| Metamath
Proof Explorer Theorem List (p. 430 of 494) | < Previous Next > | |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | (1-30937) |
(30938-32460) |
(32461-49324) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now. | ||
| Theorem | reglogcl 42901 | General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 26816 instead. |
| ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → ((log‘𝐴) / (log‘𝐵)) ∈ ℝ) | ||
| Theorem | reglogltb 42902 | General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 26827 instead. |
| ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 < 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) < ((log‘𝐵) / (log‘𝐶)))) | ||
| Theorem | reglogleb 42903 | General logarithm preserves ≤. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 26826 instead. |
| ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) ≤ ((log‘𝐵) / (log‘𝐶)))) | ||
| Theorem | reglogmul 42904 | Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 26820 instead. |
| ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐴 · 𝐵)) / (log‘𝐶)) = (((log‘𝐴) / (log‘𝐶)) + ((log‘𝐵) / (log‘𝐶)))) | ||
| Theorem | reglogexp 42905 | Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 26819 instead. |
| ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐴↑𝑁)) / (log‘𝐶)) = (𝑁 · ((log‘𝐴) / (log‘𝐶)))) | ||
| Theorem | reglogbas 42906 | General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 26811 instead. |
| ⊢ ((𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1) → ((log‘𝐶) / (log‘𝐶)) = 1) | ||
| Theorem | reglog1 42907 | General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 26812 instead. |
| ⊢ ((𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1) → ((log‘1) / (log‘𝐶)) = 0) | ||
| Theorem | reglogexpbas 42908 | General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbexp 26823 instead. |
| ⊢ ((𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐶↑𝑁)) / (log‘𝐶)) = 𝑁) | ||
| Theorem | pellfund14 42909* | Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
| ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) | ||
| Theorem | pellfund14b 42910* | The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
| ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))) | ||
| Syntax | crmx 42911 | Extend class notation to include the Robertson-Matiyasevich X sequence. |
| class Xrm | ||
| Syntax | crmy 42912 | Extend class notation to include the Robertson-Matiyasevich Y sequence. |
| class Yrm | ||
| Definition | df-rmx 42913* | Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 42925 and rmxyval 42927 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ Xrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)))) | ||
| Definition | df-rmy 42914* | Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 42926 and rmxyval 42927 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ Yrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (2nd ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)))) | ||
| Theorem | rmxfval 42915* | Value of the X sequence. Not used after rmxyval 42927 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) = (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))) | ||
| Theorem | rmyfval 42916* | Value of the Y sequence. Not used after rmxyval 42927 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) = (2nd ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)))) | ||
| Theorem | rmspecsqrtnq 42917 | The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by AV, 2-Aug-2021.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | ||
| Theorem | rmspecnonsq 42918 | The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) | ||
| Theorem | qirropth 42919 | This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))) | ||
| Theorem | rmspecfund 42920 | The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (PellFund‘((𝐴↑2) − 1)) = (𝐴 + (√‘((𝐴↑2) − 1)))) | ||
| Theorem | rmxyelqirr 42921* | The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))}) | ||
| Theorem | rmxyelqirrOLD 42922* | Obsolete version of rmxyelqirr 42921 as of 23-Dec-2024. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))}) | ||
| Theorem | rmxypairf1o 42923* | The function used to extract rational and irrational parts in df-rmx 42913 and df-rmy 42914 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))}) | ||
| Theorem | rmxyelxp 42924* | Lemma for frmx 42925 and frmy 42926. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) ∈ (ℕ0 × ℤ)) | ||
