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Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrnfsuppbi 8301 Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.)
((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 finSupp 𝑍 ↔ (𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)))
 
Theoremfsuppmptif 8302* A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹 finSupp 𝑍)       (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 𝑍)) finSupp 𝑍)
 
Theoremfsuppcolem 8303 Lemma for fsuppco 8304. Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
(𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)    &   (𝜑𝐺:𝑋1-1𝑌)       (𝜑 → ((𝐹𝐺) “ (V ∖ {𝑍})) ∈ Fin)
 
Theoremfsuppco 8304 The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.)
(𝜑𝐹 finSupp 𝑍)    &   (𝜑𝐺:𝑋1-1𝑌)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹𝑉)       (𝜑 → (𝐹𝐺) finSupp 𝑍)
 
Theoremfsuppco2 8305 The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8306 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)
(𝜑𝑍𝑊)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐵)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐹 finSupp 𝑍)    &   (𝜑 → (𝐺𝑍) = 𝑍)       (𝜑 → (𝐺𝐹) finSupp 𝑍)
 
Theoremfsuppcor 8306 The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)
(𝜑0𝑊)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐷)    &   (𝜑𝐶𝐵)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝐹 finSupp 𝑍)    &   (𝜑 → (𝐺𝑍) = 0 )       (𝜑 → (𝐺𝐹) finSupp 0 )
 
Theoremmapfienlem1 8307* Lemma 1 for mapfien 8310. (Contributed by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       ((𝜑𝑓𝑆) → (𝐺 ∘ (𝑓𝐹)) finSupp 𝑊)
 
Theoremmapfienlem2 8308* Lemma 2 for mapfien 8310. (Contributed by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       ((𝜑𝑔𝑇) → ((𝐺𝑔) ∘ 𝐹) finSupp 𝑍)
 
Theoremmapfienlem3 8309* Lemma 3 for mapfien 8310. (Contributed by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       ((𝜑𝑔𝑇) → ((𝐺𝑔) ∘ 𝐹) ∈ 𝑆)
 
Theoremmapfien 8310* A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺𝑍)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   (𝜑𝐺:𝐵1-1-onto𝐷)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐷 ∈ V)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑓𝑆 ↦ (𝐺 ∘ (𝑓𝐹))):𝑆1-1-onto𝑇)
 
Theoremmapfien2 8311* Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
𝑆 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 0 }    &   𝑇 = {𝑥 ∈ (𝐷𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑0𝐵)    &   (𝜑𝑊𝐷)       (𝜑𝑆𝑇)
 
Theoremsniffsupp 8312* A function mapping all but one arguments to zero is finitely supported. (Contributed by AV, 8-Jul-2019.)
(𝜑𝐼𝑉)    &   (𝜑0𝑊)    &   𝐹 = (𝑥𝐼 ↦ if(𝑥 = 𝑋, 𝐴, 0 ))       (𝜑𝐹 finSupp 0 )
 
2.4.29  Finite intersections
 
Syntaxcfi 8313 Extend class notation with the function whose value is the class of all the finite intersections of the elements of a given set.
class fi
 
Definitiondf-fi 8314* Function whose value is the class of all the finite intersections of the elements of 𝑥. (Contributed by FL, 27-Apr-2008.)
fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})
 
Theoremfival 8315* The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
 
Theoremelfi 8316* Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
 
Theoremelfi2 8317* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
 
Theoremelfir 8318 Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉 ∧ (𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (fi‘𝐵))
 
Theoremintrnfi 8319 Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
 
Theoremiinfi 8320* An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐶𝑉 ∧ (∀𝑥𝐴 𝐵𝐶𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝑥𝐴 𝐵 ∈ (fi‘𝐶))
 
Theoreminelfi 8321 The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ (fi‘𝑋))
 
Theoremssfii 8322 Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(𝐴𝑉𝐴 ⊆ (fi‘𝐴))
 
Theoremfi0 8323 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘∅) = ∅
 
Theoremfieq0 8324 If 𝐴 is not empty, the class of all the finite intersections of 𝐴 is not empty either. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
 
Theoremfiin 8325 The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴𝐵) ∈ (fi‘𝐶))
 
Theoremdffi2 8326* The set of finite intersections is the smallest set that contains 𝐴 and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
 
Theoremfiss 8327 Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉𝐴𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))
 
Theoreminficl 8328* A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
 
Theoremfipwuni 8329 The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(fi‘𝐴) ⊆ 𝒫 𝐴
 
Theoremfisn 8330 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
(fi‘{𝐴}) = {𝐴}
 
Theoremfiuni 8331 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 𝐴 = (fi‘𝐴))
 
Theoremfipwss 8332 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
(𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
 
Theoremelfiun 8333* A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.)
((𝐵𝐷𝐶𝐾) → (𝐴 ∈ (fi‘(𝐵𝐶)) ↔ (𝐴 ∈ (fi‘𝐵) ∨ 𝐴 ∈ (fi‘𝐶) ∨ ∃𝑥 ∈ (fi‘𝐵)∃𝑦 ∈ (fi‘𝐶)𝐴 = (𝑥𝑦))))
 
Theoremdffi3 8334* The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))       (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
 
Theoremfifo 8335* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)       (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
 
2.4.30  Hall's marriage theorem
 
Theoremmarypha1lem 8336* Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)
(𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐𝑑) → ∃𝑒 ∈ 𝒫 𝑐𝑒:𝐴1-1→V)))
 
Theoremmarypha1 8337* (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pigeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶 ⊆ (𝐴 × 𝐵))    &   ((𝜑𝑑𝐴) → 𝑑 ≼ (𝐶𝑑))       (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴1-1𝐵)
 
Theoremmarypha2lem1 8338* Lemma for marypha2 8342. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       𝑇 ⊆ (𝐴 × ran 𝐹)
 
Theoremmarypha2lem2 8339* Lemma for marypha2 8342. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
 
Theoremmarypha2lem3 8340* Lemma for marypha2 8342. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
 
Theoremmarypha2lem4 8341* Lemma for marypha2 8342. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
 
Theoremmarypha2 8342* Version of marypha1 8337 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴⟶Fin)    &   ((𝜑𝑑𝐴) → 𝑑 (𝐹𝑑))       (𝜑 → ∃𝑔(𝑔:𝐴1-1→V ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ (𝐹𝑥)))
 
2.4.31  Supremum and infimum
 
Syntaxcsup 8343 Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
 
Syntaxcinf 8344 Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
 
Definitiondf-sup 8345* Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 13971. See dfsup2 8347 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.)
sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
 
Definitiondf-inf 8346 Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
 
Theoremdfsup2 8347 Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
 
Theoremsupeq1 8348 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
(𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1d 8349 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐵 = 𝐶)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1i 8350 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐵 = 𝐶       sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
 
Theoremsupeq2 8351 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
 
Theoremsupeq3 8352 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
 
Theoremsupeq123d 8353 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
 
Theoremnfsup 8354 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥sup(𝐴, 𝐵, 𝑅)
 
Theoremsupmo 8355* Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremsupexd 8356 A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝜑𝑅 Or 𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
 
Theoremsupeu 8357* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremsupval2 8358* Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.)
(𝜑𝑅 Or 𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
 
Theoremeqsup 8359* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
 
Theoremeqsupd 8360* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)    &   ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoremsupcl 8361* A supremum belongs to its base class (closure law). See also supub 8362 and suplub 8363. (Contributed by NM, 12-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
 
Theoremsupub 8362* A supremum is an upper bound. See also supcl 8361 and suplub 8363.

This proof demonstrates how to expand an iota-based definition (df-iota 5849) using riotacl2 6621.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
 
Theoremsuplub 8363* A supremum is the least upper bound. See also supcl 8361 and supub 8362. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
 
Theoremsuplub2 8364* Bidirectional form of suplub 8363. (Contributed by Mario Carneiro, 6-Sep-2014.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
 
Theoremsupnub 8365* An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
 
Theoremsupex 8366 A supremum is a set. (Contributed by NM, 22-May-1999.)
𝑅 Or 𝐴       sup(𝐵, 𝐴, 𝑅) ∈ V
 
Theoremsup00 8367 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐵, ∅, 𝑅) = ∅
 
Theoremsup0riota 8368* The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
(𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
 
Theoremsup0 8369* The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
 
Theoremsupmax 8370* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoremfisup2g 8371* A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremfisupcl 8372 A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)
((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
 
Theoremsupgtoreq 8373 The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶𝐵)    &   (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))       (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
 
Theoremsuppr 8374 The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
 
Theoremsupsn 8375 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
 
Theoremsupisolem 8376* Lemma for supiso 8378. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)       ((𝜑𝐷𝐴) → ((∀𝑦𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐷 → ∃𝑧𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝐷)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝐷) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
 
Theoremsupisoex 8377* Lemma for supiso 8378. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
 
Theoremsupiso 8378* Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))    &   (𝜑𝑅 Or 𝐴)       (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
 
Theoreminfeq1 8379 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 
Theoreminfeq1d 8380 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 
Theoreminfeq1i 8381 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐵 = 𝐶       inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)
 
Theoreminfeq2 8382 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
 
Theoreminfeq3 8383 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
 
Theoreminfeq123d 8384 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
 
Theoremnfinf 8385 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥inf(𝐴, 𝐵, 𝑅)
 
Theoreminfexd 8386 An infimum is a set. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
Theoremeqinf 8387* Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
 
Theoremeqinfd 8388* Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)    &   ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoreminfval 8389* Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
 
Theoreminfcllem 8390* Lemma for infcl 8391, inflb 8392, infglb 8393, etc. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoreminfcl 8391* An infimum belongs to its base class (closure law). See also inflb 8392 and infglb 8393. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
 
Theoreminflb 8392* An infimum is a lower bound. See also infcl 8391 and infglb 8393. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
 
Theoreminfglb 8393* An infimum is the greatest lower bound. See also infcl 8391 and inflb 8392. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))
 
Theoreminfglbb 8394* Bidirectional form of infglb 8393. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧𝐵 𝑧𝑅𝐶))
 
Theoreminfnlb 8395* A lower bound is not greater than the infimum. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
 
Theoreminfex 8396 An infimum is a set. (Contributed by AV, 3-Sep-2020.)
𝑅 Or 𝐴       inf(𝐵, 𝐴, 𝑅) ∈ V
 
Theoreminfmin 8397* The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoreminfmo 8398* Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by AV, 6-Oct-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
 
Theoreminfeu 8399* An infimum is unique. (Contributed by AV, 6-Oct-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
 
Theoremfimin2g 8400* A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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