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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | m1r 7701 | The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) |
⊢ -1R ∈ R | ||
Theorem | addclsr 7702 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) | ||
Theorem | mulclsr 7703 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | ||
Theorem | addcomsrg 7704 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) | ||
Theorem | addasssrg 7705 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶))) | ||
Theorem | mulcomsrg 7706 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) | ||
Theorem | mulasssrg 7707 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶))) | ||
Theorem | distrsrg 7708 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))) | ||
Theorem | m1p1sr 7709 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
⊢ (-1R +R 1R) = 0R | ||
Theorem | m1m1sr 7710 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
⊢ (-1R ·R -1R) = 1R | ||
Theorem | lttrsr 7711* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ)) | ||
Theorem | ltposr 7712 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
⊢ <R Po R | ||
Theorem | ltsosr 7713 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
⊢ <R Or R | ||
Theorem | 0lt1sr 7714 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
⊢ 0R <R 1R | ||
Theorem | 1ne0sr 7715 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
⊢ ¬ 1R = 0R | ||
Theorem | 0idsr 7716 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | ||
Theorem | 1idsr 7717 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | ||
Theorem | 00sr 7718 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | ||
Theorem | ltasrg 7719 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵))) | ||
Theorem | pn0sr 7720 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) | ||
Theorem | negexsr 7721* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
⊢ (𝐴 ∈ R → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R) | ||
Theorem | recexgt0sr 7722* | 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 7723* | 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 7724 | 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 7725 | 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 7726 | 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 7727 | 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 7728 | 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 7729 | Lemma for mulextsr1 7730. (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 7730 | 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 7731* | 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 7732* | 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 7733 | 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 7734 | 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 7735 | 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 7736 | 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 7737* | 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 7738* | Lemma for caucvgsr 7751. Terms of the sequence from caucvgsrlemgt1 7744 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 7739* | Lemma for caucvgsr 7751. 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 7740* | Lemma for caucvgsr 7751. 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 7741* | Lemma for caucvgsr 7751. 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 7742* | Lemma for caucvgsr 7751. 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 7743* | Lemma for caucvgsr 7751. 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 7744* | Lemma for caucvgsr 7751. 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 7745* | Lemma for caucvgsr 7751. 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 7746* | Lemma for caucvgsr 7751. 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 7747* | Lemma for caucvgsr 7751. 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 7748* | Lemma for caucvgsr 7751. 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 7749* | Lemma for caucvgsr 7751. 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 7750* | Lemma for caucvgsr 7751. 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 7751* |
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 7661 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 7750). 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 7746). 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 7661 to get a limit (see caucvgsrlemgt1 7744). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7744). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7749). (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 7752 | 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 7753 | 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 7754* | 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 7755* | Lemma for suplocsr 7758. 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 7756* | Lemma for suplocsr 7758. 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 7757* | Lemma for suplocsr 7758. 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 7758* | 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 7759 | Class of complex numbers. |
class ℂ | ||
Syntax | cr 7760 | Class of real numbers. |
class ℝ | ||
Syntax | cc0 7761 | Extend class notation to include the complex number 0. |
class 0 | ||
Syntax | c1 7762 | Extend class notation to include the complex number 1. |
class 1 | ||
Syntax | ci 7763 | Extend class notation to include the complex number i. |
class i | ||
Syntax | caddc 7764 | Addition on complex numbers. |
class + | ||
Syntax | cltrr 7765 | 'Less than' predicate (defined over real subset of complex numbers). |
class <ℝ | ||
Syntax | cmul 7766 | Multiplication on complex numbers. The token · is a center dot. |
class · | ||
Definition | df-c 7767 | Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ ℂ = (R × R) | ||
Definition | df-0 7768 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) |
⊢ 0 = 〈0R, 0R〉 | ||
Definition | df-1 7769 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) |
⊢ 1 = 〈1R, 0R〉 | ||
Definition | df-i 7770 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) |
⊢ i = 〈0R, 1R〉 | ||
Definition | df-r 7771 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ ℝ = (R × {0R}) | ||
Definition | df-add 7772* | 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 7773* | 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 7774* | 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 7775 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) |
⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | ||
Theorem | opelreal 7776 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) | ||
Theorem | elreal 7777* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) |
⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | ||
Theorem | elrealeu 7778* | The real number mapping in elreal 7777 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.) |
⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | ||
Theorem | elreal2 7779 | 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 7780 | 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 7781 which is a related property. (Contributed by NM, 2-May-1996.) |
⊢ ¬ ∅ ∈ ℂ | ||
Theorem | cnm 7781* | 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 7782 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) |
⊢ <ℝ ⊆ (ℝ × ℝ) | ||
Theorem | addcnsr 7783 | 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 7784 | 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 7785 | 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 7786 | 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 7787 | 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 7788 | 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 7789 | 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 7790 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ from those in R. The trick involves qsid 6574, 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 7767), 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 7791 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7790 and mulcnsrec 7792. (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 7792 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6573, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7790. (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 7793 | 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 7886. (Contributed by Jim Kingdon, 14-Jul-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V) | ||
Theorem | pitonnlem1 7794* | Lemma for pitonn 7797. 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 7795 | Lemma for pitonn 7797. 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 7796* | Lemma for pitonn 7797. 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 7797* | 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 7798* | 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 7799* | Embedding from N to ℝ. Similar to pitonn 7797 but separate in the sense that we have not proved nnssre 8869 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 7800* | 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) |
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