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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuplocexpr 7501* An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
 
Definitiondf-enr 7502* 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 7503 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 7504* 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⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
 
Definitiondf-mr 7505* 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⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
 
Definitiondf-ltr 7506* 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⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
 
Definitiondf-0r 7507 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 7508 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 7509 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 7510 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (⟨𝐴, 𝐵⟩ ~R𝐶, 𝐷⟩ ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))
 
Theoremenrer 7511 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 7512 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))
 
Theoremenrex 7513 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)
~R ∈ V
 
Theoremltrelsr 7514 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
<R ⊆ (R × R)
 
Theoremaddcmpblnr 7515 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((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 7516 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((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 7517 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((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 7518* Decomposing signed reals into positive reals. Lemma for addsrpr 7521 and mulsrpr 7522. (Contributed by Jim Kingdon, 30-Dec-2019.)
(((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))))
 
Theoremaddsrmo 7519* 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⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
 
Theoremmulsrmo 7520* 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⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
 
Theoremaddsrpr 7521 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
 
Theoremmulsrpr 7522 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )
 
Theoremltsrprg 7523 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
 
Theoremgt0srpr 7524 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
(0R <R [⟨𝐴, 𝐵⟩] ~R𝐵<P 𝐴)
 
Theorem0nsr 7525 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
¬ ∅ ∈ R
 
Theorem0r 7526 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.)
0RR
 
Theorem1sr 7527 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.)
1RR
 
Theoremm1r 7528 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
-1RR
 
Theoremaddclsr 7529 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)
 
Theoremmulclsr 7530 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) ∈ R)
 
Theoremaddcomsrg 7531 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴))
 
Theoremaddasssrg 7532 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R𝐶R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)))
 
Theoremmulcomsrg 7533 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴))
 
Theoremmulasssrg 7534 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R𝐶R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
 
Theoremdistrsrg 7535 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
 
Theoremm1p1sr 7536 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 7537 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
(-1R ·R -1R) = 1R
 
Theoremlttrsr 7538* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
((𝑓R𝑔RR) → ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R ))
 
Theoremltposr 7539 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
<R Po R
 
Theoremltsosr 7540 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
<R Or R
 
Theorem0lt1sr 7541 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
0R <R 1R
 
Theorem1ne0sr 7542 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
¬ 1R = 0R
 
Theorem0idsr 7543 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
(𝐴R → (𝐴 +R 0R) = 𝐴)
 
Theorem1idsr 7544 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)
 
Theorem00sr 7545 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
(𝐴R → (𝐴 ·R 0R) = 0R)
 
Theoremltasrg 7546 Ordering property of addition. (Contributed by NM, 10-May-1996.)
((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
 
Theorempn0sr 7547 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
(𝐴R → (𝐴 +R (𝐴 ·R -1R)) = 0R)
 
Theoremnegexsr 7548* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
(𝐴R → ∃𝑥R (𝐴 +R 𝑥) = 0R)
 
Theoremrecexgt0sr 7549* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
(0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))
 
Theoremrecexsrlem 7550* 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 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
 
Theoremaddgt0sr 7551 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))
 
Theoremltadd1sr 7552 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴R𝐴 <R (𝐴 +R 1R))
 
Theoremltm1sr 7553 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
(𝐴R → (𝐴 +R -1R) <R 𝐴)
 
Theoremmulgt0sr 7554 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 7555 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴R𝐵R ∧ ¬ (𝐴 <R 𝐵𝐵 <R 𝐴)) → 𝐴 = 𝐵)
 
Theoremmulextsr1lem 7556 Lemma for mulextsr1 7557. (Contributed by Jim Kingdon, 17-Feb-2020.)
(((𝑋P𝑌P) ∧ (𝑍P𝑊P) ∧ (𝑈P𝑉P)) → ((((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 7557 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
((𝐴R𝐵R𝐶R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵𝐵 <R 𝐴)))
 
Theoremarchsr 7558* 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.)
(𝐴R → ∃𝑥N 𝐴 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
 
Theoremsrpospr 7559* Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
 
Theoremprsrcl 7560 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴P → [⟨(𝐴 +P 1P), 1P⟩] ~RR)
 
Theoremprsrpos 7561 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴P → 0R <R [⟨(𝐴 +P 1P), 1P⟩] ~R )
 
Theoremprsradd 7562 Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴P𝐵P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
 
