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Type | Label | Description |
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Statement | ||
Theorem | recexgt0sr 7801* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
⊢ (0R <R 𝐴 → ∃𝑥 ∈ R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)) | ||
Theorem | recexsrlem 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 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) | ||
Theorem | addgt0sr 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)) | ||
Theorem | ltadd1sr 7804 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
⊢ (𝐴 ∈ R → 𝐴 <R (𝐴 +R 1R)) | ||
Theorem | ltm1sr 7805 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
⊢ (𝐴 ∈ R → (𝐴 +R -1R) <R 𝐴) | ||
Theorem | mulgt0sr 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)) | ||
Theorem | aptisr 7807 | 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) | ||
Theorem | mulextsr1lem 7808 | Lemma for mulextsr1 7809. (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)))) | ||
Theorem | mulextsr1 7809 | 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))) | ||
Theorem | archsr 7810* | For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [〈(〈{𝑙 ∣ 𝑙 <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 ∈ R → ∃𝑥 ∈ N 𝐴 <R [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑥, 1o〉] ~Q }, {𝑢 ∣ [〈𝑥, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R ) | ||
Theorem | srpospr 7811* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) | ||
Theorem | prsrcl 7812 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
⊢ (𝐴 ∈ P → [〈(𝐴 +P 1P), 1P〉] ~R ∈ R) | ||
Theorem | prsrpos 7813 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
⊢ (𝐴 ∈ P → 0R <R [〈(𝐴 +P 1P), 1P〉] ~R ) | ||
Theorem | prsradd 7814 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [〈((𝐴 +P 𝐵) +P 1P), 1P〉] ~R = ([〈(𝐴 +P 1P), 1P〉] ~R +R [〈(𝐵 +P 1P), 1P〉] ~R )) | ||
Theorem | prsrlt 7815 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ [〈(𝐴 +P 1P), 1P〉] ~R <R [〈(𝐵 +P 1P), 1P〉] ~R )) | ||
Theorem | prsrriota 7816* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉] ~R = 𝐴) | ||
Theorem | caucvgsrlemcl 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.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [〈(𝑦 +P 1P), 1P〉] ~R ) ∈ P) | ||
Theorem | caucvgsrlemasr 7818* | Lemma for caucvgsr 7830. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → 𝐴 ∈ R) | ||
Theorem | caucvgsrlemfv 7819* | Lemma for caucvgsr 7830. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [〈((𝐺‘𝐴) +P 1P), 1P〉] ~R = (𝐹‘𝐴)) | ||
Theorem | caucvgsrlemf 7820* | Lemma for caucvgsr 7830. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) ⇒ ⊢ (𝜑 → 𝐺:N⟶P) | ||
Theorem | caucvgsrlemcau 7821* | Lemma for caucvgsr 7830. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛)<P ((𝐺‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐺‘𝑘)<P ((𝐺‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉)))) | ||
Theorem | caucvgsrlembound 7822* | Lemma for caucvgsr 7830. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [〈(𝑦 +P 1P), 1P〉] ~R )) ⇒ ⊢ (𝜑 → ∀𝑚 ∈ N 1P<P (𝐺‘𝑚)) | ||
Theorem | caucvgsrlemgt1 7823* | Lemma for caucvgsr 7830. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑖 ∈ N (𝑗 <N 𝑖 → ((𝐹‘𝑖) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑖) +R 𝑥))))) | ||
Theorem | caucvgsrlemoffval 7824* | Lemma for caucvgsr 7830. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +R 𝐴) = ((𝐹‘𝐽) +R 1R)) | ||
Theorem | caucvgsrlemofff 7825* | Lemma for caucvgsr 7830. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) ⇒ ⊢ (𝜑 → 𝐺:N⟶R) | ||
Theorem | caucvgsrlemoffcau 7826* | Lemma for caucvgsr 7830. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) | ||
Theorem | caucvgsrlemoffgt1 7827* | Lemma for caucvgsr 7830. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) ⇒ ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐺‘𝑚)) | ||
Theorem | caucvgsrlemoffres 7828* | Lemma for caucvgsr 7830. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥))))) | ||
Theorem | caucvgsrlembnd 7829* | Lemma for caucvgsr 7830. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥))))) | ||
Theorem | caucvgsr 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.) |
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R )))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥))))) | ||
Theorem | ltpsrprg 7831 | Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵)) | ||
Theorem | mappsrprg 7832 | Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ R) → (𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R )) | ||
Theorem | map2psrprg 7833* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
⊢ (𝐶 ∈ R → ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [〈𝑥, 1P〉] ~R ) = 𝐴)) | ||
Theorem | suplocsrlemb 7834* | Lemma for suplocsr 7837. The set 𝐵 is located. (Contributed by Jim Kingdon, 18-Jan-2024.) |
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [〈𝑤, 1P〉] ~R ) ∈ 𝐴} & ⊢ (𝜑 → 𝐴 ⊆ R) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) & ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) ⇒ ⊢ (𝜑 → ∀𝑢 ∈ P ∀𝑣 ∈ P (𝑢<P 𝑣 → (∃𝑞 ∈ 𝐵 𝑢<P 𝑞 ∨ ∀𝑞 ∈ 𝐵 𝑞<P 𝑣))) | ||
Theorem | suplocsrlempr 7835* | Lemma for suplocsr 7837. The set 𝐵 has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.) |
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [〈𝑤, 1P〉] ~R ) ∈ 𝐴} & ⊢ (𝜑 → 𝐴 ⊆ R) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) & ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢))) | ||
Theorem | suplocsrlem 7836* | Lemma for suplocsr 7837. The set 𝐴 has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.) |
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R [〈𝑤, 1P〉] ~R ) ∈ 𝐴} & ⊢ (𝜑 → 𝐴 ⊆ R) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) & ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) | ||
Theorem | suplocsr 7837* | An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.) |
⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) & ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) | ||
Syntax | cc 7838 | Class of complex numbers. |
class ℂ | ||
Syntax | cr 7839 | Class of real numbers. |
class ℝ | ||
Syntax | cc0 7840 | Extend class notation to include the complex number 0. |
class 0 | ||
Syntax | c1 7841 | Extend class notation to include the complex number 1. |
class 1 | ||
Syntax | ci 7842 | Extend class notation to include the complex number i. |
class i | ||
Syntax | caddc 7843 | Addition on complex numbers. |
class + | ||
Syntax | cltrr 7844 | 'Less than' predicate (defined over real subset of complex numbers). |
class <ℝ | ||
Syntax | cmul 7845 | Multiplication on complex numbers. The token · is a center dot. |
class · | ||
Definition | df-c 7846 | Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ ℂ = (R × R) | ||
Definition | df-0 7847 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) |
⊢ 0 = 〈0R, 0R〉 | ||
Definition | df-1 7848 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) |
⊢ 1 = 〈1R, 0R〉 | ||
Definition | df-i 7849 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) |
⊢ i = 〈0R, 1R〉 | ||
Definition | df-r 7850 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ ℝ = (R × {0R}) | ||
Definition | df-add 7851* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) |
⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} | ||
Definition | df-mul 7852* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) |
⊢ · = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))〉))} | ||
Definition | df-lt 7853* | Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | ||
Theorem | opelcn 7854 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) |
⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | ||
Theorem | opelreal 7855 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) | ||
Theorem | elreal 7856* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) |
⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | ||
Theorem | elrealeu 7857* | The real number mapping in elreal 7856 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.) |
⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | ||
Theorem | elreal2 7858 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) |
⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | ||
Theorem | 0ncn 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 | ||
Theorem | cnm 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 𝑥 ∈ 𝐴) | ||
Theorem | ltrelre 7861 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ <ℝ ⊆ (ℝ × ℝ) | ||
Theorem | addcnsr 7862 | 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)〉) | ||
Theorem | mulcnsr 7863 | 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))〉) | ||
Theorem | eqresr 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) | ||
Theorem | addresr 7865 | 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〉) | ||
Theorem | mulresr 7866 | 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〉) | ||
Theorem | ltresr 7867 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
⊢ (〈𝐴, 0R〉 <ℝ 〈𝐵, 0R〉 ↔ 𝐴 <R 𝐵) | ||
Theorem | ltresr2 7868 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (1st ‘𝐴) <R (1st ‘𝐵))) | ||
Theorem | dfcnqs 7869 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ 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 ℂ 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 ) | ||
Theorem | addcnsrec 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.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ([〈𝐴, 𝐵〉]◡ E + [〈𝐶, 𝐷〉]◡ E ) = [〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉]◡ E ) | ||
Theorem | mulcnsrec 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.) |
⊢ (((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 ) | ||
Theorem | addvalex 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V) | ||
Theorem | pitonnlem1 7873* | Lemma for pitonn 7876. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.) |
⊢ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈1o, 1o〉] ~Q }, {𝑢 ∣ [〈1o, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 = 1 | ||
Theorem | pitonnlem1p1 7874 | Lemma for pitonn 7876. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.) |
⊢ (𝐴 ∈ P → [〈(𝐴 +P (1P +P 1P)), (1P +P 1P)〉] ~R = [〈(𝐴 +P 1P), 1P〉] ~R ) | ||
Theorem | pitonnlem2 7875* | Lemma for pitonn 7876. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.) |
⊢ (𝐾 ∈ N → (〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 + 1) = 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈(𝐾 +N 1o), 1o〉] ~Q }, {𝑢 ∣ [〈(𝐾 +N 1o), 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) | ||
Theorem | pitonn 7876* | Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.) |
⊢ (𝑁 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) | ||
Theorem | pitoregt0 7877* | Embedding from N to ℝ yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → 0 <ℝ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) | ||
Theorem | pitore 7878* | Embedding from N to ℝ. Similar to pitonn 7876 but separate in the sense that we have not proved nnssre 8952 yet. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ℝ) | ||
Theorem | recnnre 7879* | Embedding the reciprocal of a natural number into ℝ. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → 〈[〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ∈ ℝ) | ||
Theorem | peano1nnnn 7880* | One is an element of ℕ. 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 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ 1 ∈ 𝑁 | ||
Theorem | peano2nnnn 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 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁) | ||
Theorem | ltrennb 7882* | Ordering of natural numbers with <N or <ℝ. (Contributed by Jim Kingdon, 13-Jul-2021.) |
⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 <ℝ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉)) | ||
Theorem | ltrenn 7883* | Ordering of natural numbers with <N or <ℝ. (Contributed by Jim Kingdon, 12-Jul-2021.) |
⊢ (𝐽 <N 𝐾 → 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 <ℝ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) | ||
Theorem | recidpipr 7884* | Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.) |
⊢ (𝑁 ∈ N → (〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 ·P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉) = 1P) | ||
Theorem | recidpirqlemcalc 7885 | Lemma for recidpirq 7886. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.) |
⊢ (𝜑 → 𝐴 ∈ P) & ⊢ (𝜑 → 𝐵 ∈ P) & ⊢ (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))) | ||
Theorem | recidpirq 7886* | A real number times its reciprocal is one, where reciprocal is expressed with *Q. (Contributed by Jim Kingdon, 15-Jul-2021.) |
⊢ (𝑁 ∈ N → (〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝑁, 1o〉] ~Q }, {𝑢 ∣ [〈𝑁, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 · 〈[〈(〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑁, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑁, 1o〉] ~Q ) <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉) = 1) | ||
Theorem | axcnex 7887 | The complex numbers form a set. Use cnex 7964 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
⊢ ℂ ∈ V | ||
Theorem | axresscn 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 | ||
Theorem | ax1cn 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 | ||
Theorem | ax1re 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 | ||
Theorem | axicn 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 | ||
Theorem | axaddcl 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) | ||
Theorem | axaddrcl 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) | ||
Theorem | axmulcl 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) | ||
Theorem | axmulrcl 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) | ||
Theorem | axaddf 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 | ||
Theorem | axmulf 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 | ||
Theorem | axaddcom 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)) | ||
Theorem | axmulcom 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)) | ||
Theorem | axaddass 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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