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

Theoremgenpcdl 6501* Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)    &   ((((A P g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))       ((A P B P) → (f (1st ‘(A𝐹B)) → (x <Q fx (1st ‘(A𝐹B)))))

Theoremgenpcuu 6502* Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)    &   ((((A P g (2ndA)) (B P (2ndB))) x Q) → ((g𝐺) <Q xx (2nd ‘(A𝐹B))))       ((A P B P) → (f (2nd ‘(A𝐹B)) → (f <Q xx (2nd ‘(A𝐹B)))))

Theoremgenprndl 6503* 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 Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) 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 (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))       ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))

Theoremgenprndu 6504* 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 Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) 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 (2ndA)) (B P (2ndB))) x Q) → ((g𝐺) <Q xx (2nd ‘(A𝐹B))))       ((A P B P) → 𝑟 Q (𝑟 (2nd ‘(A𝐹B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A𝐹B)))))

Theoremgenpdisj 6505* 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 Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) 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))))

Theoremgenpassl 6506* Associativity of lower cuts. Lemma for genpassg 6508. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)    &   dom 𝐹 = (P × P)    &   ((f P g P) → (f𝐹g) P)    &   ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))       ((A P B P 𝐶 P) → (1st ‘((A𝐹B)𝐹𝐶)) = (1st ‘(A𝐹(B𝐹𝐶))))

Theoremgenpassu 6507* Associativity of upper cuts. Lemma for genpassg 6508. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)    &   dom 𝐹 = (P × P)    &   ((f P g P) → (f𝐹g) P)    &   ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))       ((A P B P 𝐶 P) → (2nd ‘((A𝐹B)𝐹𝐶)) = (2nd ‘(A𝐹(B𝐹𝐶))))

Theoremgenpassg 6508* Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)    &   ((y Q z Q) → (y𝐺z) Q)    &   dom 𝐹 = (P × P)    &   ((f P g P) → (f𝐹g) P)    &   ((f Q g Q Q) → ((f𝐺g)𝐺) = (f𝐺(g𝐺)))       ((A P B P 𝐶 P) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶)))

Theoremaddnqprllem 6509 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 (*Q𝑆)) ·Q 𝐺) 𝐿))

Theoremaddnqprulem 6510 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 (*Q𝑆)) ·Q 𝐺) 𝑈))

Theoremaddnqprl 6511 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 (1st ‘(A +P B))))

Theoremaddnqpru 6512 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 (2nd ‘(A +P B))))

Theoremaddlocprlemlt 6513 Lemma for addlocpr 6518. The 𝑄 <Q (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6514 Lemma for addlocpr 6518. 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.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6515 Lemma for addlocpr 6518. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6516 Lemma for addlocpr 6518. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6517 Lemma for addlocpr 6518. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
(φA P)    &   (φB P)    &   (φ𝑄 <Q 𝑅)    &   (φ𝑃 Q)    &   (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 6518* 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 6485 to both A and B, and uses nqtri3or 6380 rather than prloc 6473 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)))))

Theoremaddclpr 6519 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)

Theoremplpvlu 6520* Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((A P B P) → (A +P B) = ⟨{x Qy (1stA)z (1stB)x = (y +Q z)}, {x Qy (2ndA)z (2ndB)x = (y +Q z)}⟩)

Theoremmpvlu 6521* Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((A P B P) → (A ·P B) = ⟨{x Qy (1stA)z (1stB)x = (y ·Q z)}, {x Qy (2ndA)z (2ndB)x = (y ·Q z)}⟩)

Theoremdmplp 6522 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
dom +P = (P × P)

Theoremdmmp 6523 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
dom ·P = (P × P)

Theoremnqprm 6524* A cut produced from a rational is inhabited. Lemma for nqprlu 6529. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → (𝑞 Q 𝑞 {xx <Q A} 𝑟 Q 𝑟 {xA <Q x}))

Theoremnqprrnd 6525* A cut produced from a rational is rounded. Lemma for nqprlu 6529. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → (𝑞 Q (𝑞 {xx <Q A} ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 {xx <Q A})) 𝑟 Q (𝑟 {xA <Q x} ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 {xA <Q x}))))

Theoremnqprdisj 6526* A cut produced from a rational is disjoint. Lemma for nqprlu 6529. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q𝑞 Q ¬ (𝑞 {xx <Q A} 𝑞 {xA <Q x}))

