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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecexgt0sr 7801* 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 7802* 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 7803 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 7804 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴R𝐴 <R (𝐴 +R 1R))
 
Theoremltm1sr 7805 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 7806 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 7807 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴R𝐵R ∧ ¬ (𝐴 <R 𝐵𝐵 <R 𝐴)) → 𝐴 = 𝐵)
 
Theoremmulextsr1lem 7808 Lemma for mulextsr1 7809. (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 7809 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 7810* 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 7811* 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 7812 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴P → [⟨(𝐴 +P 1P), 1P⟩] ~RR)
 
Theoremprsrpos 7813 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 7814 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 7815 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 7816* 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 7817* Lemma for caucvgsr 7830. Terms of the sequence from caucvgsrlemgt1 7823 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 7818* Lemma for caucvgsr 7830. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
(𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))       (𝜑𝐴R)
 
Theoremcaucvgsrlemfv 7819* Lemma for caucvgsr 7830. 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 7820* Lemma for caucvgsr 7830. 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 7821* Lemma for caucvgsr 7830. 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 7822* Lemma for caucvgsr 7830. 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 7823* Lemma for caucvgsr 7830. 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 7824* Lemma for caucvgsr 7830. 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 7825* Lemma for caucvgsr 7830. 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 7826* Lemma for caucvgsr 7830. 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 7827* Lemma for caucvgsr 7830. 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 7828* Lemma for caucvgsr 7830. 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 7829* Lemma for caucvgsr 7830. 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 7830* 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 7740 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 7829).

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 7825).

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 7740 to get a limit (see caucvgsrlemgt1 7823).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7823).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7828). (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 7831 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 7832 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 7833* 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 7834* Lemma for suplocsr 7837. 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 7835* Lemma for suplocsr 7837. 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 7836* Lemma for suplocsr 7837. 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 7837* 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 7838 Class of complex numbers.
class
 
Syntaxcr 7839 Class of real numbers.
class
 
Syntaxcc0 7840 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 7841 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 7842 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 7843 Addition on complex numbers.
class +
 
Syntaxcltrr 7844 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 7845 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 7846 Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
ℂ = (R × R)
 
Definitiondf-0 7847 Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
0 = ⟨0R, 0R
 
Definitiondf-1 7848 Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
1 = ⟨1R, 0R
 
Definitiondf-i 7849 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
i = ⟨0R, 1R
 
Definitiondf-r 7850 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
ℝ = (R × {0R})
 
Definitiondf-add 7851* 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 7852* 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 7853* 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 7854 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
(⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
 
Theoremopelreal 7855 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
(⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)
 
Theoremelreal 7856* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
(𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
 
Theoremelrealeu 7857* The real number mapping in elreal 7856 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝐴 ∈ ℝ ↔ ∃!𝑥R𝑥, 0R⟩ = 𝐴)
 
Theoremelreal2 7858 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
(𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
 
Theorem0ncn 7859 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. See also cnm 7860 which is a related property. (Contributed by NM, 2-May-1996.)
¬ ∅ ∈ ℂ
 
Theoremcnm 7860* A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
(𝐴 ∈ ℂ → ∃𝑥 𝑥𝐴)
 
Theoremltrelre 7861 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
< ⊆ (ℝ × ℝ)
 
Theoremaddcnsr 7862 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 7863 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 7864 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 7865 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 7866 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 7867 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 7868 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 7869 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in R. The trick involves qsid 6625, 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 7846), 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 7870 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7869 and mulcnsrec 7871. (Contributed by NM, 13-Aug-1995.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → ([⟨𝐴, 𝐵⟩] E + [⟨𝐶, 𝐷⟩] E ) = [⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩] E )
 
Theoremmulcnsrec 7871 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6624, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7869. (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 7872 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 7965. (Contributed by Jim Kingdon, 14-Jul-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 + 𝐵) ∈ V)
 
Theorempitonnlem1 7873* Lemma for pitonn 7876. 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 7874 Lemma for pitonn 7876. 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 7875* Lemma for pitonn 7876. 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 7876* 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 7877* 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 7878* Embedding from N to . Similar to pitonn 7876 but separate in the sense that we have not proved nnssre 8952 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 7879* 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 7880* One is an element of . This is a counterpart to 1nn 8959 designed for real number axioms which involve natural numbers (notably, axcaucvg 7928). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       1 ∈ 𝑁
 
Theorempeano2nnnn 7881* A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8960 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7928). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}       (𝐴𝑁 → (𝐴 + 1) ∈ 𝑁)
 
Theoremltrennb 7882* 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 7883* 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 7884* 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 7885 Lemma for recidpirq 7886. 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 7886* 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)
 
4.1.2  Final derivation of real and complex number postulates
 
Theoremaxcnex 7887 The complex numbers form a set. Use cnex 7964 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremaxresscn 7888 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 7932. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
ℝ ⊆ ℂ
 
Theoremax1cn 7889 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 7933. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
1 ∈ ℂ
 
Theoremax1re 7890 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 7934.

In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7933 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 7891 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 7935. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
i ∈ ℂ
 
Theoremaxaddcl 7892 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 7936 be used later. Instead, in most cases use addcl 7965. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
 
Theoremaxaddrcl 7893 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 7937 be used later. Instead, in most cases use readdcl 7966. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremaxmulcl 7894 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 7938 be used later. Instead, in most cases use mulcl 7967. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
 
Theoremaxmulrcl 7895 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 7939 be used later. Instead, in most cases use remulcl 7968. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
 
Theoremaxaddf 7896 Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 7892. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 7962. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
+ :(ℂ × ℂ)⟶ℂ
 
Theoremaxmulf 7897 Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 7894. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 7963. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
· :(ℂ × ℂ)⟶ℂ
 
Theoremaxaddcom 7898 Addition commutes. 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-addcom 7940 be used later. Instead, use addcom 8123.

In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremaxmulcom 7899 Multiplication of complex numbers is commutative. 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-mulcom 7941 be used later. Instead, use mulcom 7969. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaxaddass 7900 Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. 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-addass 7942 be used later. Instead, use addass 7970. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
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