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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdistrsrg 6901 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 6902 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 6903 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
(-1R ·R -1R) = 1R
 
Theoremlttrsr 6904* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
((𝑓R𝑔RR) → ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R ))
 
Theoremltposr 6905 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
<R Po R
 
Theoremltsosr 6906 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
<R Or R
 
Theorem0lt1sr 6907 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
0R <R 1R
 
Theorem1ne0sr 6908 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
¬ 1R = 0R
 
Theorem0idsr 6909 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 6910 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)
 
Theorem00sr 6911 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
(𝐴R → (𝐴 ·R 0R) = 0R)
 
Theoremltasrg 6912 Ordering property of addition. (Contributed by NM, 10-May-1996.)
((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
 
Theorempn0sr 6913 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
(𝐴R → (𝐴 +R (𝐴 ·R -1R)) = 0R)
 
Theoremnegexsr 6914* 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 6915* 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 6916* 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 6917 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 6918 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴R𝐴 <R (𝐴 +R 1R))
 
Theoremmulgt0sr 6919 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 6920 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴R𝐵R ∧ ¬ (𝐴 <R 𝐵𝐵 <R 𝐴)) → 𝐴 = 𝐵)
 
Theoremmulextsr1lem 6921 Lemma for mulextsr1 6922. (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 6922 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 6923* 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 6924* 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 6925 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴P → [⟨(𝐴 +P 1P), 1P⟩] ~RR)
 
Theoremprsrpos 6926 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 6927 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 6928 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 6929* 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 6930* Lemma for caucvgsr 6943. Terms of the sequence from caucvgsrlemgt1 6936 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 6931* Lemma for caucvgsr 6943. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
(𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))       (𝜑𝐴R)
 
Theoremcaucvgsrlemfv 6932* Lemma for caucvgsr 6943. 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 6933* Lemma for caucvgsr 6943. 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 6934* Lemma for caucvgsr 6943. 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 6935* Lemma for caucvgsr 6943. 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 6936* Lemma for caucvgsr 6943. 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 6937* Lemma for caucvgsr 6943. 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 6938* Lemma for caucvgsr 6943. 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 6939* Lemma for caucvgsr 6943. 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 6940* Lemma for caucvgsr 6943. 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 6941* Lemma for caucvgsr 6943. 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 6942* Lemma for caucvgsr 6943. 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 6943* 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 6867 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 6942).

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 6938).

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 6867 to get a limit (see caucvgsrlemgt1 6936).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6936).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6941). (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)))))
 
Syntaxcc 6944 Class of complex numbers.
class
 
Syntaxcr 6945 Class of real numbers.
class
 
Syntaxcc0 6946 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 6947 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 6948 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 6949 Addition on complex numbers.
class +
 
Syntaxcltrr 6950 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 6951 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 6952 Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
ℂ = (R × R)
 
Definitiondf-0 6953 Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
0 = ⟨0R, 0R
 
Definitiondf-1 6954 Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
1 = ⟨1R, 0R
 
Definitiondf-i 6955 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
i = ⟨0R, 1R
 
Definitiondf-r 6956 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
ℝ = (R × {0R})
 
Definitiondf-add 6957* 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 6958* 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))⟩))}
 
Definitiondf-lt 6959* Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
< = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
 
Theoremopelcn 6960 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
(⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
 
Theoremopelreal 6961 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
(⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)
 
Theoremelreal 6962* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
(𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
 
Theoremelrealeu 6963* The real number mapping in elreal 6962 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝐴 ∈ ℝ ↔ ∃!𝑥R𝑥, 0R⟩ = 𝐴)
 
Theoremelreal2 6964 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
(𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
 
Theorem0ncn 6965 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.)
¬ ∅ ∈ ℂ
 
Theoremltrelre 6966 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
< ⊆ (ℝ × ℝ)
 
Theoremaddcnsr 6967 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
 
Theoremmulcnsr 6968 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩)
 
