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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnmulge 30401 | Multiplying by a positive integer 𝑀 yields greater than or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑀 · 𝑁)) | ||
Theorem | lt2addrd 30402* | If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵 ∧ 𝑐 < 𝐶)) | ||
Theorem | xrlelttric 30403 | Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | xaddeq0 30404 | Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) = 0 ↔ 𝐴 = -𝑒𝐵)) | ||
Theorem | xrinfm 30405 | The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.) |
⊢ inf(ℝ*, ℝ*, < ) = -∞ | ||
Theorem | le2halvesd 30406 | A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ (𝐶 / 2)) & ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 2)) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ≤ 𝐶) | ||
Theorem | xraddge02 30407 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +𝑒 𝐵))) | ||
Theorem | xrge0addge 30408 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 𝐴 ≤ (𝐴 +𝑒 𝐵)) | ||
Theorem | xlt2addrd 30409* | If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ≠ -∞) & ⊢ (𝜑 → 𝐶 ≠ -∞) & ⊢ (𝜑 → 𝐴 < (𝐵 +𝑒 𝐶)) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ℝ* ∃𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵 ∧ 𝑐 < 𝐶)) | ||
Theorem | xrsupssd 30410 | Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐶) & ⊢ (𝜑 → 𝐶 ⊆ ℝ*) ⇒ ⊢ (𝜑 → sup(𝐵, ℝ*, < ) ≤ sup(𝐶, ℝ*, < )) | ||
Theorem | xrge0infss 30411* | Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | ||
Theorem | xrge0infssd 30412 | Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ (0[,]+∞)) ⇒ ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) | ||
Theorem | xrge0addcld 30413 | Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.) |
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞)) | ||
Theorem | xrge0subcld 30414 | Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.) |
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) | ||
Theorem | infxrge0lb 30415 | A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) | ||
Theorem | infxrge0glb 30416* | The infimum of a set of nonnegative extended reals is the greatest lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) | ||
Theorem | infxrge0gelb 30417* | The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) | ||
Theorem | dfrp2 30418 | Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.) |
⊢ ℝ+ = (0(,)+∞) | ||
Theorem | xrofsup 30419 | The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.) |
⊢ (𝜑 → 𝑋 ⊆ ℝ*) & ⊢ (𝜑 → 𝑌 ⊆ ℝ*) & ⊢ (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞) & ⊢ (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞) & ⊢ (𝜑 → 𝑍 = ( +𝑒 “ (𝑋 × 𝑌))) ⇒ ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < ))) | ||
Theorem | supxrnemnf 30420 | The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞) | ||
Theorem | xnn0gt0 30421 | Nonzero extended nonnegative integers are strictly greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
⊢ ((𝑁 ∈ ℕ0* ∧ 𝑁 ≠ 0) → 0 < 𝑁) | ||
Theorem | xnn01gt 30422 | An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than 1. (Contributed by Thierry Arnoux, 21-Nov-2023.) |
⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ {0, 1} ↔ 1 < 𝑁)) | ||
Theorem | nn0xmulclb 30423 | Finite multiplication in the extended nonnegative integers. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 ·e 𝐵) ∈ ℕ0 ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0))) | ||
Theorem | joiniooico 30424 | Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶))) | ||
Theorem | ubico 30425 | A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴[,)𝐵)) | ||
Theorem | xeqlelt 30426 | 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 30427 | 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 30428 | 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 30429 | 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 30430 | 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 30431 | 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 30432 | 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 30433 | 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 30434 | 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 30435 | 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 30436 | Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.) |
⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1))) | ||
Theorem | ssnnssfz 30437* | 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 30438 | 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 30439 | 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 30440 | Split the last element of a finite set of sequential integers. (more generic than fzsuc 12944) (Contributed by Thierry Arnoux, 7-Nov-2016.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) | ||
Theorem | fzdif2 30441 | Split the last element of a finite set of sequential integers. (more generic than fzsuc 12944) (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) | ||
Theorem | fzodif2 30442 | 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 30443 | 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 30444 | 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 30445 | 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 30446* | Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 24078. (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 30447* | A disjoint union is disjoint, finite version. Cf. iundisj2 24079. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ Disj 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
Theorem | iundisjcnt 30448* | 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 30449* | A countable disjoint union is disjoint. Cf. iundisj2 24079. (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 30450 | 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 30451 | 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 30452* | 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 30448 or iundisj2cnt 30449. (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 30453 | 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 30454 | 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 30455* | The cardinality of a disjoint finite union of finite sets. Cf. hashuni 15171. (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 30456 | The size of the Cartesian product of two finite sets is the product of their sizes. This is a version of hashxp 13785 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 30457 | 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 30458 | 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 30459 | 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 30460 | 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 30461 | Calculate the reduced form of a quotient using gcd. This version extends divnumden 16078 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 30462* | 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 | nnindd 30463* | Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
⊢ (𝑥 = 1 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂) | ||
Theorem | nn0min 30464* | Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12066. (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 30465 | 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 30466 | Subtracting an integer number from another number decreases it. See ltsubrpd 12453. (Contributed by Thierry Arnoux, 18-Apr-2017.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀) | ||
Theorem | fprodeq02 30467* | 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 30468 | The range of the indicator function is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.) |
⊢ {0, 1} ⊆ ℝ | ||
Theorem | fprodex01 30469* | 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 30470* | 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 30471* | 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 30472* | 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 30473* | 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 30474 | 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 30475 | Constant used for decimal fraction constructor. See df-dp2 30476. |
class _𝐴𝐵 | ||
Definition | df-dp2 30476 | Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12088. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | ||
Theorem | dp2eq1 30477 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶) | ||
Theorem | dp2eq2 30478 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵) | ||
Theorem | dp2eq1i 30479 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ _𝐴𝐶 = _𝐵𝐶 | ||
Theorem | dp2eq2i 30480 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ _𝐶𝐴 = _𝐶𝐵 | ||
Theorem | dp2eq12i 30481 | Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ _𝐴𝐶 = _𝐵𝐷 | ||
Theorem | dp20u 30482 | Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ _𝐴0 = 𝐴 | ||
Theorem | dp20h 30483 | Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ _0𝐴 = (𝐴 / ;10) | ||
Theorem | dp2cl 30484 | 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 30485 | Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℚ ⇒ ⊢ _𝐴𝐵 ∈ ℚ | ||
Theorem | rpdp2cl 30486 | 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 30487 | 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 30488 | 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 30489 | 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 30490 | 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 30491 | 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 30493 and df-dp2 30476 for more information; dpval2 30497 and dpfrac1 30496 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12088. | ||
Syntax | cdp 30492 | Decimal point operator. See df-dp 30493. |
class . | ||
Definition | df-dp 30493* |
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 30494 | Define the value of the decimal point operator. See df-dp 30493. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | ||
Theorem | dpcl 30495 | Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 30493. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ) | ||
Theorem | dpfrac1 30496 | Prove a simple equivalence involving the decimal point. See df-dp 30493 and dpcl 30495. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) | ||
Theorem | dpval2 30497 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) | ||
Theorem | dpval3 30498 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
Theorem | dpmul10 30499 | Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵 | ||
Theorem | decdiv10 30500 | Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵) |
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