| Metamath
Proof Explorer Theorem List (p. 485 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 | ||
| Theorem | islininds2 48401* | Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 → ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠))) | ||
| Theorem | isldepslvec2 48402* | Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 48400 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑍 = (0g‘𝑀) & ⊢ 𝑅 = (Scalar‘𝑀) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ 𝒫 𝐵) → (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ 𝑆 linDepS 𝑀)) | ||
| Theorem | lindssnlvec 48403 | A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.) |
| ⊢ ((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g‘𝑀)) → {𝑆} linIndS 𝑀) | ||
| Theorem | lmod1lem1 48404* | Lemma 1 for lmod1 48409. (Contributed by AV, 28-Apr-2019.) |
| ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)𝐼) ∈ {𝐼}) | ||
| Theorem | lmod1lem2 48405* | Lemma 2 for lmod1 48409. (Contributed by AV, 28-Apr-2019.) |
| ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠 ‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) | ||
| Theorem | lmod1lem3 48406* | Lemma 3 for lmod1 48409. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) ⇒ ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = ((𝑞( ·𝑠 ‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) | ||
| Theorem | lmod1lem4 48407* | Lemma 4 for lmod1 48409. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) ⇒ ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠 ‘𝑀)𝐼) = (𝑞( ·𝑠 ‘𝑀)(𝑟( ·𝑠 ‘𝑀)𝐼))) | ||
| Theorem | lmod1lem5 48408* | Lemma 5 for lmod1 48409. (Contributed by AV, 28-Apr-2019.) |
| ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝐼) = 𝐼) | ||
| Theorem | lmod1 48409* | The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑀 ∈ LMod) | ||
| Theorem | lmod1zr 48410 | The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑅 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} & ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑍, 𝐼〉, 𝐼〉}〉}) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑀 ∈ LMod) | ||
| Theorem | lmod1zrnlvec 48411 | There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑅 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} & ⊢ 𝑀 = ({〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), {〈〈𝑍, 𝐼〉, 𝐼〉}〉}) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑀 ∉ LVec) | ||
| Theorem | lmodn0 48412 | Left modules exist. (Contributed by AV, 29-Apr-2019.) |
| ⊢ LMod ≠ ∅ | ||
| Theorem | zlmodzxzequa 48413 | Example of an equation within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 0 = {〈0, 0〉, 〈1, 0〉} & ⊢ ∙ = ( ·𝑠 ‘𝑍) & ⊢ − = (-g‘𝑍) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} ⇒ ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 | ||
| Theorem | zlmodzxznm 48414 | Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 0 = {〈0, 0〉, 〈1, 0〉} & ⊢ ∙ = ( ·𝑠 ‘𝑍) & ⊢ − = (-g‘𝑍) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} ⇒ ⊢ ∀𝑖 ∈ ℤ ((𝑖 ∙ 𝐴) ≠ 𝐵 ∧ (𝑖 ∙ 𝐵) ≠ 𝐴) | ||
| Theorem | zlmodzxzldeplem 48415 | A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
| Theorem | zlmodzxzequap 48416 | Example of an equation within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} & ⊢ 0 = {〈0, 0〉, 〈1, 0〉} & ⊢ + = (+g‘𝑍) & ⊢ ∙ = ( ·𝑠 ‘𝑍) ⇒ ⊢ ((2 ∙ 𝐴) + (-3 ∙ 𝐵)) = 0 | ||
| Theorem | zlmodzxzldeplem1 48417 | Lemma 1 for zlmodzxzldep 48421. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} & ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} ⇒ ⊢ 𝐹 ∈ (ℤ ↑m {𝐴, 𝐵}) | ||
| Theorem | zlmodzxzldeplem2 48418 | Lemma 2 for zlmodzxzldep 48421. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} & ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} ⇒ ⊢ 𝐹 finSupp 0 | ||
| Theorem | zlmodzxzldeplem3 48419 | Lemma 3 for zlmodzxzldep 48421. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} & ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} ⇒ ⊢ (𝐹( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) | ||
| Theorem | zlmodzxzldeplem4 48420* | Lemma 4 for zlmodzxzldep 48421. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} & ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} ⇒ ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 | ||
| Theorem | zlmodzxzldep 48421 | { A , B } is a linearly dependent set within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} ⇒ ⊢ {𝐴, 𝐵} linDepS 𝑍 | ||
| Theorem | ldepsnlinclem1 48422 | Lemma 1 for ldepsnlinc 48425. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} ⇒ ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴) | ||
| Theorem | ldepsnlinclem2 48423 | Lemma 2 for ldepsnlinc 48425. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.) |
| ⊢ 𝑍 = (ℤring freeLMod {0, 1}) & ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} & ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} ⇒ ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) | ||
| Theorem | lvecpsslmod 48424 | The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 21105) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 48411. (Contributed by AV, 29-Apr-2019.) |
| ⊢ LVec ⊊ LMod | ||