Theoremeqresr 6969 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 6970 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
((𝐴R𝐵R) → (⟨𝐴, 0R⟩ + ⟨𝐵, 0R⟩) = ⟨(𝐴 +R 𝐵), 0R⟩)
 
Theoremmulresr 6971 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
((𝐴R𝐵R) → (⟨𝐴, 0R⟩ · ⟨𝐵, 0R⟩) = ⟨(𝐴 ·R 𝐵), 0R⟩)
 
Theoremltresr 6972 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 6973 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (1st𝐴) <R (1st𝐵)))
 
Theoremdfcnqs 6974 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 6201, 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 6952), 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 6975 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 6974 and mulcnsrec 6976. (Contributed by NM, 13-Aug-1995.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E )
 
Theoremmulcnsrec 6976 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6200, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 6974. (Contributed by NM, 13-Aug-1995.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E · [⟨𝐶, 𝐷⟩] E ) = [⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩] E )
 
Theoremaddvalex 6977 Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7063. (Contributed by Jim Kingdon, 14-Jul-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 + 𝐵) ∈ V)
 
Theorempitonnlem1 6978* Lemma for pitonn 6981. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
⟨[⟨(⟨{𝑙𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
 
Theorempitonnlem1p1 6979 Lemma for pitonn 6981. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴P → [⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R )
 
Theorempitonnlem2 6980* Lemma for pitonn 6981. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
(𝐾N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
Theorempitonn 6981* Mapping from N to . (Contributed by Jim Kingdon, 22-Apr-2020.)
(𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
 
Theorempitoregt0 6982* Embedding from N to yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → 0 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
Theorempitore 6983* Embedding from N to . Similar to pitonn 6981 but separate in the sense that we have not proved nnssre 7993 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
 
Theoremrecnnre 6984* Embedding the reciprocal of a natural number into . (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
 
Theorempeano1nnnn 6985* One is an element of . This is a counterpart to 1nn 8000 designed for real number axioms which involve natural numbers (notably, axcaucvg 7031). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       1 ∈ 𝑁
 
Theorempeano2nnnn 6986* A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8001 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7031). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       (𝐴𝑁 → (𝐴 + 1) ∈ 𝑁)
 
Theoremltrennb 6987* Ordering of natural numbers with <N or <. (Contributed by Jim Kingdon, 13-Jul-2021.)
((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
 
Theoremltrenn 6988* Ordering of natural numbers with <N or <. (Contributed by Jim Kingdon, 12-Jul-2021.)
(𝐽 <N 𝐾 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
Theoremrecidpipr 6989* Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
(𝑁N → (⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = 1P)
 
Theoremrecidpirqlemcalc 6990 Lemma for recidpirq 6991. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
(𝜑𝐴P)    &   (𝜑𝐵P)    &   (𝜑 → (𝐴 ·P 𝐵) = 1P)       (𝜑 → ((((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) +P 1P) = ((((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) +P (1P +P 1P)))
 
Theoremrecidpirq 6991* A real number times its reciprocal is one, where reciprocal is expressed with *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = 1)
 
3.1.2  Final derivation of real and complex number postulates
 
Theoremaxcnex 6992 The complex numbers form a set. Use cnex 7062 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremaxresscn 6993 The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 7033. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
ℝ ⊆ ℂ
 
Theoremax1cn 6994 1 is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 7034. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
1 ∈ ℂ
 
Theoremax1re 6995 1 is a real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1re 7035.

In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7034 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)

1 ∈ ℝ
 
Theoremaxicn 6996 i is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 7036. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
i ∈ ℂ
 
Theoremaxaddcl 6997 Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 7037 be used later. Instead, in most cases use addcl 7063. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremaxaddrcl 6998 Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 7038 be used later. Instead, in most cases use readdcl 7064. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremaxmulcl 6999 Closure law for multiplication of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 7039 be used later. Instead, in most cases use mulcl 7065. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremaxmulrcl 7000 Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 7040 be used later. Instead, in most cases use remulcl 7066. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
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