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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xeqlelt 30501 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵))) | ||
Theorem | eliccelico 30502 | Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵))) | ||
Theorem | elicoelioo 30503 | Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)))) | ||
Theorem | iocinioc2 30504 | Intersection between two open-below, closed-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶)) | ||
Theorem | xrdifh 30505 | Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.) |
⊢ 𝐴 ∈ ℝ* ⇒ ⊢ (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴) | ||
Theorem | iocinif 30506 | Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶))) | ||
Theorem | difioo 30507 | The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶)) | ||
Theorem | difico 30508 | The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵)) | ||
Theorem | uzssico 30509 | Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) | ||
Theorem | fz2ssnn0 30510 | A finite set of sequential integers that is a subset of ℕ0. (Contributed by Thierry Arnoux, 8-Dec-2021.) |
⊢ (𝑀 ∈ ℕ0 → (𝑀...𝑁) ⊆ ℕ0) | ||
Theorem | nndiffz1 30511 | Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.) |
⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1))) | ||
Theorem | ssnnssfz 30512* | For any finite subset of ℕ, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
⊢ (𝐴 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝐴 ⊆ (1...𝑛)) | ||
Theorem | fzne1 30513 | Elementhood in a finite set of sequential integers, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁)) | ||
Theorem | fzm1ne1 30514 | Elementhood of an integer and its predecessor in finite intervals of integers. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀...(𝑁 − 1))) | ||
Theorem | fzspl 30515 | Split the last element of a finite set of sequential integers. (more generic than fzsuc 12957) (Contributed by Thierry Arnoux, 7-Nov-2016.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) | ||
Theorem | fzdif2 30516 | Split the last element of a finite set of sequential integers. (more generic than fzsuc 12957) (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) | ||
Theorem | fzodif2 30517 | Split the last element of a half-open range of sequential integers. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀..^(𝑁 + 1)) ∖ {𝑁}) = (𝑀..^𝑁)) | ||
Theorem | fzodif1 30518 | Set difference of two half-open range of sequential integers sharing the same starting value. (Contributed by Thierry Arnoux, 2-Oct-2023.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝑀..^𝑁) ∖ (𝑀..^𝐾)) = (𝐾..^𝑁)) | ||
Theorem | fzsplit3 30519 | Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) | ||
Theorem | bcm1n 30520 | The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.) |
⊢ ((𝐾 ∈ (0...(𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (((𝑁 − 1)C𝐾) / (𝑁C𝐾)) = ((𝑁 − 𝐾) / 𝑁)) | ||
Theorem | iundisjfi 30521* | Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 24151. (Contributed by Thierry Arnoux, 15-Feb-2017.) |
⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ ∪ 𝑛 ∈ (1..^𝑁)𝐴 = ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
Theorem | iundisj2fi 30522* | A disjoint union is disjoint, finite version. Cf. iundisj2 24152. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ Disj 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
Theorem | iundisjcnt 30523* | Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀))) ⇒ ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) | ||
Theorem | iundisj2cnt 30524* | A countable disjoint union is disjoint. Cf. iundisj2 24152. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀))) ⇒ ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) | ||
Theorem | fzone1 30525 | Elementhood in a half-open interval, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)..^𝑁)) | ||
Theorem | fzom1ne1 30526 | Elementhood in a half-open interval, except the lower bound, shifted by one. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1))) | ||
Theorem | f1ocnt 30527* | Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with ℕ or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 30523 or iundisj2cnt 30524. (Contributed by Thierry Arnoux, 25-Jul-2020.) |
⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))) | ||
Theorem | fz1nnct 30528 | NN and integer ranges starting from 1 are countable. (Contributed by Thierry Arnoux, 25-Jul-2020.) |
⊢ ((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) → 𝐴 ≼ ω) | ||
Theorem | fz1nntr 30529 | NN and integer ranges starting from 1 are a transitive family of set. (Contributed by Thierry Arnoux, 25-Jul-2020.) |
⊢ (((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) ∧ 𝑁 ∈ 𝐴) → (1..^𝑁) ⊆ 𝐴) | ||
Theorem | hashunif 30530* | The cardinality of a disjoint finite union of finite sets. Cf. hashuni 15183. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) ⇒ ⊢ (𝜑 → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥)) | ||
Theorem | hashxpe 30531 | The size of the Cartesian product of two finite sets is the product of their sizes. This is a version of hashxp 13798 valid for infinite sets, which uses extended real numbers. (Contributed by Thierry Arnoux, 27-May-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) ·e (♯‘𝐵))) | ||
Theorem | hashgt1 30532 | Restate "set contains at least two elements" in terms of elementhood. (Contributed by Thierry Arnoux, 21-Nov-2023.) |
⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ (◡♯ “ {0, 1}) ↔ 1 < (♯‘𝐴))) | ||
Theorem | dvdszzq 30533 | Divisibility for an integer quotient. (Contributed by Thierry Arnoux, 17-Sep-2023.) |
⊢ 𝑁 = (𝐴 / 𝐵) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝑃 ∥ 𝐴) & ⊢ (𝜑 → ¬ 𝑃 ∥ 𝐵) ⇒ ⊢ (𝜑 → 𝑃 ∥ 𝑁) | ||
Theorem | prmdvdsbc 30534 | Condition for a prime number to divide a binomial coefficient. (Contributed by Thierry Arnoux, 17-Sep-2023.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑁)) | ||
Theorem | numdenneg 30535 | Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.) |
⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) | ||
Theorem | divnumden2 30536 | Calculate the reduced form of a quotient using gcd. This version extends divnumden 16090 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ -𝐵 ∈ ℕ) → ((numer‘(𝐴 / 𝐵)) = -(𝐴 / (𝐴 gcd 𝐵)) ∧ (denom‘(𝐴 / 𝐵)) = -(𝐵 / (𝐴 gcd 𝐵)))) | ||
Theorem | nnindf 30537* | Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ ℕ → 𝜏) | ||
Theorem | nn0min 30538* | Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12080. (Contributed by Thierry Arnoux, 6-May-2018.) |
⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒)) & ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃)) & ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏)) | ||
Theorem | subne0nn 30539 | A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.) |
⊢ (𝜑 → 𝑀 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℂ) & ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ≠ 𝑁) ⇒ ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℕ) | ||
Theorem | ltesubnnd 30540 | Subtracting an integer number from another number decreases it. See ltsubrpd 12466. (Contributed by Thierry Arnoux, 18-Apr-2017.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀) | ||
Theorem | fprodeq02 30541* | If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 = 0) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) | ||
Theorem | pr01ssre 30542 | The range of the indicator function is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.) |
⊢ {0, 1} ⊆ ℝ | ||
Theorem | fprodex01 30543* | A product of factors equal to zero or one is zero exactly when one of the factors is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
⊢ (𝑘 = 𝑙 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {0, 1}) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = if(∀𝑙 ∈ 𝐴 𝐶 = 1, 1, 0)) | ||
Theorem | prodpr 30544* | A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) & ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) | ||
Theorem | prodtp 30545* | A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) & ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺)) | ||
Theorem | fsumub 30546* | An upper bound for a term of a positive finite sum. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ (𝑘 = 𝐾 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐷 ≤ 𝐶) | ||
Theorem | fsumiunle 30547* | Upper bound for a sum of nonnegative terms over an indexed union. The inequality may be strict if the indexed union is non-disjoint, since in the right hand side, a summand may be counted several times. (Contributed by Thierry Arnoux, 1-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶) | ||
Theorem | dfdec100 30548 | Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) | ||
Define a decimal expansion constructor. The decimal expansions built with this constructor are not meant to be used alone outside of this chapter. Rather, they are meant to be used exclusively as part of a decimal number with a decimal fraction, for example (3._1_4_1_59). That decimal point operator is defined in the next section. The bulk of these constructions have originally been proposed by David A. Wheeler on 12-May-2015, and discussed with Mario Carneiro in this thread: https://groups.google.com/g/metamath/c/2AW7T3d2YiQ. | ||
Syntax | cdp2 30549 | Constant used for decimal fraction constructor. See df-dp2 30550. |
class _𝐴𝐵 | ||
Definition | df-dp2 30550 | Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12102. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | ||
Theorem | dp2eq1 30551 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶) | ||
Theorem | dp2eq2 30552 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵) | ||
Theorem | dp2eq1i 30553 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ _𝐴𝐶 = _𝐵𝐶 | ||
Theorem | dp2eq2i 30554 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ _𝐶𝐴 = _𝐶𝐵 | ||
Theorem | dp2eq12i 30555 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ _𝐴𝐶 = _𝐵𝐷 | ||
Theorem | dp20u 30556 | Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ _𝐴0 = 𝐴 | ||
Theorem | dp20h 30557 | Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ _0𝐴 = (𝐴 / ;10) | ||
Theorem | dp2cl 30558 | Closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ) | ||
Theorem | dp2clq 30559 | Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℚ ⇒ ⊢ _𝐴𝐵 ∈ ℚ | ||
Theorem | rpdp2cl 30560 | Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ _𝐴𝐵 ∈ ℝ+ | ||
Theorem | rpdp2cl2 30561 | Closure for a decimal fraction with no decimal expansion in the positive real numbers. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ _𝐴0 ∈ ℝ+ | ||
Theorem | dp2lt10 30562 | Decimal fraction builds real numbers less than 10. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐴 < ;10 & ⊢ 𝐵 < ;10 ⇒ ⊢ _𝐴𝐵 < ;10 | ||
Theorem | dp2lt 30563 | Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℝ+ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ _𝐴𝐵 < _𝐴𝐶 | ||
Theorem | dp2ltsuc 30564 | Comparing a decimal fraction with the next integer. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐵 < ;10 & ⊢ (𝐴 + 1) = 𝐶 ⇒ ⊢ _𝐴𝐵 < 𝐶 | ||
Theorem | dp2ltc 30565 | Comparing two decimal expansions (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ+ & ⊢ 𝐵 < ;10 & ⊢ 𝐴 < 𝐶 ⇒ ⊢ _𝐴𝐵 < _𝐶𝐷 | ||
Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 30567 and df-dp2 30550 for more information; dpval2 30571 and dpfrac1 30570 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12102. | ||
Syntax | cdp 30566 | Decimal point operator. See df-dp 30567. |
class . | ||
Definition | df-dp 30567* |
Define the . (decimal point) operator. For example,
(1.5) = (3 / 2), and
-(;32._7_18) =
-(;;;;32718 / ;;;1000)
Unary minus, if applied, should normally be applied in front of the
parentheses.
Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that Metamath has no built-in way to parse and handle decimal numbers as traditionally written, e.g., "2.54". Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers. The RHS is ℝ, not ℚ; this should simplify some proofs. The LHS is ℕ0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression -(𝐴.𝐵) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) | ||
Theorem | dpval 30568 | Define the value of the decimal point operator. See df-dp 30567. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | ||
Theorem | dpcl 30569 | Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 30567. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ) | ||
Theorem | dpfrac1 30570 | Prove a simple equivalence involving the decimal point. See df-dp 30567 and dpcl 30569. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) | ||
Theorem | dpval2 30571 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) | ||
Theorem | dpval3 30572 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
Theorem | dpmul10 30573 | Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵 | ||
Theorem | decdiv10 30574 | Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵) | ||
Theorem | dpmul100 30575 | Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 | ||
Theorem | dp3mul10 30576 | Multiply by 10 a decimal expansion with 3 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶) | ||
Theorem | dpmul1000 30577 | Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ ⇒ ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 | ||
Theorem | dpval3rp 30578 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
Theorem | dp0u 30579 | Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (𝐴.0) = 𝐴 | ||
Theorem | dp0h 30580 | Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ (0.𝐴) = (𝐴 / ;10) | ||
Theorem | rpdpcl 30581 | Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ (𝐴.𝐵) ∈ ℝ+ | ||
Theorem | dplt 30582 | Comparing two decimal expansions (equal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℝ+ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ (𝐴.𝐵) < (𝐴.𝐶) | ||
Theorem | dplti 30583 | Comparing a decimal expansions with the next higher integer. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐵 < ;10 & ⊢ (𝐴 + 1) = 𝐶 ⇒ ⊢ (𝐴.𝐵) < 𝐶 | ||
Theorem | dpgti 30584 | Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ 𝐴 < (𝐴.𝐵) | ||
Theorem | dpltc 30585 | Comparing two decimal integers (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ+ & ⊢ 𝐴 < 𝐶 & ⊢ 𝐵 < ;10 ⇒ ⊢ (𝐴.𝐵) < (𝐶.𝐷) | ||
Theorem | dpexpp1 30586 | Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ (𝑃 + 1) = 𝑄 & ⊢ 𝑃 ∈ ℤ & ⊢ 𝑄 ∈ ℤ ⇒ ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) | ||
Theorem | 0dp2dp 30587 | Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ ((0._𝐴𝐵) · ;10) = (𝐴.𝐵) | ||
Theorem | dpadd2 30588 | Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ+ & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℝ+ & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 & ⊢ (𝐺 + 𝐻) = 𝐼 & ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) ⇒ ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) | ||
Theorem | dpadd 30589 | Addition with one decimal. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ (;𝐴𝐵 + ;𝐶𝐷) = ;𝐸𝐹 ⇒ ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) | ||
Theorem | dpadd3 30590 | Addition with two decimals. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 & ⊢ 𝐼 ∈ ℕ0 & ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 ⇒ ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) | ||
Theorem | dpmul 30591 | Multiplication with one decimal point. (Contributed by Thierry Arnoux, 26-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐽 ∈ ℕ0 & ⊢ 𝐾 ∈ ℕ0 & ⊢ (𝐴 · 𝐶) = 𝐹 & ⊢ (𝐴 · 𝐷) = 𝑀 & ⊢ (𝐵 · 𝐶) = 𝐿 & ⊢ (𝐵 · 𝐷) = ;𝐸𝐾 & ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽 & ⊢ (𝐹 + 𝐺) = 𝐼 ⇒ ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾) | ||
Theorem | dpmul4 30592 | An upper bound to multiplication of decimal numbers with 4 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐽 ∈ ℕ0 & ⊢ 𝐾 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 & ⊢ 𝐼 ∈ ℕ0 & ⊢ 𝐿 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑂 ∈ ℕ0 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝑄 ∈ ℕ0 & ⊢ 𝑅 ∈ ℕ0 & ⊢ 𝑆 ∈ ℕ0 & ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝑈 ∈ ℕ0 & ⊢ 𝑊 ∈ ℕ0 & ⊢ 𝑋 ∈ ℕ0 & ⊢ 𝑌 ∈ ℕ0 & ⊢ 𝑍 ∈ ℕ0 & ⊢ 𝑈 < ;10 & ⊢ 𝑃 < ;10 & ⊢ 𝑄 < ;10 & ⊢ (;;𝐿𝑀𝑁 + 𝑂) = ;;;𝑅𝑆𝑇𝑈 & ⊢ ((𝐴.𝐵) · (𝐸.𝐹)) = (𝐼._𝐽𝐾) & ⊢ ((𝐶.𝐷) · (𝐺.𝐻)) = (𝑂._𝑃𝑄) & ⊢ (;;;𝐼𝐽𝐾1 + ;;𝑅𝑆𝑇) = ;;;𝑊𝑋𝑌𝑍 & ⊢ (((𝐴.𝐵) + (𝐶.𝐷)) · ((𝐸.𝐹) + (𝐺.𝐻))) = (((𝐼._𝐽𝐾) + (𝐿._𝑀𝑁)) + (𝑂._𝑃𝑄)) ⇒ ⊢ ((𝐴._𝐵_𝐶𝐷) · (𝐸._𝐹_𝐺𝐻)) < (𝑊._𝑋_𝑌𝑍) | ||
Theorem | threehalves 30593 | Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ (3 / 2) = (1.5) | ||
Theorem | 1mhdrd 30594 | Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
⊢ ((0._99) + (0._01)) = 1 | ||
Syntax | cxdiv 30595 | Extend class notation to include division of extended reals. |
class /𝑒 | ||
Definition | df-xdiv 30596* | Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥)) | ||
Theorem | xdivval 30597* | Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) | ||
Theorem | xrecex 30598* | Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1) | ||
Theorem | xmulcand 30599 | Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | xreceu 30600* | Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) |
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