| Theorem | ldepsnlinc 48425* | The reverse implication of islindeps2 48400 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combination of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 48413 and zlmodzxznm 48414. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣)) | ||
| Theorem | ldepslinc 48426* | For (left) vector spaces, isldepslvec2 48402 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 48425 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.) |
| ⊢ (∀𝑚 ∈ LVec ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣 ∈ 𝑠 ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∧ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣 ∈ 𝑠 ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))) | ||
| Theorem | suppdm 48427 | If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.) |
| ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹) | ||
| Theorem | eluz2cnn0n1 48428 | An integer greater than 1 is a complex number not equal to 0 or 1. (Contributed by AV, 23-May-2020.) |
| ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ (ℂ ∖ {0, 1})) | ||
| Theorem | divge1b 48429 | The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.) |
| ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 1 ≤ (𝐵 / 𝐴))) | ||
| Theorem | divgt1b 48430 | The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.) |
| ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 1 < (𝐵 / 𝐴))) | ||
| Theorem | ltsubaddb 48431 | Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐶) < (𝐵 − 𝐷) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶))) | ||
| Theorem | ltsubsubb 48432 | Equivalence for the "less than" relation between differences. (Contributed by AV, 6-Jun-2020.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐶) < (𝐵 − 𝐷) ↔ (𝐴 − 𝐵) < (𝐶 − 𝐷))) | ||
| Theorem | ltsubadd2b 48433 | Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐷 − 𝐶) < (𝐵 − 𝐴) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶))) | ||
| Theorem | divsub1dir 48434 | Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) − 1) = ((𝐴 − 𝐵) / 𝐵)) | ||
| Theorem | expnegico01 48435 | An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.) |
| ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵↑𝑁) ∈ (0[,)1)) | ||
| Theorem | elfzolborelfzop1 48436 | An element of a half-open integer interval is either equal to the left bound of the interval or an element of a half-open integer interval with a lower bound increased by 1. (Contributed by AV, 2-Jun-2020.) |
| ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁))) | ||
| Theorem | pw2m1lepw2m1 48437 | 2 to the power of a positive integer decreased by 1 is less than or equal to 2 to the power of the integer minus 1. (Contributed by AV, 30-May-2020.) |
| ⊢ (𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1)) | ||
| Theorem | zgtp1leeq 48438 | If an integer is between another integer and its predecessor, the integer is equal to the other integer. (Contributed by AV, 7-Jun-2020.) |
| ⊢ ((𝐼 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((𝐴 − 1) < 𝐼 ∧ 𝐼 ≤ 𝐴) → 𝐼 = 𝐴)) | ||
| Theorem | flsubz 48439 | An integer can be moved in and out of the floor of a difference. (Contributed by AV, 29-May-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 − 𝑁)) = ((⌊‘𝐴) − 𝑁)) | ||
| Theorem | mod0mul 48440* | If an integer is 0 modulo a positive integer, this integer must be the product of another integer and the modulus. (Contributed by AV, 7-Jun-2020.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁))) | ||
| Theorem | modn0mul 48441* | If an integer is not 0 modulo a positive integer, this integer must be the sum of the product of another integer and the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) ≠ 0 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦))) | ||
| Theorem | m1modmmod 48442 | An integer decreased by 1 modulo a positive integer minus the integer modulo the same modulus is either -1 or the modulus minus 1. (Contributed by AV, 7-Jun-2020.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) mod 𝑁) − (𝐴 mod 𝑁)) = if((𝐴 mod 𝑁) = 0, (𝑁 − 1), -1)) | ||
| Theorem | difmodm1lt 48443 | The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd 𝐴 and 𝑁 = 2, since ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) would be (1 − 0) = 1 which is not less than (𝑁 − 1) = 1. (Contributed by AV, 6-Jun-2012.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) < (𝑁 − 1)) | ||
| Theorem | nn0onn0ex 48444* | For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1)) | ||
| Theorem | nn0enn0ex 48445* | For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = (2 · 𝑚)) | ||
| Theorem | nnennex 48446* | For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚)) | ||
| Theorem | nneop 48447 | A positive integer is even or odd. (Contributed by AV, 30-May-2020.) |
| ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) | ||
| Theorem | nneom 48448 | A positive integer is even or odd. (Contributed by AV, 30-May-2020.) |
| ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
| Theorem | nn0eo 48449 | A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.) |
| ⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0)) | ||
| Theorem | nnpw2even 48450 | 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) |
| ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) / 2) ∈ ℕ) | ||
