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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecexprlemloc 6601* B is located. 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 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1stB) 𝑟 (2ndB))))
 
Theoremrecexprlempr 6602* B is a positive real. 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 PB P)
 
Theoremrecexprlem1ssl 6603* The lower cut of one is a subset of the lower cut of A ·P 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))}⟩       (A P → (1st ‘1P) ⊆ (1st ‘(A ·P B)))
 
Theoremrecexprlem1ssu 6604* The upper cut of one is a subset of the upper cut of A ·P 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))}⟩       (A P → (2nd ‘1P) ⊆ (2nd ‘(A ·P B)))
 
Theoremrecexprlemss1l 6605* The lower cut of A ·P B is a subset of the lower cut of one. 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 → (1st ‘(A ·P B)) ⊆ (1st ‘1P))
 
Theoremrecexprlemss1u 6606* The upper cut of A ·P B is a subset of the upper cut of one. 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 → (2nd ‘(A ·P B)) ⊆ (2nd ‘1P))
 
Theoremrecexprlemex 6607* B is the reciprocal of A. 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 → (A ·P B) = 1P)
 
Theoremrecexpr 6608* 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)
 
Theoremaptiprleml 6609 Lemma for aptipr 6611. (Contributed by Jim Kingdon, 28-Jan-2020.)
((A P B P ¬ B<P A) → (1stA) ⊆ (1stB))
 
Theoremaptiprlemu 6610 Lemma for aptipr 6611. (Contributed by Jim Kingdon, 28-Jan-2020.)
((A P B P ¬ B<P A) → (2ndB) ⊆ (2ndA))
 
Theoremaptipr 6611 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
((A P B P ¬ (A<P B B<P A)) → A = B)
 
Theoremltmprr 6612 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
((A P B P 𝐶 P) → ((𝐶 ·P A)<P (𝐶 ·P B) → A<P B))
 
Theoremarchpr 6613* For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer x is embedded into the reals as described at nnprlu 6533. (Contributed by Jim Kingdon, 22-Apr-2020.)
(A Px N A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
 
Definitiondf-enr 6614* 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 6615 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 6616* 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 6617* 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 6618* 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 6619 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 6620 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 6621 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 6622 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 6623 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 6624 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 6625 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)
~R V
 
Theoremltrelsr 6626 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
<R ⊆ (R × R)
 
Theoremaddcmpblnr 6627 Lemma showing compatibility of addition. (Contributed by NM, 3-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), (B +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))
 
Theoremmulcmpblnrlemg 6628 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 6629 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 6630* Decomposing signed reals into positive reals. Lemma for addsrpr 6633 and mulsrpr 6634. (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 6631* 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 6632* 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 6633 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 6634 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 6635 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 6636 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
(0R <R [⟨A, B⟩] ~RB<P A)
 
Theorem0nsr 6637 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
¬ ∅ R
 
Theorem0r 6638 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.)
0R R
 
Theorem1sr 6639 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.)
1R R
 
Theoremm1r 6640 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
-1R R
 
Theoremaddclsr 6641 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
((A R B R) → (A +R B) R)
 
Theoremmulclsr 6642 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
((A R B R) → (A ·R B) R)
 
Theoremaddcomsrg 6643 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((A R B R) → (A +R B) = (B +R A))
 
Theoremaddasssrg 6644 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((A R B R 𝐶 R) → ((A +R B) +R 𝐶) = (A +R (B +R 𝐶)))
 
Theoremmulcomsrg 6645 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((A R B R) → (A ·R B) = (B ·R A))
 
Theoremmulasssrg 6646 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((A R B R 𝐶 R) → ((A ·R B) ·R 𝐶) = (A ·R (B ·R 𝐶)))
 
Theoremdistrsrg 6647 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
((A R B R 𝐶 R) → (A ·R (B +R 𝐶)) = ((A ·R B) +R (A ·R 𝐶)))
 
Theoremm1p1sr 6648 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 6649 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
(-1R ·R -1R) = 1R
 
Theoremlttrsr 6650* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
((f R g R R) → ((f <R g g <R ) → f <R ))
 
Theoremltposr 6651 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
<R Po R
 
Theoremltsosr 6652 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
<R Or R
 
Theorem0lt1sr 6653 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
0R <R 1R
 
Theorem1ne0sr 6654 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
¬ 1R = 0R
 
Theorem0idsr 6655 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
(A R → (A +R 0R) = A)
 
Theorem1idsr 6656 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
(A R → (A ·R 1R) = A)
 
Theorem00sr 6657 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
(A R → (A ·R 0R) = 0R)
 
Theoremltasrg 6658 Ordering property of addition. (Contributed by NM, 10-May-1996.)
((A R B R 𝐶 R) → (A <R B ↔ (𝐶 +R A) <R (𝐶 +R B)))
 
Theorempn0sr 6659 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
(A R → (A +R (A ·R -1R)) = 0R)
 
Theoremnegexsr 6660* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
(A Rx R (A +R x) = 0R)
 
Theoremrecexgt0sr 6661* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
(0R <R Ax R (0R <R x (A ·R x) = 1R))
 
Theoremrecexsrlem 6662* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.)
(0R <R Ax R (A ·R x) = 1R)
 
Theoremaddgt0sr 6663 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)
((0R <R A 0R <R B) → 0R <R (A +R B))
 
