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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnmulge 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))
 
20.3.5.2  Ordering on reals - misc additions
 
Theoremlt2addrd 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵 + 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))
 
20.3.5.3  Extended reals - misc additions
 
Theoremxrlelttric 30403 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))
 
Theoremxaddeq0 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))
 
Theoremxrinfm 30405 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
inf(ℝ*, ℝ*, < ) = -∞
 
Theoremle2halvesd 30406 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐶 / 2))    &   (𝜑𝐵 ≤ (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) ≤ 𝐶)
 
Theoremxraddge02 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 ≤ 𝐵𝐴 ≤ (𝐴 +𝑒 𝐵)))
 
Theoremxrge0addge 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))
 
Theoremxlt2addrd 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ≠ -∞)    &   (𝜑𝐶 ≠ -∞)    &   (𝜑𝐴 < (𝐵 +𝑒 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ*𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))
 
Theoremxrsupssd 30410 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℝ*)       (𝜑 → sup(𝐵, ℝ*, < ) ≤ sup(𝐶, ℝ*, < ))
 
Theoremxrge0infss 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[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
 
Theoremxrge0infssd 30412 Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐶𝐵)    &   (𝜑𝐵 ⊆ (0[,]+∞))       (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < ))
 
Theoremxrge0addcld 30413 Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))
 
Theoremxrge0subcld 30414 Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))
 
Theoreminfxrge0lb 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.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵)
 
Theoreminfxrge0glb 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.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥𝐴 𝑥 < 𝐵))
 
Theoreminfxrge0gelb 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.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥𝐴 𝐵𝑥))
 
Theoremdfrp2 30418 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
+ = (0(,)+∞)
 
Theoremxrofsup 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.)
(𝜑𝑋 ⊆ ℝ*)    &   (𝜑𝑌 ⊆ ℝ*)    &   (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞)    &   (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞)    &   (𝜑𝑍 = ( +𝑒 “ (𝑋 × 𝑌)))       (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < )))
 
Theoremsupxrnemnf 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*, < ) ≠ -∞)
 
20.3.5.4  Extended nonnegative integers - misc additions
 
Theoremxnn0gt0 30421 Nonzero extended nonnegative integers are strictly greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
((𝑁 ∈ ℕ0*𝑁 ≠ 0) → 0 < 𝑁)
 
Theoremxnn01gt 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))
 
Theoremnn0xmulclb 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 ∈ ℕ0𝐵 ∈ ℕ0)))
 
20.3.5.5  Real number intervals - misc additions
 
Theoremjoiniooico 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 < 𝐵𝐵𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))
 
Theoremubico 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))
 
Theoremxeqlelt 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 ↔ (𝐴𝐵 ∧ ¬ 𝐴 < 𝐵)))
 
Theoremeliccelico 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*𝐴𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))
 
Theoremelicoelioo 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 = 𝐴𝐶 ∈ (𝐴(,)𝐵))))
 
Theoremiocinioc2 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*) ∧ 𝐴𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))
 
Theoremxrdifh 30430 Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
𝐴 ∈ ℝ*       (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴)
 
Theoremiocinif 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)))
 
Theoremdifioo 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))
 
Theoremdifico 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*) ∧ (𝐴𝐵𝐵𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))
 
20.3.5.6  Finite intervals of integers - misc additions
 
Theoremuzssico 30434 Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.)
(𝑀 ∈ ℤ → (ℤ𝑀) ⊆ (𝑀[,)+∞))
 
Theoremfz2ssnn0 30435 A finite set of sequential integers that is a subset of 0. (Contributed by Thierry Arnoux, 8-Dec-2021.)
(𝑀 ∈ ℕ0 → (𝑀...𝑁) ⊆ ℕ0)
 
Theoremnndiffz1 30436 Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.)
(𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ‘(𝑁 + 1)))
 
Theoremssnnssfz 30437* For any finite subset of , 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))
 
Theoremfzne1 30438 Elementhood in a finite set of sequential integers, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁))
 
Theoremfzm1ne1 30439 Elementhood of an integer and its predecessor in finite intervals of integers. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾𝑀) → (𝐾 − 1) ∈ (𝑀...(𝑁 − 1)))
 
Theoremfzspl 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 ∈ (ℤ𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
 
Theoremfzdif2 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 ∈ (ℤ𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1)))
 
Theoremfzodif2 30442 Split the last element of a half-open range of sequential integers. (Contributed by Thierry Arnoux, 5-Dec-2021.)
(𝑁 ∈ (ℤ𝑀) → ((𝑀..^(𝑁 + 1)) ∖ {𝑁}) = (𝑀..^𝑁))
 
Theoremfzodif1 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))
 
Theoremfzsplit3 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)))
 
Theorembcm1n 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)) = ((𝑁𝐾) / 𝑁))
 
20.3.5.7  Half-open integer ranges - misc additions
 
Theoremiundisjfi 30446* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 24078. (Contributed by Thierry Arnoux, 15-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ (1..^𝑁)𝐴 = 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremiundisj2fi 30447* A disjoint union is disjoint, finite version. Cf. iundisj2 24079. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremiundisjcnt 30448* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))       (𝜑 𝑛𝑁 𝐴 = 𝑛𝑁 (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
 
Theoremiundisj2cnt 30449* A countable disjoint union is disjoint. Cf. iundisj2 24079. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))       (𝜑Disj 𝑛𝑁 (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
 
Theoremfzone1 30450 Elementhood in a half-open interval, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾𝑀) → 𝐾 ∈ ((𝑀 + 1)..^𝑁))
 
Theoremfzom1ne1 30451 Elementhood in a half-open interval, except the lower bound, shifted by one. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1)))
 
Theoremf1ocnt 30452* Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with 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 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
 
Theoremfz1nnct 30453 NN and integer ranges starting from 1 are countable. (Contributed by Thierry Arnoux, 25-Jul-2020.)
((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) → 𝐴 ≼ ω)
 
Theoremfz1nntr 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)) ∧ 𝑁𝐴) → (1..^𝑁) ⊆ 𝐴)
 
20.3.5.8  The ` # ` (set size) function - misc additions
 
Theoremhashunif 30455* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 15171. (Contributed by Thierry Arnoux, 17-Feb-2017.)
𝑥𝜑    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ Fin)    &   (𝜑Disj 𝑥𝐴 𝑥)       (𝜑 → (♯‘ 𝐴) = Σ𝑥𝐴 (♯‘𝑥))
 
Theoremhashxpe 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.)
((𝐴𝑉𝐵𝑊) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) ·e (♯‘𝐵)))
 
Theoremhashgt1 30457 Restate "set contains at least two elements" in terms of elementhood. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝐴)))
 
20.3.5.9  The greatest common divisor operator - misc. add
 
Theoremdvdszzq 30458 Divisibility for an integer quotient. (Contributed by Thierry Arnoux, 17-Sep-2023.)
𝑁 = (𝐴 / 𝐵)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑃𝐴)    &   (𝜑 → ¬ 𝑃𝐵)       (𝜑𝑃𝑁)
 
Theoremprmdvdsbc 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))
 
Theoremnumdenneg 30460 Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))
 
Theoremdivnumden2 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))))
 
20.3.5.10  Integers
 
Theoremnnindf 30462* Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
𝑦𝜑    &   (𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ → (𝜒𝜃))       (𝐴 ∈ ℕ → 𝜏)
 
Theoremnnindd 30463* Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝑥 = 1 → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 + 1) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   (((𝜑𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏)       ((𝜑𝐴 ∈ ℕ) → 𝜂)
 
Theoremnn0min 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 → (𝜓𝜒))    &   (𝑛 = 𝑚 → (𝜓𝜃))    &   (𝑛 = (𝑚 + 1) → (𝜓𝜏))    &   (𝜑 → ¬ 𝜒)    &   (𝜑 → ∃𝑛 ∈ ℕ 𝜓)       (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))
 
Theoremsubne0nn 30465 A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.)
(𝜑𝑀 ∈ ℂ)    &   (𝜑𝑁 ∈ ℂ)    &   (𝜑 → (𝑀𝑁) ∈ ℕ0)    &   (𝜑𝑀𝑁)       (𝜑 → (𝑀𝑁) ∈ ℕ)
 
Theoremltesubnnd 30466 Subtracting an integer number from another number decreases it. See ltsubrpd 12453. (Contributed by Thierry Arnoux, 18-Apr-2017.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)
 
Theoremfprodeq02 30467* If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝑘 = 𝐾𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐾𝐴)    &   (𝜑𝐶 = 0)       (𝜑 → ∏𝑘𝐴 𝐵 = 0)
 
Theorempr01ssre 30468 The range of the indicator function is a subset of . (Contributed by Thierry Arnoux, 14-Aug-2017.)
{0, 1} ⊆ ℝ
 
Theoremfprodex01 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 = 𝑙𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ {0, 1})       (𝜑 → ∏𝑘𝐴 𝐵 = if(∀𝑙𝐴 𝐶 = 1, 1, 0))
 
Theoremprodpr 30470* A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐹 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹))
 
Theoremprodtp 30471* A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐹 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   (𝑘 = 𝐶𝐷 = 𝐺)    &   (𝜑𝐶𝑋)    &   (𝜑𝐺 ∈ ℂ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺))
 
Theoremfsumub 30472* An upper bound for a term of a positive finite sum. (Contributed by Thierry Arnoux, 27-Dec-2021.)
(𝑘 = 𝐾𝐵 = 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → Σ𝑘𝐴 𝐵 = 𝐶)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)    &   (𝜑𝐾𝐴)       (𝜑𝐷𝐶)
 
Theoremfsumiunle 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 0 ≤ 𝐶)       (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 ≤ Σ𝑥𝐴 Σ𝑘𝐵 𝐶)
 
20.3.5.11  Decimal numbers
 
Theoremdfdec100 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)
 
20.3.6  Decimal expansion

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 (3.14159).

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.

 
Syntaxcdp2 30475 Constant used for decimal fraction constructor. See df-dp2 30476.
class 𝐴𝐵
 
Definitiondf-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))
 
Theoremdp2eq1 30477 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 
Theoremdp2eq2 30478 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 
Theoremdp2eq1i 30479 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶
 
Theoremdp2eq2i 30480 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵
 
Theoremdp2eq12i 30481 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷
 
Theoremdp20u 30482 Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       𝐴0 = 𝐴
 
Theoremdp20h 30483 Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       0𝐴 = (𝐴 / 10)
 
Theoremdp2cl 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)
 
Theoremdp2clq 30485 Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℚ       𝐴𝐵 ∈ ℚ
 
Theoremrpdp2cl 30486 Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴𝐵 ∈ ℝ+
 
Theoremrpdp2cl2 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+
 
Theoremdp2lt10 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
 
Theoremdp2lt 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
 
Theoremdp2ltsuc 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
 
Theoremdp2ltc 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
 
20.3.6.1  Decimal point

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.

 
Syntaxcdp 30492 Decimal point operator. See df-dp 30493.
class .
 
Definitiondf-dp 30493* Define the . (decimal point) operator. For example, (1.5) = (3 / 2), and -(32.718) = -(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 , not ; this should simplify some proofs. The LHS is 0, 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)
 
Theoremdpval 30494 Define the value of the decimal point operator. See df-dp 30493. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
 
Theoremdpcl 30495 Prove that the closure of the decimal point is as we have defined it. See df-dp 30493. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
 
Theoremdpfrac1 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 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))
 
Theoremdpval2 30497 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = (𝐴 + (𝐵 / 10))
 
Theoremdpval3 30498 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = 𝐴𝐵
 
Theoremdpmul10 30499 Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       ((𝐴.𝐵) · 10) = 𝐴𝐵
 
Theoremdecdiv10 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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