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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnqprlu 6401* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
(A Q → ⟨{xx <Q A}, {xA <Q x}⟩ P)

Theoremltnqex 6402 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xx <Q A} V

Theoremgtnqex 6403 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{xA <Q x} V

Theorem1pr 6404 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
1P P

Theorem1prl 6405 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(1st ‘1P) = {xx <Q 1Q}

Theorem1pru 6406 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(2nd ‘1P) = {x ∣ 1Q <Q x}

Theoremappdivnq 6407* 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 6408* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6407 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 6409 Calculations for prmuloc 6410. (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 6410* 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 6411* Positive reals are multiplicatively located. This is a variation of prmuloc 6410 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 6412 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 6413 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 6414 Calculations for mullocpr 6415. (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 6415* 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 6416 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 6417 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 6418 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 6419 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 6420 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 6421 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 6422 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 6423* 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 6424* 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 6425 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 6426 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 6427 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 6428 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 6429 Lemma for 1idpr 6431. (Contributed by Jim Kingdon, 13-Dec-2019.)
(A P → (1st ‘(A ·P 1P)) = (1stA))

Theorem1idpru 6430 Lemma for 1idpr 6431. (Contributed by Jim Kingdon, 13-Dec-2019.)
(A P → (2nd ‘(A ·P 1P)) = (2ndA))

Theorem1idpr 6431 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 6432 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 6433. (Contributed by Jim Kingdon, 15-Dec-2019.)
<P Po P

Theoremltsopr 6433 Positive real 'less than' is a weak linear order (in the sense of df-iso 4008). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
<P Or P

Theoremltaddpr 6434 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 6435* Element in lower cut of the constructed difference. Lemma for ltexpri 6450. (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 6436* Element in upper cut of the constructed difference. Lemma for ltexpri 6450. (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 6437* Our constructed difference is inhabited. Lemma for ltexpri 6450. (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 6438* The lower cut of our constructed difference is open. Lemma for ltexpri 6450. (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 6439* The lower cut of our constructed difference is lower. Lemma for ltexpri 6450. (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 6440* The upper cut of our constructed difference is open. Lemma for ltexpri 6450. (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 6441* The upper cut of our constructed difference is upper. Lemma for ltexpri 6450. (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 6442* Our constructed difference is rounded. Lemma for ltexpri 6450. (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 6443* Our constructed difference is disjoint. Lemma for ltexpri 6450. (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 6444* Our constructed difference is located. Lemma for ltexpri 6450. (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 6445* Our constructed difference is a positive real. Lemma for ltexpri 6450. (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 6446* Lemma for ltexpri 6450. 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 6447* Lemma for ltexpri 6450. 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 6448* Lemma for ltexpri 6450. 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 6449* Lemma for ltexpri 6450. 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 6450* 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 6453 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 6454 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 6455 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. Part of Definition 11.2.7(vi) of [HoTT]], p. (varies). (Contributed by Jim Kingdon, 26-Dec-2019.)
((A P B P 𝐶 P) → (A<P B ↔ (𝐶 +P A)<P (𝐶 +P B)))

Theoremrecexprlemell 6456* Membership in the lower cut of B. Lemma for recexpr 6472. (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 6457* Membership in the upper cut of B. Lemma for recexpr 6472. (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 6458* B is inhabited. Lemma for recexpr 6472. (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 6459* The lower cut of B is open. Lemma for recexpr 6472. (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 6460* The lower cut of B is lower. Lemma for recexpr 6472. (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 6461* The upper cut of B is open. Lemma for recexpr 6472. (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 6462* The upper cut of B is upper. Lemma for recexpr 6472. (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 6463* B is rounded. Lemma for recexpr 6472. (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 6464* B is disjoint. Lemma for recexpr 6472. (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)))

Theoremrecexprlemloc 6465* B is located. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1stB) 𝑟 (2ndB))))

Theoremrecexprlempr 6466* B is a positive real. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A PB P)

Theoremrecexprlem1ssl 6467* The lower cut of one is a subset of the lower cut of A ·P B. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (1st ‘1P) ⊆ (1st ‘(A ·P B)))

Theoremrecexprlem1ssu 6468* The upper cut of one is a subset of the upper cut of A ·P B. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (2nd ‘1P) ⊆ (2nd ‘(A ·P B)))

Theoremrecexprlemss1l 6469* The lower cut of A ·P B is a subset of the lower cut of one. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (1st ‘(A ·P B)) ⊆ (1st ‘1P))

Theoremrecexprlemss1u 6470* The upper cut of A ·P B is a subset of the upper cut of one. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (2nd ‘(A ·P B)) ⊆ (2nd ‘1P))

Theoremrecexprlemex 6471* B is the reciprocal of A. Lemma for recexpr 6472. (Contributed by Jim Kingdon, 27-Dec-2019.)
B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩       (A P → (A ·P B) = 1P)

Theoremrecexpr 6472* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
(A Px P (A ·P x) = 1P)

Definitiondf-enr 6473* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
~R = {⟨x, y⟩ ∣ ((x (P × P) y (P × P)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v)))}

Definitiondf-nr 6474 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
R = ((P × P) / ~R )

Definitiondf-plr 6475* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
+R = {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨(w +P u), (v +P f)⟩] ~R ))}

Definitiondf-mr 6476* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
·R = {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvuf((x = [⟨w, v⟩] ~R y = [⟨u, f⟩] ~R ) z = [⟨((w ·P u) +P (v ·P f)), ((w ·P f) +P (v ·P u))⟩] ~R ))}

Definitiondf-ltr 6477* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.)
<R = {⟨x, y⟩ ∣ ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v)))}

Definitiondf-0r 6478 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)
0R = [⟨1P, 1P⟩] ~R

Definitiondf-1r 6479 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)
1R = [⟨(1P +P 1P), 1P⟩] ~R

Definitiondf-m1r 6480 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.)
-1R = [⟨1P, (1P +P 1P)⟩] ~R

Theoremenrbreq 6481 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)
(((A P B P) (𝐶 P 𝐷 P)) → (⟨A, B⟩ ~R𝐶, 𝐷⟩ ↔ (A +P 𝐷) = (B +P 𝐶)))

Theoremenrer 6482 The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
~R Er (P × P)

Theoremenreceq 6483 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
(((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (A +P 𝐷) = (B +P 𝐶)))

Theoremenrex 6484 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)
~R V

Theoremltrelsr 6485 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
<R ⊆ (R × R)

((((A P B P) (𝐶 P 𝐷 P)) ((𝐹 P 𝐺 P) (𝑅 P 𝑆 P))) → (((A +P 𝐷) = (B +P 𝐶) (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨(A +P 𝐹), (B +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))

Theoremmulcmpblnrlemg 6487 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
((((A P B P) (𝐶 P 𝐷 P)) ((𝐹 P 𝐺 P) (𝑅 P 𝑆 P))) → (((A +P 𝐷) = (B +P 𝐶) (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((A ·P 𝐹) +P (B ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((A ·P 𝐺) +P (B ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))))

Theoremmulcmpblnr 6488 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
((((A P B P) (𝐶 P 𝐷 P)) ((𝐹 P 𝐺 P) (𝑅 P 𝑆 P))) → (((A +P 𝐷) = (B +P 𝐶) (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨((A ·P 𝐹) +P (B ·P 𝐺)), ((A ·P 𝐺) +P (B ·P 𝐹))⟩ ~R ⟨((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)), ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))⟩))

Theoremprsrlem1 6489* Decomposing signed reals into positive reals. Lemma for addsrpr 6492 and mulsrpr 6493. (Contributed by Jim Kingdon, 30-Dec-2019.)
(((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((((w P v P) (𝑠 P f P)) ((u P 𝑡 P) (g P P))) ((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g))))

Theoremaddsrmo 6490* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((A ((P × P) / ~R ) B ((P × P) / ~R )) → ∃*zwvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))

Theoremmulsrmo 6491* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((A ((P × P) / ~R ) B ((P × P) / ~R )) → ∃*zwvu𝑡((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) z = [⟨((w ·P u) +P (v ·P 𝑡)), ((w ·P 𝑡) +P (v ·P u))⟩] ~R ))

Theoremaddsrpr 6492 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )

Theoremmulsrpr 6493 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((A ·P 𝐶) +P (B ·P 𝐷)), ((A ·P 𝐷) +P (B ·P 𝐶))⟩] ~R )

Theoremltsrprg 6494 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
(((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (A +P 𝐷)<P (B +P 𝐶)))

Theoremgt0srpr 6495 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
(0R <R [⟨A, B⟩] ~RB<P A)

Theorem0nsr 6496 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
¬ ∅ R

Theorem0r 6497 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.)
0R R

Theorem1sr 6498 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.)
1R R

Theoremm1r 6499 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
-1R R

Theoremaddclsr 6500 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
((A R B R) → (A +R B) R)

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