Theoremnqprloc 6527* A cut produced from a rational is located. Lemma for nqprlu 6529. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 {xx <Q A} 𝑟 {xA <Q x})))

Theoremnqprxx 6528* 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 → ⟨{xx <Q A}, {xA <Q x}⟩ P)

Theoremnqprlu 6529* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
(A Q → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)

Theoremltnqex 6530 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xx <Q A} V

Theoremgtnqex 6531 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xA <Q x} V

Theoremnqprl 6532* 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 (1stB) ↔ ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B))

Theoremnnprlu 6533* The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
(A N → ⟨{𝑙𝑙 <Q [⟨A, 1𝑜⟩] ~Q }, {u ∣ [⟨A, 1𝑜⟩] ~Q <Q u}⟩ P)

Theorem1pr 6534 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
1P P

Theorem1prl 6535 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(1st ‘1P) = {xx <Q 1Q}

Theorem1pru 6536 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(2nd ‘1P) = {x ∣ 1Q <Q x}

Theoremaddnqprlemrl 6537* Lemma for addnqpr 6541. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((A Q B Q) → (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 6538* Lemma for addnqpr 6541. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((A Q B Q) → (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 6539* Lemma for addnqpr 6541. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((A Q B Q) → (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 6540* Lemma for addnqpr 6541. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((A Q B Q) → (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 6541* 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) → ⟨{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 6542* 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 6541. (Contributed by Jim Kingdon, 26-Apr-2020.)
(A Q → ⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩ = (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))

Theoremappdivnq 6543* 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))

Theoremappdiv0nq 6544* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6543 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)

Theoremprmuloclemcalc 6545 Calculations for prmuloc 6546. (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))    &   (φA Q)    &   (φB Q)    &   (φ𝐷 Q)    &   (φ𝑃 Q)    &   (φ𝑋 Q)       (φ → (𝑈 ·Q A) <Q (𝐷 ·Q B))

Theoremprmuloc 6546* 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)))

Theoremprmuloc2 6547* Positive reals are multiplicatively located. This is a variation of prmuloc 6546 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)

Theoremmulnqprl 6548 Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 (1st ‘(A ·P B))))

Theoremmulnqpru 6549 Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝑋 (2nd ‘(A ·P B))))

Theoremmullocprlem 6550 Calculations for mullocpr 6551. (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 (1stA) 𝑈 (2ndA)))    &   (φ → (𝐸 Q 𝑇 Q))       (φ → (𝑄 (1st ‘(A ·P B)) 𝑅 (2nd ‘(A ·P B))))

Theoremmullocpr 6551* 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)))))

Theoremmulclpr 6552 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)

Theoremaddcomprg 6553 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))

Theoremaddassprg 6554 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)))

Theoremmulcomprg 6555 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))

Theoremmulassprg 6556 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)))

Theoremdistrlem1prl 6557 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))))

Theoremdistrlem1pru 6558 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))))

Theoremdistrlem4prl 6559* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
(((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))

Theoremdistrlem4pru 6560* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
(((A P B P 𝐶 P) ((x (2ndA) y (2ndB)) (f (2ndA) z (2nd𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (2nd ‘(A ·P (B +P 𝐶))))

Theoremdistrlem5prl 6561 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))))

Theoremdistrlem5pru 6562 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))))

Theoremdistrprg 6563 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)))

Theoremltprordil 6564 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 6565 Lemma for 1idpr 6567. (Contributed by Jim Kingdon, 13-Dec-2019.)
(A P → (1st ‘(A ·P 1P)) = (1stA))

Theorem1idpru 6566 Lemma for 1idpr 6567. (Contributed by Jim Kingdon, 13-Dec-2019.)
(A P → (2nd ‘(A ·P 1P)) = (2ndA))

Theorem1idpr 6567 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)

Theoremltpopr 6568 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 6569. (Contributed by Jim Kingdon, 15-Dec-2019.)
<P Po P

Theoremltsopr 6569 Positive real 'less than' is a weak linear order (in the sense of df-iso 4025). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
<P Or P

Theoremltaddpr 6570 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))

Theoremltexprlemell 6571* Element in lower cut of the constructed difference. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))