Theoremmulgt0sr 6664 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
((0R <R A 0R <R B) → 0R <R (A ·R B))
 
Theoremaptisr 6665 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
((A R B R ¬ (A <R B B <R A)) → A = B)
 
Theoremmulextsr1lem 6666 Lemma for mulextsr1 6667. (Contributed by Jim Kingdon, 17-Feb-2020.)
(((𝑋 P 𝑌 P) (𝑍 P 𝑊 P) (𝑈 P 𝑉 P)) → ((((𝑋 ·P 𝑈) +P (𝑌 ·P 𝑉)) +P ((𝑍 ·P 𝑉) +P (𝑊 ·P 𝑈)))<P (((𝑋 ·P 𝑉) +P (𝑌 ·P 𝑈)) +P ((𝑍 ·P 𝑈) +P (𝑊 ·P 𝑉))) → ((𝑋 +P 𝑊)<P (𝑌 +P 𝑍) (𝑍 +P 𝑌)<P (𝑊 +P 𝑋))))
 
Theoremmulextsr1 6667 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
((A R B R 𝐶 R) → ((A ·R 𝐶) <R (B ·R 𝐶) → (A <R B B <R A)))
 
Theoremarchsr 6668* For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R is the embedding of the positive integer x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
(A Rx N A <R [⟨(⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ +P 1P), 1P⟩] ~R )
 
Syntaxcc 6669 Class of complex numbers.
class
 
Syntaxcr 6670 Class of real numbers.
class
 
Syntaxcc0 6671 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 6672 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 6673 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 6674 Addition on complex numbers.
class +
 
Syntaxcltrr 6675 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 6676 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 6677 Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
ℂ = (R × R)
 
Definitiondf-0 6678 Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
0 = ⟨0R, 0R
 
Definitiondf-1 6679 Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
1 = ⟨1R, 0R
 
Definitiondf-i 6680 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
i = ⟨0R, 1R
 
Definitiondf-r 6681 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
ℝ = (R × {0R})
 
Definitiondf-add 6682* Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
+ = {⟨⟨x, y⟩, z⟩ ∣ ((x y ℂ) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩))}
 
Definitiondf-mul 6683* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
· = {⟨⟨x, y⟩, z⟩ ∣ ((x y ℂ) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·R u) +R (-1R ·R (v ·R f))), ((v ·R u) +R (w ·R f))⟩))}
 
Definitiondf-lt 6684* Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
< = {⟨x, y⟩ ∣ ((x y ℝ) zw((x = ⟨z, 0R y = ⟨w, 0R⟩) z <R w))}
 
Theoremopelcn 6685 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
(⟨A, B ℂ ↔ (A R B R))
 
Theoremopelreal 6686 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
(⟨A, 0R ℝ ↔ A R)
 
Theoremelreal 6687* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
(A ℝ ↔ x Rx, 0R⟩ = A)
 
Theoremelreal2 6688 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
(A ℝ ↔ ((1stA) R A = ⟨(1stA), 0R⟩))
 
Theorem0ncn 6689 The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.)
¬ ∅
 
Theoremltrelre 6690 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
< ⊆ (ℝ × ℝ)
 
Theoremaddcnsr 6691 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.)
(((A R B R) (𝐶 R 𝐷 R)) → (⟨A, B⟩ + ⟨𝐶, 𝐷⟩) = ⟨(A +R 𝐶), (B +R 𝐷)⟩)
 
Theoremmulcnsr 6692 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.)
(((A R B R) (𝐶 R 𝐷 R)) → (⟨A, B⟩ · ⟨𝐶, 𝐷⟩) = ⟨((A ·R 𝐶) +R (-1R ·R (B ·R 𝐷))), ((B ·R 𝐶) +R (A ·R 𝐷))⟩)
 
Theoremeqresr 6693 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
A V       (⟨A, 0R⟩ = ⟨B, 0R⟩ ↔ A = B)
 
Theoremaddresr 6694 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
((A R B R) → (⟨A, 0R⟩ + ⟨B, 0R⟩) = ⟨(A +R B), 0R⟩)
 
Theoremmulresr 6695 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
((A R B R) → (⟨A, 0R⟩ · ⟨B, 0R⟩) = ⟨(A ·R B), 0R⟩)
 
Theoremltresr 6696 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
(⟨A, 0R⟩ <B, 0R⟩ ↔ A <R B)
 
Theoremltresr2 6697 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
((A B ℝ) → (A < B ↔ (1stA) <R (1stB)))
 
Theoremdfcnqs 6698 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 6107, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that is a quotient set, even though it is not (compare df-c 6677), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.)
ℂ = ((R × R) / E )
 
Theoremaddcnsrec 6699 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 6698 and mulcnsrec 6700. (Contributed by NM, 13-Aug-1995.)
(((A R B R) (𝐶 R 𝐷 R)) → ([⟨A, B⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(A +R 𝐶), (B +R 𝐷)⟩] E )
 
Theoremmulcnsrec 6700 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6106, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 6698. (Contributed by NM, 13-Aug-1995.)
(((A R B R) (𝐶 R 𝐷 R)) → ([⟨A, B⟩] E · [⟨𝐶, 𝐷⟩] E ) = [⟨((A ·R 𝐶) +R (-1R ·R (B ·R 𝐷))), ((B ·R 𝐶) +R (A ·R 𝐷))⟩] E )
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