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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgenpmu 6501* 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 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 B P) → 𝑞 Q 𝑞 (2nd ‘(A𝐹B)))
 
Theoremgenpcdl 6502* 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 6503* 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 6504* 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 6505* 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 6506* 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 6507* Associativity of lower cuts. Lemma for genpassg 6509. (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 6508* Associativity of upper cuts. Lemma for genpassg 6509. (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 6509* 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 6510 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 6511 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 6512 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 6513 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 6514 Lemma for addlocpr 6519. 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 6515 Lemma for addlocpr 6519. 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 6516 Lemma for addlocpr 6519. 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 6517 Lemma for addlocpr 6519. 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 6518 Lemma for addlocpr 6519. 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 6519* 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 6520 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 6521* 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 6522* 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 6523 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
dom +P = (P × P)
 
Theoremdmmp 6524 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
dom ·P = (P × P)
 
Theoremnqprm 6525* A cut produced from a rational is inhabited. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → (𝑞 Q 𝑞 {xx <Q A} 𝑟 Q 𝑟 {xA <Q x}))
 
Theoremnqprrnd 6526* A cut produced from a rational is rounded. Lemma for nqprlu 6530. (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 6527* A cut produced from a rational is disjoint. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q𝑞 Q ¬ (𝑞 {xx <Q A} 𝑞 {xA <Q x}))
 
Theoremnqprloc 6528* A cut produced from a rational is located. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 {xx <Q A} 𝑟 {xA <Q x})))
 
Theoremnqprxx 6529* 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 6530* 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 6531 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xx <Q A} V
 
Theoremgtnqex 6532 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xA <Q x} V
 
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}
 
Theoremaddnqpr1lemrl 6537* Lemma for addnqpr1 6541. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 28-Apr-2020.)
(A Q → (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
 
Theoremaddnqpr1lemru 6538* Lemma for addnqpr1 6541. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 28-Apr-2020.)
(A Q → (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
 
Theoremaddnqpr1lemil 6539* Lemma for addnqpr1 6541. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 28-Apr-2020.)
(A Q → (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
 
Theoremaddnqpr1lemiu 6540* Lemma for addnqpr1 6541. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 28-Apr-2020.)
(A Q → (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
 
Theoremaddnqpr1 6541* 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. (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 6542* 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 6543* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6542 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 6544 Calculations for prmuloc 6545. (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 6545* 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 6546* Positive reals are multiplicatively located. This is a variation of prmuloc 6545 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 6547 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 6548 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 6549 Calculations for mullocpr 6550. (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 6550* 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 6551 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 6552 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 6553 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 6554 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 6555 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 6556 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 6557 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 6558* 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 6559* 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 6560 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 6561 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 6562 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 6563 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 6564 Lemma for 1idpr 6566. (Contributed by Jim Kingdon, 13-Dec-2019.)
(A P → (1st ‘(A ·P 1P)) = (1stA))
 
Theorem1idpru 6565 Lemma for 1idpr 6566. (Contributed by Jim Kingdon, 13-Dec-2019.)
(A P → (2nd ‘(A ·P 1P)) = (2ndA))
 
Theorem1idpr 6566 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 6567 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 6568. (Contributed by Jim Kingdon, 15-Dec-2019.)
<P Po P
 
Theoremltsopr 6568 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 6569 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 6570* Element in lower cut of the constructed difference. Lemma for ltexpri 6585. (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 6571* Element in upper cut of the constructed difference. Lemma for ltexpri 6585. (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 6572* Our constructed difference is inhabited. Lemma for ltexpri 6585. (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 6573* The lower cut of our constructed difference is open. Lemma for ltexpri 6585. (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 6574* The lower cut of our constructed difference is lower. Lemma for ltexpri 6585. (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 6575* The upper cut of our constructed difference is open. Lemma for ltexpri 6585. (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 6576* The upper cut of our constructed difference is upper. Lemma for ltexpri 6585. (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 6577* Our constructed difference is rounded. Lemma for ltexpri 6585. (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 6578* Our constructed difference is disjoint. Lemma for ltexpri 6585. (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 6579* Our constructed difference is located. Lemma for ltexpri 6585. (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 6580* Our constructed difference is a positive real. Lemma for ltexpri 6585. (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 6581* Lemma for ltexpri 6585. 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 6582* Lemma for ltexpri 6585. 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 6583* Lemma for ltexpri 6585. 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 6584* Lemma for ltexpri 6585. 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 6585* 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)
 
Theoremaddcanprleml 6586 Lemma for addcanprg 6588. (Contributed by Jim Kingdon, 25-Dec-2019.)
(((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) ⊆ (1st𝐶))
 
Theoremaddcanprlemu 6587 Lemma for addcanprg 6588. (Contributed by Jim Kingdon, 25-Dec-2019.)
(((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) ⊆ (2nd𝐶))
 
Theoremaddcanprg 6588 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 6589 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 6590 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 6591 Strong extensionality of addition (ordering version). This is similar to addext 7354 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 6592* Membership in the lower cut of B. Lemma for recexpr 6608. (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 6593* Membership in the upper cut of B. Lemma for recexpr 6608. (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 6594* B is inhabited. Lemma for recexpr 6608. (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 6595* The lower cut of B is open. Lemma for recexpr 6608. (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 6596* The lower cut of B is lower. Lemma for recexpr 6608. (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 6597* The upper cut of B is open. Lemma for recexpr 6608. (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 6598* The upper cut of B is upper. Lemma for recexpr 6608. (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 6599* B is rounded. Lemma for recexpr 6608. (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)))))
 
Theoremrecexprlemdisj 6600* B is disjoint. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P𝑞 Q ¬ (𝑞 (1stB) 𝑞 (2ndB)))
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