Theoremltexprlemelu 6572* Element in upper cut of the constructed difference. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (𝑟 (2nd𝐶) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))

Theoremltexprlemm 6573* Our constructed difference is inhabited. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B → (𝑞 Q 𝑞 (1st𝐶) 𝑟 Q 𝑟 (2nd𝐶)))

Theoremltexprlemopl 6574* The lower cut of our constructed difference is open. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       ((A<P B 𝑞 Q 𝑞 (1st𝐶)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))

Theoremltexprlemlol 6575* The lower cut of our constructed difference is lower. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       ((A<P B 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)) → 𝑞 (1st𝐶)))

Theoremltexprlemopu 6576* The upper cut of our constructed difference is open. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       ((A<P B 𝑟 Q 𝑟 (2nd𝐶)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))

Theoremltexprlemupu 6577* The upper cut of our constructed difference is upper. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       ((A<P B 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)) → 𝑟 (2nd𝐶)))

Theoremltexprlemrnd 6578* Our constructed difference is rounded. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B → (𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))) 𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))))

Theoremltexprlemdisj 6579* Our constructed difference is disjoint. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B𝑞 Q ¬ (𝑞 (1st𝐶) 𝑞 (2nd𝐶)))

Theoremltexprlemloc 6580* Our constructed difference is located. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐶) 𝑟 (2nd𝐶))))

Theoremltexprlempr 6581* Our constructed difference is a positive real. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B𝐶 P)

Theoremltexprlemfl 6582* Lemma for ltexpri 6586. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B → (1st ‘(A +P 𝐶)) ⊆ (1stB))

Theoremltexprlemrl 6583* Lemma for ltexpri 6586. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B → (1stB) ⊆ (1st ‘(A +P 𝐶)))

Theoremltexprlemfu 6584* Lemma for ltexpri 6586. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B → (2nd ‘(A +P 𝐶)) ⊆ (2ndB))

Theoremltexprlemru 6585* Lemma for ltexpri 6586. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩       (A<P B → (2ndB) ⊆ (2nd ‘(A +P 𝐶)))

Theoremltexpri 6586* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
(A<P Bx P (A +P x) = B)

(((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) ⊆ (1st𝐶))

(((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) ⊆ (2nd𝐶))

Theoremaddcanprg 6589 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
((A P B P 𝐶 P) → ((A +P B) = (A +P 𝐶) → B = 𝐶))

Theoremltaprlem 6590 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
(𝐶 P → (A<P B → (𝐶 +P A)<P (𝐶 +P B)))

Theoremltaprg 6591 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
((A P B P 𝐶 P) → (A<P B ↔ (𝐶 +P A)<P (𝐶 +P B)))

Theoremaddextpr 6592 Strong extensionality of addition (ordering version). This is similar to addext 7374 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
(((A P B P) (𝐶 P 𝐷 P)) → ((A +P B)<P (𝐶 +P 𝐷) → (A<P 𝐶 B<P 𝐷)))

Theoremrecexprlemell 6593* Membership in the lower cut of B. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (𝐶 (1stB) ↔ y(𝐶 <Q y (*Qy) (2ndA)))

Theoremrecexprlemelu 6594* Membership in the upper cut of B. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (𝐶 (2ndB) ↔ y(y <Q 𝐶 (*Qy) (1stA)))

Theoremrecexprlemm 6595* B is inhabited. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (𝑞 Q 𝑞 (1stB) 𝑟 Q 𝑟 (2ndB)))

Theoremrecexprlemopl 6596* The lower cut of B is open. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 28-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       ((A P 𝑞 Q 𝑞 (1stB)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))

Theoremrecexprlemlol 6597* The lower cut of B is lower. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 28-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       ((A P 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)) → 𝑞 (1stB)))

Theoremrecexprlemopu 6598* The upper cut of B is open. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 28-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       ((A P 𝑟 Q 𝑟 (2ndB)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))

Theoremrecexprlemupu 6599* The upper cut of B is upper. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 28-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       ((A P 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)) → 𝑟 (2ndB)))

Theoremrecexprlemrnd 6600* B is rounded. Lemma for recexpr 6609. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (𝑞 Q (𝑞 (1stB) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))) 𝑟 Q (𝑟 (2ndB) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2ndB)))))

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