Theoremprsrlt 7563 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
 
Theoremprsrriota 7564* Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴R ∧ 0R <R 𝐴) → [⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)
 
Theoremcaucvgsrlemcl 7565* Lemma for caucvgsr 7578. Terms of the sequence from caucvgsrlemgt1 7571 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))       ((𝜑𝐴N) → (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
 
Theoremcaucvgsrlemasr 7566* Lemma for caucvgsr 7578. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
(𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))       (𝜑𝐴R)
 
Theoremcaucvgsrlemfv 7567* Lemma for caucvgsr 7578. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))
 
Theoremcaucvgsrlemf 7568* Lemma for caucvgsr 7578. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       (𝜑𝐺:NP)
 
Theoremcaucvgsrlemcau 7569* Lemma for caucvgsr 7578. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
 
Theoremcaucvgsrlembound 7570* Lemma for caucvgsr 7578. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))    &   𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))       (𝜑 → ∀𝑚N 1P<P (𝐺𝑚))
 
Theoremcaucvgsrlemgt1 7571* Lemma for caucvgsr 7578. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑖N (𝑗 <N 𝑖 → ((𝐹𝑖) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑖) +R 𝑥)))))
 
Theoremcaucvgsrlemoffval 7572* Lemma for caucvgsr 7578. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       ((𝜑𝐽N) → ((𝐺𝐽) +R 𝐴) = ((𝐹𝐽) +R 1R))
 
Theoremcaucvgsrlemofff 7573* Lemma for caucvgsr 7578. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑𝐺:NR)
 
Theoremcaucvgsrlemoffcau 7574* Lemma for caucvgsr 7578. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
 
Theoremcaucvgsrlemoffgt1 7575* Lemma for caucvgsr 7578. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑 → ∀𝑚N 1R <R (𝐺𝑚))
 
Theoremcaucvgsrlemoffres 7576* Lemma for caucvgsr 7578. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))    &   𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))
 
Theoremcaucvgsrlembnd 7577* Lemma for caucvgsr 7578. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))
 
Theoremcaucvgsr 7578* A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

This is similar to caucvgprpr 7488 but is for signed reals rather than positive reals.

Here is an outline of how we prove it:

1. Choose a lower bound for the sequence (see caucvgsrlembnd 7577).

2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7573).

3. Since a signed real (element of R) which is greater than zero can be mapped to a positive real (element of P), perform that mapping on each element of the sequence and invoke caucvgprpr 7488 to get a limit (see caucvgsrlemgt1 7571).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7571).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7576). (Contributed by Jim Kingdon, 20-Jun-2021.)

(𝜑𝐹:NR)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))       (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))
 
Theoremltpsrprg 7579 Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
((𝐴P𝐵P𝐶R) → ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵))
 
Theoremmappsrprg 7580 Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
((𝐴P𝐶R) → (𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ))
 
Theoremmap2psrprg 7581* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
(𝐶R → ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
 
Theoremsuplocsrlemb 7582* Lemma for suplocsr 7585. The set 𝐵 is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
𝐵 = {𝑤P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑𝐴R)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∀𝑢P𝑣P (𝑢<P 𝑣 → (∃𝑞𝐵 𝑢<P 𝑞 ∨ ∀𝑞𝐵 𝑞<P 𝑣)))
 
Theoremsuplocsrlempr 7583* Lemma for suplocsr 7585. The set 𝐵 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
𝐵 = {𝑤P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑𝐴R)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∃𝑣P (∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)))
 
Theoremsuplocsrlem 7584* Lemma for suplocsr 7585. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
𝐵 = {𝑤P ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   (𝜑𝐴R)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 
Theoremsuplocsr 7585* An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)    &   (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))       (𝜑 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 
Syntaxcc 7586 Class of complex numbers.
class
 
Syntaxcr 7587 Class of real numbers.
class
 
Syntaxcc0 7588 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 7589 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 7590 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 7591 Addition on complex numbers.
class +
 
Syntaxcltrr 7592 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 7593 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 7594 Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
ℂ = (R × R)
 
Definitiondf-0 7595 Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
0 = ⟨0R, 0R
 
Definitiondf-1 7596 Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
1 = ⟨1R, 0R
 
Definitiondf-i 7597 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
i = ⟨0R, 1R
 
Definitiondf-r 7598 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
ℝ = (R × {0R})
 
Definitiondf-add 7599* Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
+ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
 
Definitiondf-mul 7600* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
· = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 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))⟩))}
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