl 7672	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrlem5pru 7673	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrprg 7674	Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
Theorem	ltprordil 7675	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 → (1st ‘𝐴) ⊆ (1st ‘𝐵))
Theorem	1idprl 7676	Lemma for 1idpr 7678. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 1P)) = (1st ‘𝐴))
Theorem	1idpru 7677	Lemma for 1idpr 7678. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))
Theorem	1idpr 7678	1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.)
⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴)
Theorem	ltnqpr 7679*	We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
Theorem	ltnqpri 7680*	We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
⊢ (𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)
Theorem	ltpopr 7681	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 7682. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ <P Po P
Theorem	ltsopr 7682	Positive real 'less than' is a weak linear order (in the sense of df-iso 4333). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
⊢ <P Or P
Theorem	ltaddpr 7683	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.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<P (𝐴 +P 𝐵))
Theorem	ltexprlemell 7684*	Element in lower cut of the constructed difference. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
Theorem	ltexprlemelu 7685*	Element in upper cut of the constructed difference. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
Theorem	ltexprlemm 7686*	Our constructed difference is inhabited. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemopl 7687*	The lower cut of our constructed difference is open. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)))
Theorem	ltexprlemlol 7688*	The lower cut of our constructed difference is lower. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)) → 𝑞 ∈ (1st ‘𝐶)))
Theorem	ltexprlemopu 7689*	The upper cut of our constructed difference is open. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemupu 7690*	The upper cut of our constructed difference is upper. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) → 𝑟 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemrnd 7691*	Our constructed difference is rounded. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))))
Theorem	ltexprlemdisj 7692*	Our constructed difference is disjoint. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemloc 7693*	Our constructed difference is located. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
Theorem	ltexprlempr 7694*	Our constructed difference is a positive real. Lemma for ltexpri 7699. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
Theorem	ltexprlemfl 7695*	Lemma for ltexpri 7699. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st ‘𝐵))
Theorem	ltexprlemrl 7696*	Lemma for ltexpri 7699. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (1st ‘𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
Theorem	ltexprlemfu 7697*	Lemma for ltexpri 7699. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))
Theorem	ltexprlemru 7698*	Lemma for ltexpri 7699. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
Theorem	ltexpri 7699*	Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)
Theorem	addcanprleml 7700	Lemma for addcanprg 7702. (Contributed by Jim Kingdon, 25-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘𝐶))