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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.3.5.2  Ordering on reals - misc additions

Theoremlt2addrd 29501* 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 29502 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))

Theoremxaddeq0 29503 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 29504 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
inf(ℝ*, ℝ*, < ) = -∞

Theoremle2halvesd 29505 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 29506 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 29507 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 29508* 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 29509 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 29510* 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 29511 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 29512 Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))

Theoremxrge0subcld 29513 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 29514 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 29515* 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 29516* 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 29517 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
+ = (0(,)+∞)

Theoremxrofsup 29518 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 29519 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*, < ) ≠ -∞)

Theoremxrhaus 29520 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
(ordTop‘ ≤ ) ∈ Haus

20.3.5.4  Real number intervals - misc additions

Theoremjoiniooico 29521 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 29522 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 29523 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 29524 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 29525 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 29526 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 29527 Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
𝐴 ∈ ℝ*       (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴)

Theoremiocinif 29528 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 29529 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 29530 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))

20.3.5.5  Finite intervals of integers - misc additions

Theoremuzssico 29531 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 29532 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 29533 Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.)
(𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ‘(𝑁 + 1)))

Theoremssnnssfz 29534* 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))

Theoremfzspl 29535 Split the last element of a finite set of sequential integers. (more generic than fzsuc 12385) (Contributed by Thierry Arnoux, 7-Nov-2016.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))

Theoremfzdif2 29536 Split the last element of a finite set of sequential integers. (more generic than fzsuc 12385) (Contributed by Thierry Arnoux, 22-Aug-2020.)
(𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1)))

Theoremfzodif2 29537 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))

Theoremfzsplit3 29538 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 29539 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))

20.3.5.6  Half-open integer ranges - misc additions

Theoremiundisjfi 29540* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 23310. (Contributed by Thierry Arnoux, 15-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ (1..^𝑁)𝐴 = 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisj2fi 29541* A disjoint union is disjoint, finite version. Cf. iundisj2 23311. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisjcnt 29542* 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 29543* A countable disjoint union is disjoint. Cf. iundisj2 23311. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))       (𝜑Disj 𝑛𝑁 (𝐴 𝑘 ∈ (1..^𝑛)𝐵))

Theoremf1ocnt 29544* 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 29542 or iundisj2cnt 29543. (Contributed by Thierry Arnoux, 25-Jul-2020.)
(𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))

Theoremfz1nnct 29545 NN and integer ranges starting from 1 are countable. (Contributed by Thierry Arnoux, 25-Jul-2020.)
((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) → 𝐴 ≼ ω)

Theoremfz1nntr 29546 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)

20.3.5.7  The ` # ` (set size) function - misc additions

Theoremhashunif 29547* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 14552. (Contributed by Thierry Arnoux, 17-Feb-2017.)
𝑥𝜑    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ Fin)    &   (𝜑Disj 𝑥𝐴 𝑥)       (𝜑 → (#‘ 𝐴) = Σ𝑥𝐴 (#‘𝑥))

20.3.5.8  The greatest common divisor operator - misc. add

Theoremnumdenneg 29548 Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))

Theoremdivnumden2 29549 Calculate the reduced form of a quotient using gcd. This version extends divnumden 15450 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))))

20.3.5.9  Integers

Theoremnnindf 29550* 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 29551* 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 29552* Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 11469. (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))

Theoremltesubnnd 29553 Subtracting an integer number from another number decreases it. See ltsubrpd 11901. (Contributed by Thierry Arnoux, 18-Apr-2017.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)

Theoremfprodeq02 29554* 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 29555 The range of the indicator function is a subset of . (Contributed by Thierry Arnoux, 14-Aug-2017.)
{0, 1} ⊆ ℝ

Theoremfprodex01 29556* 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 29557* 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 29558* 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 29559* 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 29560* 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)

20.3.5.10  Decimal numbers

Theoremdfdec100 29561 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 29562 Constant used for decimal fraction constructor. See df-dp2 29563.
class 𝐴𝐵

Definitiondf-dp2 29563 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 11491. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))

Theoremdfdp2OLD 29564 Obsolete version of df-dp2 29563 as of 9-Sep-2021. (Contributed by David A. Wheeler, 15-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 = (𝐴 + (𝐵 / 10))

Theoremdp2eq1 29565 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Theoremdp2eq2 29566 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)

Theoremdp2eq1i 29567 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶

Theoremdp2eq2i 29568 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵

Theoremdp2eq12i 29569 Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷

Theoremdp20u 29570 Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       𝐴0 = 𝐴

Theoremdp20h 29571 Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       0𝐴 = (𝐴 / 10)

Theoremdp2cl 29572 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 29573 Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℚ       𝐴𝐵 ∈ ℚ

Theoremrpdp2cl 29574 Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴𝐵 ∈ ℝ+

Theoremrpdp2cl2 29575 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 29576 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 29577 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 29578 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 29579 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 29581 and df-dp2 29563 for more information; dpval2 29586 and dpfrac1 29584 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 11491.

Syntaxcdp 29580 Decimal point operator. See df-dp 29581.
class .

Definitiondf-dp 29581* 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 29582 Define the value of the decimal point operator. See df-dp 29581. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)

Theoremdpcl 29583 Prove that the closure of the decimal point is as we have defined it. See df-dp 29581. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)

Theoremdpfrac1 29584 Prove a simple equivalence involving the decimal point. See df-dp 29581 and dpcl 29583. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))

Theoremdpfrac1OLD 29585 Obsolete version of dpfrac1 29584 as of 9-Sep-2021. (Contributed by David A. Wheeler, 15-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = (𝐴𝐵 / 10))

Theoremdpval2 29586 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = (𝐴 + (𝐵 / 10))

Theoremdpval3 29587 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴.𝐵) = 𝐴𝐵

Theoremdpmul10 29588 Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       ((𝐴.𝐵) · 10) = 𝐴𝐵

Theoremdecdiv10 29589 Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ       (𝐴𝐵 / 10) = (𝐴.𝐵)

Theoremdpmul100 29590 Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℝ       ((𝐴.𝐵𝐶) · 100) = 𝐴𝐵𝐶

Theoremdp3mul10 29591 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)

Theoremdpmul1000 29592 Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℝ       ((𝐴.𝐵𝐶𝐷) · 1000) = 𝐴𝐵𝐶𝐷

Theoremdpval3rp 29593 Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) = 𝐴𝐵

Theoremdp0u 29594 Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0       (𝐴.0) = 𝐴

Theoremdp0h 29595 Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℝ+       (0.𝐴) = (𝐴 / 10)

Theoremrpdpcl 29596 Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       (𝐴.𝐵) ∈ ℝ+

Theoremdplt 29597 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)

Theoremdplti 29598 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

Theoremdpgti 29599 Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℝ+       𝐴 < (𝐴.𝐵)

Theoremdpltc 29600 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)

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