| Theorem | zefldiv2 48451 | The floor of an even integer divided by 2 is equal to the integer divided by 2. (Contributed by AV, 7-Jun-2020.) |
| ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = (𝑁 / 2)) | ||
| Theorem | zofldiv2 48452 | The floor of an odd integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) |
| ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) | ||
| Theorem | nn0ofldiv2 48453 | The floor of an odd nonnegative integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) (Proof shortened by AV, 7-Jun-2020.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) | ||
| Theorem | flnn0div2ge 48454 | The floor of a positive integer divided by 2 is greater than or equal to the integer decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) |
| ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) / 2) ≤ (⌊‘(𝑁 / 2))) | ||
| Theorem | flnn0ohalf 48455 | The floor of the half of an odd positive integer is equal to the floor of the half of the integer decreased by 1. (Contributed by AV, 5-Jun-2012.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = (⌊‘((𝑁 − 1) / 2))) | ||
| Theorem | logcxp0 48456 | Logarithm of a complex power. Generalization of logcxp 26711. (Contributed by AV, 22-May-2020.) |
| ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℂ ∧ (𝐵 · (log‘𝐴)) ∈ ran log) → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴))) | ||
| Theorem | regt1loggt0 48457 | The natural logarithm for a real number greater than 1 is greater than 0. (Contributed by AV, 25-May-2020.) |
| ⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵)) | ||
| Syntax | cfdiv 48458 | Extend class notation with the division operator of two functions. |
| class /f | ||
| Definition | df-fdiv 48459* | Define the division of two functions into the complex numbers. (Contributed by AV, 15-May-2020.) |
| ⊢ /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0))) | ||
| Theorem | fdivval 48460 | The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 /f 𝐺) = ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0))) | ||
| Theorem | fdivmpt 48461* | The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.) |
| ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) | ||
| Theorem | fdivmptf 48462 | The quotient of two functions into the complex numbers is a function into the complex numbers. (Contributed by AV, 16-May-2020.) |
| ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ) | ||
| Theorem | refdivmptf 48463 | The quotient of two functions into the real numbers is a function into the real numbers. (Contributed by AV, 16-May-2020.) |
| ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ) | ||
| Theorem | fdivpm 48464 | The quotient of two functions into the complex numbers is a partial function. (Contributed by AV, 16-May-2020.) |
| ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴)) | ||
| Theorem | refdivpm 48465 | The quotient of two functions into the real numbers is a partial function. (Contributed by AV, 16-May-2020.) |
| ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℝ ↑pm 𝐴)) | ||
| Theorem | fdivmptfv 48466 | The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.) |
| ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋))) | ||
| Theorem | refdivmptfv 48467 | The function value of a quotient of two functions into the real numbers. (Contributed by AV, 19-May-2020.) |
| ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋))) | ||
| Syntax | cbigo 48468 | Extend class notation with the class of the "big-O" function. |
| class Ο | ||
| Definition | df-bigo 48469* | Define the function "big-O", mapping a real function g to the set of real functions "of order g(x)". Definition in section 1.1 of [AhoHopUll] p. 2. This is a generalization of "big-O of one", see df-o1 15526 and df-lo1 15527. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 48478. (Contributed by AV, 15-May-2020.) |
| ⊢ Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) | ||
| Theorem | bigoval 48470* | Set of functions of order G(x). (Contributed by AV, 15-May-2020.) |
| ⊢ (𝐺 ∈ (ℝ ↑pm ℝ) → (Ο‘𝐺) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))}) | ||
| Theorem | elbigofrcl 48471 | Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.) |
| ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) | ||
| Theorem | elbigo 48472* | Properties of a function of order G(x). (Contributed by AV, 16-May-2020.) |
| ⊢ (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) | ||
| Theorem | elbigo2 48473* | Properties of a function of order G(x) under certain assumptions. (Contributed by AV, 17-May-2020.) |
| ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))) | ||
| Theorem | elbigo2r 48474* | Sufficient condition for a function to be of order G(x). (Contributed by AV, 18-May-2020.) |
| ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → 𝐹 ∈ (Ο‘𝐺)) | ||
| Theorem | elbigof 48475 | A function of order G(x) is a function. (Contributed by AV, 18-May-2020.) |
| ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ) | ||
| Theorem | elbigodm 48476 | The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.) |
| ⊢ (𝐹 ∈ (Ο‘𝐺) → dom 𝐹 ⊆ ℝ) | ||
| Theorem | elbigoimp 48477* | The defining property of a function of order G(x). (Contributed by AV, 18-May-2020.) |
| ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) | ||
| Theorem | elbigolo1 48478 | A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.) (Proof shortened by II, 16-Feb-2023.) |
| ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ+ ∧ 𝐹:𝐴⟶ℝ+) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /f 𝐺) ∈ ≤𝑂(1))) | ||
| Theorem | rege1logbrege0 48479 | The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.) |
| ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋)) | ||
| Theorem | rege1logbzge0 48480 | The general logarithm, with an integer base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.) |
| ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋)) | ||
| Theorem | fllogbd 48481 | A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.) |
| ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) & ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋)) ⇒ ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1)))) | ||
| Theorem | relogbmulbexp 48482 | The logarithm of the product of a positive real number and the base to the power of a real number is the logarithm of the positive real number plus the real number. (Contributed by AV, 29-May-2020.) |
| ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶)) | ||
| Theorem | relogbdivb 48483 | The logarithm of the quotient of a positive real number and the base is the logarithm of the number minus 1. (Contributed by AV, 29-May-2020.) |
| ⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ 𝐴 ∈ ℝ+) → (𝐵 logb (𝐴 / 𝐵)) = ((𝐵 logb 𝐴) − 1)) | ||
| Theorem | logbge0b 48484 | The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.) |
| ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (0 ≤ (𝐵 logb 𝑋) ↔ 1 ≤ 𝑋)) | ||
| Theorem | logblt1b 48485 | The logarithm of a number is less than 1 iff the number is less than the base of the logarithm. (Contributed by AV, 30-May-2020.) |
| ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → ((𝐵 logb 𝑋) < 1 ↔ 𝑋 < 𝐵)) | ||
If the binary logarithm is used more often, a separate symbol/definition could be provided for it, e.g., log2 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (2 logb 𝑋)). Then we can write "( log2 ` x )" (analogous to (log𝑥) for the natural logarithm) instead of (2 logb 𝑥). | ||
| Theorem | fldivexpfllog2 48486 | The floor of a positive real number divided by 2 to the power of the floor of the logarithm to base 2 of the number is 1. (Contributed by AV, 26-May-2020.) |
| ⊢ (𝑋 ∈ ℝ+ → (⌊‘(𝑋 / (2↑(⌊‘(2 logb 𝑋))))) = 1) | ||
| Theorem | nnlog2ge0lt1 48487 | A positive integer is 1 iff its binary logarithm is between 0 and 1. (Contributed by AV, 30-May-2020.) |
| ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1))) | ||
| Theorem | logbpw2m1 48488 | The floor of the binary logarithm of 2 to the power of a positive integer minus 1 is equal to the integer minus 1. (Contributed by AV, 31-May-2020.) |
| ⊢ (𝐼 ∈ ℕ → (⌊‘(2 logb ((2↑𝐼) − 1))) = (𝐼 − 1)) | ||
| Theorem | fllog2 48489 | The floor of the binary logarithm of 2 to the power of an element of a half-open integer interval bounded by powers of 2 is equal to the integer. (Contributed by AV, 31-May-2020.) |
| ⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ((2↑𝐼)..^(2↑(𝐼 + 1)))) → (⌊‘(2 logb 𝑁)) = 𝐼) | ||
| Syntax | cblen 48490 | Extend class notation with the class of the binary length function. |
| class #b | ||
| Definition | df-blen 48491 | Define the binary length of an integer. Definition in section 1.3 of [AhoHopUll] p. 12. Although not restricted to integers, this definition is only meaningful for 𝑛 ∈ ℤ or even for 𝑛 ∈ ℂ. (Contributed by AV, 16-May-2020.) |
| ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1))) | ||
| Theorem | blenval 48492 | The binary length of an integer. (Contributed by AV, 20-May-2020.) |
| ⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1))) | ||
| Theorem | blen0 48493 | The binary length of 0. (Contributed by AV, 20-May-2020.) |
| ⊢ (#b‘0) = 1 | ||
| Theorem | blenn0 48494 | The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.) |
| ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 0) → (#b‘𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1)) | ||
| Theorem | blenre 48495 | The binary length of a positive real number. (Contributed by AV, 20-May-2020.) |
| ⊢ (𝑁 ∈ ℝ+ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | ||
| Theorem | blennn 48496 | The binary length of a positive integer. (Contributed by AV, 21-May-2020.) |
| ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | ||
| Theorem | blennnelnn 48497 | The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.) |
| ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | ||
| Theorem | blennn0elnn 48498 | The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.) |
| ⊢ (𝑁 ∈ ℕ0 → (#b‘𝑁) ∈ ℕ) | ||
| Theorem | blenpw2 48499 | The binary length of a power of 2 is the exponent plus 1. (Contributed by AV, 30-May-2020.) |
| ⊢ (𝐼 ∈ ℕ0 → (#b‘(2↑𝐼)) = (𝐼 + 1)) | ||
| Theorem | blenpw2m1 48500 | The binary length of a power of 2 minus 1 is the exponent. (Contributed by AV, 31-May-2020.) |
| ⊢ (𝐼 ∈ ℕ → (#b‘((2↑𝐼) − 1)) = 𝐼) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |