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Type | Label | Description |
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Statement | ||
Theorem | lmod1 44901* | 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 44902 | 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 44903 | 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 44904 | Left modules exist. (Contributed by AV, 29-Apr-2019.) |
⊢ LMod ≠ ∅ | ||
Theorem | zlmodzxzequa 44905 | 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 44906 | 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 44907 | 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 44908 | 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 44909 | Lemma 1 for zlmodzxzldep 44913. (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 44910 | Lemma 2 for zlmodzxzldep 44913. (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 44911 | Lemma 3 for zlmodzxzldep 44913. (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 44912* | Lemma 4 for zlmodzxzldep 44913. (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 44913 | { 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 44914 | Lemma 1 for ldepsnlinc 44917. (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 44915 | Lemma 2 for ldepsnlinc 44917. (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 44916 | 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 19871) 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 44903. (Contributed by AV, 29-Apr-2019.) |
⊢ LVec ⊊ LMod | ||
Theorem | ldepsnlinc 44917* | The reverse implication of islindeps2 44892 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 combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 44905 and zlmodzxznm 44906. (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 44918* | For (left) vector spaces, isldepslvec2 44894 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 44917 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 44919 | 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 44920 | 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 44921 | 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 44922 | 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 44923 | 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 44924 | 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 44925 | 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 44926 | 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 44927 | 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 44928 | 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 44929 | 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 44930 | 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 44931 | 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 | fldivmod 44932 | Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) = ((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) | ||
Theorem | mod0mul 44933* | 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 44934* | 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 44935 | 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 44936 | 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 44937* | 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 44938* | 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 44939* | 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 44940 | A positive integer is even or odd. (Contributed by AV, 30-May-2020.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) | ||
Theorem | nneom 44941 | A positive integer is even or odd. (Contributed by AV, 30-May-2020.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
Theorem | nn0eo 44942 | A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.) |
⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0)) | ||
Theorem | nnpw2even 44943 | 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → ((2↑𝑁) / 2) ∈ ℕ) | ||
Theorem | zefldiv2 44944 | 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 44945 | 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 44946 | 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 44947 | 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 44948 | 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 44949 | Logarithm of a complex power. Generalization of logcxp 25260. (Contributed by AV, 22-May-2020.) |
⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℂ ∧ (𝐵 · (log‘𝐴)) ∈ ran log) → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴))) | ||
Theorem | regt1loggt0 44950 | 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 44951 | Extend class notation with the division operator of two functions. |
class /f | ||
Definition | df-fdiv 44952* | 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 44953 | 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 44954* | 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 44955 | 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 44956 | 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 44957 | 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 44958 | 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 44959 | 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 44960 | 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 44961 | Extend class notation with the class of the "big-O" function. |
class Ο | ||
Definition | df-bigo 44962* | 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 14839 and df-lo1 14840. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 44971. (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 44963* | 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 44964 | Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.) |
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) | ||
Theorem | elbigo 44965* | 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 44966* | 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 44967* | 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 44968 | A function of order G(x) is a function. (Contributed by AV, 18-May-2020.) |
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ) | ||
Theorem | elbigodm 44969 | 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 44970* | 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 44971 | 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 44972 | 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 44973 | 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 44974 | 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 44975 | 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 44976 | 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 44977 | 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 44978 | 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 44979 | 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 44980 | 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 44981 | 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 44982 | 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 44983 | Extend class notation with the class of the binary length function. |
class #b | ||
Definition | df-blen 44984 | 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 44985 | 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 44986 | The binary length of 0. (Contributed by AV, 20-May-2020.) |
⊢ (#b‘0) = 1 | ||
Theorem | blenn0 44987 | 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 44988 | The binary length of a positive real number. (Contributed by AV, 20-May-2020.) |
⊢ (𝑁 ∈ ℝ+ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | ||
Theorem | blennn 44989 | The binary length of a positive integer. (Contributed by AV, 21-May-2020.) |
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | ||
Theorem | blennnelnn 44990 | The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.) |
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | ||
Theorem | blennn0elnn 44991 | The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#b‘𝑁) ∈ ℕ) | ||
Theorem | blenpw2 44992 | 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 44993 | 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) | ||
Theorem | nnpw2blen 44994 | A positive integer is between 2 to the power of its binary length minus 1 and 2 to the power of its binary length. (Contributed by AV, 31-May-2020.) |
⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁)))) | ||
Theorem | nnpw2blenfzo 44995 | A positive integer is between 2 to the power of the binary length of the integer minus 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ((2↑((#b‘𝑁) − 1))..^(2↑(#b‘𝑁)))) | ||
Theorem | nnpw2blenfzo2 44996 | A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → (𝑁 = (2↑((#b‘𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#b‘𝑁) − 1)) + 1)..^(2↑(#b‘𝑁))))) | ||
Theorem | nnpw2pmod 44997 | Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
⊢ (𝑁 ∈ ℕ → 𝑁 = ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1))))) | ||
Theorem | blen1 44998 | The binary length of 1. (Contributed by AV, 21-May-2020.) |
⊢ (#b‘1) = 1 | ||
Theorem | blen2 44999 | The binary length of 2. (Contributed by AV, 21-May-2020.) |
⊢ (#b‘2) = 2 | ||
Theorem | nnpw2p 45000* | Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
⊢ (𝑁 ∈ ℕ → ∃𝑖 ∈ ℕ0 ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟)) |
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