| Theorem | frmx 42925 | The X sequence is a nonnegative integer. See rmxnn 42963 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | ||
| Theorem | frmy 42926 | The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | ||
| Theorem | rmxyval 42927 | Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm 𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) | ||
| Theorem | rmspecpos 42928 | The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℝ+) | ||
| Theorem | rmxycomplete 42929* | The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. This is Metamath 100 proof #39. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℤ) → (((𝑋↑2) − (((𝐴↑2) − 1) · (𝑌↑2))) = 1 ↔ ∃𝑛 ∈ ℤ (𝑋 = (𝐴 Xrm 𝑛) ∧ 𝑌 = (𝐴 Yrm 𝑛)))) | ||
| Theorem | rmxynorm 42930 | The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1) | ||
| Theorem | rmbaserp 42931 | The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 + (√‘((𝐴↑2) − 1))) ∈ ℝ+) | ||
| Theorem | rmxyneg 42932 | Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain ℕ0 or ℤ; we use ℤ consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm -𝑁) = (𝐴 Xrm 𝑁) ∧ (𝐴 Yrm -𝑁) = -(𝐴 Yrm 𝑁))) | ||
| Theorem | rmxyadd 42933 | Addition formula for X and Y sequences. See rmxadd 42939 and rmyadd 42943 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm (𝑀 + 𝑁)) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑁)) + (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑁)))) ∧ (𝐴 Yrm (𝑀 + 𝑁)) = (((𝐴 Yrm 𝑀) · (𝐴 Xrm 𝑁)) + ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁))))) | ||
| Theorem | rmxy1 42934 | Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 Xrm 1) = 𝐴 ∧ (𝐴 Yrm 1) = 1)) | ||
| Theorem | rmxy0 42935 | Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 Xrm 0) = 1 ∧ (𝐴 Yrm 0) = 0)) | ||
| Theorem | rmxneg 42936 | Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 42932, rmxyadd 42933, rmxy0 42935, and rmxy1 42934 via qirropth 42919 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm -𝑁) = (𝐴 Xrm 𝑁)) | ||
| Theorem | rmx0 42937 | Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Xrm 0) = 1) | ||
| Theorem | rmx1 42938 | Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Xrm 1) = 𝐴) | ||
| Theorem | rmxadd 42939 | Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑀 + 𝑁)) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑁)) + (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑁))))) | ||
| Theorem | rmyneg 42940 | Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm -𝑁) = -(𝐴 Yrm 𝑁)) | ||
| Theorem | rmy0 42941 | Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Yrm 0) = 0) | ||
| Theorem | rmy1 42942 | Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Yrm 1) = 1) | ||
| Theorem | rmyadd 42943 | Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑀 + 𝑁)) = (((𝐴 Yrm 𝑀) · (𝐴 Xrm 𝑁)) + ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)))) | ||
| Theorem | rmxp1 42944 | Special addition-of-1 formula for X sequence. Part 1 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((𝐴 Xrm 𝑁) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)))) | ||
| Theorem | rmyp1 42945 | Special addition of 1 formula for Y sequence. Part 2 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = (((𝐴 Yrm 𝑁) · 𝐴) + (𝐴 Xrm 𝑁))) | ||
| Theorem | rmxm1 42946 | Subtraction of 1 formula for X sequence. Part 1 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 − 1)) = ((𝐴 · (𝐴 Xrm 𝑁)) − (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁)))) | ||
| Theorem | rmym1 42947 | Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 − 1)) = (((𝐴 Yrm 𝑁) · 𝐴) − (𝐴 Xrm 𝑁))) | ||
| Theorem | rmxluc 42948 | The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Xrm 𝑁)) − (𝐴 Xrm (𝑁 − 1)))) | ||
| Theorem | rmyluc 42949 | The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 42941 and rmy1 42942. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain (ℤ × ℤ), which simplifies some later theorems. It may shorten the derivation to use this as our initial definition. Incidentally, the X sequence satisfies the exact same recurrence. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = ((2 · ((𝐴 Yrm 𝑁) · 𝐴)) − (𝐴 Yrm (𝑁 − 1)))) | ||
| Theorem | rmyluc2 42950 | Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Yrm 𝑁)) − (𝐴 Yrm (𝑁 − 1)))) | ||
| Theorem | rmxdbl 42951 | "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (2 · 𝑁)) = ((2 · ((𝐴 Xrm 𝑁)↑2)) − 1)) | ||
| Theorem | rmydbl 42952 | "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (2 · 𝑁)) = ((2 · (𝐴 Xrm 𝑁)) · (𝐴 Yrm 𝑁))) | ||
| Theorem | monotuz 42953* | A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
| ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐻) → 𝐹 < 𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ) & ⊢ 𝐻 = (ℤ≥‘𝐼) & ⊢ (𝑥 = (𝑦 + 1) → 𝐶 = 𝐺) & ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐹) & ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻)) → (𝐴 < 𝐵 ↔ 𝐷 < 𝐸)) | ||
| Theorem | monotoddzzfi 42954* | A function which is odd and monotonic on ℕ0 is monotonic on ℤ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦))) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐹‘𝐴) < (𝐹‘𝐵))) | ||
| Theorem | monotoddzz 42955* | A function (given implicitly) which is odd and monotonic on ℕ0 is monotonic on ℤ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → 𝐸 < 𝐹)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 = -𝐹) & ⊢ (𝑥 = 𝐴 → 𝐸 = 𝐶) & ⊢ (𝑥 = 𝐵 → 𝐸 = 𝐷) & ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹) & ⊢ (𝑥 = -𝑦 → 𝐸 = 𝐺) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐶 < 𝐷)) | ||
| Theorem | oddcomabszz 42956* | An odd function which takes nonnegative values on nonnegative arguments commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ ∧ 0 ≤ 𝑥) → 0 ≤ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐶 = -𝐵) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = -𝑦 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝐷 → 𝐴 = 𝐸) & ⊢ (𝑥 = (abs‘𝐷) → 𝐴 = 𝐹) ⇒ ⊢ ((𝜑 ∧ 𝐷 ∈ ℤ) → (abs‘𝐸) = 𝐹) | ||
| Theorem | 2nn0ind 42957* | Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ 𝜓 & ⊢ 𝜒 & ⊢ (𝑦 ∈ ℕ → ((𝜃 ∧ 𝜏) → 𝜂)) & ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 − 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜏)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜂)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜌)) ⇒ ⊢ (𝐴 ∈ ℕ0 → 𝜌) | ||
| Theorem | zindbi 42958* | Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ (𝑦 ∈ ℤ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝐴 ∈ ℤ → (𝜃 ↔ 𝜏)) | ||
| Theorem | rmxypos 42959 | For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (0 < (𝐴 Xrm 𝑁) ∧ 0 ≤ (𝐴 Yrm 𝑁))) | ||
| Theorem | ltrmynn0 42960 | The Y-sequence is strictly monotonic on ℕ0. Strengthened by ltrmy 42964. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Yrm 𝑀) < (𝐴 Yrm 𝑁))) | ||
| Theorem | ltrmxnn0 42961 | The X-sequence is strictly monotonic on ℕ0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Xrm 𝑀) < (𝐴 Xrm 𝑁))) | ||
| Theorem | lermxnn0 42962 | The X-sequence is monotonic on ℕ0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝐴 Xrm 𝑀) ≤ (𝐴 Xrm 𝑁))) | ||
| Theorem | rmxnn 42963 | The X-sequence is defined to range over ℕ0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ) | ||
| Theorem | ltrmy 42964 | The Y-sequence is strictly monotonic over ℤ. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝐴 Yrm 𝑀) < (𝐴 Yrm 𝑁))) | ||
| Theorem | rmyeq0 42965 | Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ↔ (𝐴 Yrm 𝑁) = 0)) | ||
| Theorem | rmyeq 42966 | Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 𝑁 ↔ (𝐴 Yrm 𝑀) = (𝐴 Yrm 𝑁))) | ||
| Theorem | lermy 42967 | Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝐴 Yrm 𝑀) ≤ (𝐴 Yrm 𝑁))) | ||
| Theorem | rmynn 42968 | Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) → (𝐴 Yrm 𝑁) ∈ ℕ) | ||
| Theorem | rmynn0 42969 | Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 Yrm 𝑁) ∈ ℕ0) | ||
| Theorem | rmyabs 42970 | Yrm commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℤ) → (abs‘(𝐴 Yrm 𝐵)) = (𝐴 Yrm (abs‘𝐵))) | ||
| Theorem | jm2.24nn 42971 | X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to ℕ. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) → ((𝐴 Yrm (𝑁 − 1)) + (𝐴 Yrm 𝑁)) < (𝐴 Xrm 𝑁)) | ||
| Theorem | jm2.17a 42972 | First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (((2 · 𝐴) − 1)↑𝑁) ≤ (𝐴 Yrm (𝑁 + 1))) | ||
| Theorem | jm2.17b 42973 | Weak form of the second half of lemma 2.17 of [JonesMatijasevic] p. 696, allowing induction to start lower. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 Yrm (𝑁 + 1)) ≤ ((2 · 𝐴)↑𝑁)) | ||
| Theorem | jm2.17c 42974 | Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) → (𝐴 Yrm ((𝑁 + 1) + 1)) < ((2 · 𝐴)↑(𝑁 + 1))) | ||
| Theorem | jm2.24 42975 | Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 43013. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm (𝑁 − 1)) + (𝐴 Yrm 𝑁)) < (𝐴 Xrm 𝑁)) | ||
| Theorem | rmygeid 42976 | Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝐴 Yrm 𝑁)) | ||
| Theorem | congtr 42977 | A wff of the form 𝐴 ∥ (𝐵 − 𝐶) is interpreted as a congruential equation. This is similar to (𝐵 mod 𝐴) = (𝐶 mod 𝐴), but is defined such that behavior is regular for zero and negative values of 𝐴. To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐶 − 𝐷))) → 𝐴 ∥ (𝐵 − 𝐷)) | ||
| Theorem | congadd 42978 | If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐷 ∈ ℤ ∧ 𝐸 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐷 − 𝐸))) → 𝐴 ∥ ((𝐵 + 𝐷) − (𝐶 + 𝐸))) | ||
| Theorem | congmul 42979 | If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐷 ∈ ℤ ∧ 𝐸 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐷 − 𝐸))) → 𝐴 ∥ ((𝐵 · 𝐷) − (𝐶 · 𝐸))) | ||
| Theorem | congsym 42980 | Congruence mod 𝐴 is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ (𝐵 − 𝐶))) → 𝐴 ∥ (𝐶 − 𝐵)) | ||
| Theorem | congneg 42981 | If two integers are congruent mod 𝐴, so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ (𝐵 − 𝐶))) → 𝐴 ∥ (-𝐵 − -𝐶)) | ||
| Theorem | congsub 42982 | If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐷 ∈ ℤ ∧ 𝐸 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐷 − 𝐸))) → 𝐴 ∥ ((𝐵 − 𝐷) − (𝐶 − 𝐸))) | ||
| Theorem | congid 42983 | Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐵 − 𝐵)) | ||
| Theorem | mzpcong 42984* | Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.) |
| ⊢ ((𝐹 ∈ (mzPoly‘𝑉) ∧ (𝑋 ∈ (ℤ ↑m 𝑉) ∧ 𝑌 ∈ (ℤ ↑m 𝑉)) ∧ (𝑁 ∈ ℤ ∧ ∀𝑘 ∈ 𝑉 𝑁 ∥ ((𝑋‘𝑘) − (𝑌‘𝑘)))) → 𝑁 ∥ ((𝐹‘𝑋) − (𝐹‘𝑌))) | ||
| Theorem | congrep 42985* | Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℤ) → ∃𝑎 ∈ (0...(𝐴 − 1))𝐴 ∥ (𝑎 − 𝑁)) | ||
| Theorem | congabseq 42986 | If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 − 𝐶)) → ((abs‘(𝐵 − 𝐶)) < 𝐴 ↔ 𝐵 = 𝐶)) | ||
| Theorem | acongid 42987 |
A wff like that in this theorem will be known as an "alternating
congruence". A special symbol might be considered if more uses come
up.
They have many of the same properties as normal congruences, starting with
reflexivity.
JonesMatijasevic uses "a ≡ ± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ (𝐵 − 𝐵) ∨ 𝐴 ∥ (𝐵 − -𝐵))) | ||
| Theorem | acongsym 42988 | Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶))) → (𝐴 ∥ (𝐶 − 𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵))) | ||
| Theorem | acongneg2 42989 | Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − -𝐶) ∨ 𝐴 ∥ (𝐵 − --𝐶))) → (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶))) | ||
| Theorem | acongtr 42990 | Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ((𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)) ∧ (𝐴 ∥ (𝐶 − 𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷)))) → (𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷))) | ||
| Theorem | acongeq12d 42991 | Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
| ⊢ (𝜑 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐷 = 𝐸) ⇒ ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸)))) | ||
| Theorem | acongrep 42992* | Every integer is alternating-congruent to some number in the first half of the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℤ) → ∃𝑎 ∈ (0...𝐴)((2 · 𝐴) ∥ (𝑎 − 𝑁) ∨ (2 · 𝐴) ∥ (𝑎 − -𝑁))) | ||
| Theorem | fzmaxdif 42993 | Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶 − 𝐸) ≤ (𝐹 − 𝐵)) → (abs‘(𝐴 − 𝐷)) ≤ (𝐹 − 𝐵)) | ||
| Theorem | fzneg 42994 | Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (𝐵...𝐶) ↔ -𝐴 ∈ (-𝐶...-𝐵))) | ||
| Theorem | acongeq 42995 | Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 43014. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶)))) | ||
| Theorem | dvdsacongtr 42996 | Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
| ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐷 ∥ 𝐴 ∧ (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)))) → (𝐷 ∥ (𝐵 − 𝐶) ∨ 𝐷 ∥ (𝐵 − -𝐶))) | ||
| Theorem | coprmdvdsb 42997 | Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
| ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝐾 gcd 𝑀) = 1)) → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ (𝑀 · 𝑁))) | ||
| Theorem | modabsdifz 42998 | Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
| ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ) | ||
| Theorem | dvdsabsmod0 42999 | Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.) (Proof shortened by OpenAI, 3-Jul-2020.) |
| ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0)) | ||
| Theorem | jm2.18 43000 | Theorem 2.18 of [JonesMatijasevic] p. 696. Direct relationship of the exponential function to X and Y sequences. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
| ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∥ (((𝐴 Xrm 𝑁) − ((𝐴 − 𝐾) · (𝐴 Yrm 𝑁))) − (𝐾↑𝑁))) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |