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Type | Label | Description |
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Statement | ||
Theorem | fsuppco2 9401 | 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 9402 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.) |
⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) | ||
Theorem | fsuppcor 9402 | 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.) |
⊢ (𝜑 → 0 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) & ⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) & ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) | ||
Theorem | mapfienlem1 9403* | Lemma 1 for mapfien 9406. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊) | ||
Theorem | mapfienlem2 9404* | Lemma 2 for mapfien 9406. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍) | ||
Theorem | mapfienlem3 9405* | Lemma 3 for mapfien 9406. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆) | ||
Theorem | mapfien 9406* | 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.) (Revised by AV, 28-Jul-2024.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ 𝑊 = (𝐺‘𝑍) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇) | ||
Theorem | mapfien2 9407* | Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.) |
⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 } & ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} & ⊢ (𝜑 → 𝐴 ≈ 𝐶) & ⊢ (𝜑 → 𝐵 ≈ 𝐷) & ⊢ (𝜑 → 0 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
Syntax | cfi 9408 | Extend class notation with the function whose value is the class of finite intersections of the elements of a given set. |
class fi | ||
Definition | df-fi 9409* | Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 9412). (Contributed by FL, 27-Apr-2008.) |
⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) | ||
Theorem | fival 9410* | 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.) |
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) | ||
Theorem | elfi 9411* | Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | ||
Theorem | elfi2 9412* | The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥)) | ||
Theorem | elfir 9413 | Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵)) | ||
Theorem | intrnfi 9414 | 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.) |
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵)) | ||
Theorem | iinfi 9415* | An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (fi‘𝐶)) | ||
Theorem | inelfi 9416 | The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ (fi‘𝑋)) | ||
Theorem | ssfii 9417 | 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.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | ||
Theorem | fi0 9418 | The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ (fi‘∅) = ∅ | ||
Theorem | fieq0 9419 | A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) | ||
Theorem | fiin 9420 | The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶)) | ||
Theorem | dffi2 9421* | 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.) |
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)}) | ||
Theorem | fiss 9422 | Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) | ||
Theorem | inficl 9423* | 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.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴)) | ||
Theorem | fipwuni 9424 | 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 | ||
Theorem | fisn 9425 | A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (fi‘{𝐴}) = {𝐴} | ||
Theorem | fiuni 9426 | 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.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴)) | ||
Theorem | fipwss 9427 | 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) | ||
Theorem | elfiun 9428* | 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.) |
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐾) → (𝐴 ∈ (fi‘(𝐵 ∪ 𝐶)) ↔ (𝐴 ∈ (fi‘𝐵) ∨ 𝐴 ∈ (fi‘𝐶) ∨ ∃𝑥 ∈ (fi‘𝐵)∃𝑦 ∈ (fi‘𝐶)𝐴 = (𝑥 ∩ 𝑦)))) | ||
Theorem | dffi3 9429* | 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 (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧))) ⇒ ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω)) | ||
Theorem | fifo 9430* | Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) | ||
Theorem | marypha1lem 9431* | 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 ≼ (𝑐 “ 𝑑) → ∃𝑒 ∈ 𝒫 𝑐𝑒:𝐴–1-1→V))) | ||
Theorem | marypha1 9432* | (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.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵)) & ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑)) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵) | ||
Theorem | marypha2lem1 9433* | Lemma for marypha2 9437. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹) | ||
Theorem | marypha2lem2 9434* | Lemma for marypha2 9437. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} | ||
Theorem | marypha2lem3 9435* | Lemma for marypha2 9437. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥))) | ||
Theorem | marypha2lem4 9436* | Lemma for marypha2 9437. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋)) | ||
Theorem | marypha2 9437* | Version of marypha1 9432 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶Fin) & ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ ∪ (𝐹 “ 𝑑)) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))) | ||
Syntax | csup 9438 | 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) | ||
Syntax | cinf 9439 | 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) | ||
Definition | df-sup 9440* | 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 15189. See dfsup2 9442 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.) |
⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))} | ||
Definition | df-inf 9441 | 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) | ||
Theorem | dfsup2 9442 | Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.) |
⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) | ||
Theorem | supeq1 9443 | Equality theorem for supremum. (Contributed by NM, 22-May-1999.) |
⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | ||
Theorem | supeq1d 9444 | Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | ||
Theorem | supeq1i 9445 | Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅) | ||
Theorem | supeq2 9446 | Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅)) | ||
Theorem | supeq3 9447 | Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆)) | ||
Theorem | supeq123d 9448 | Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹)) | ||
Theorem | nfsup 9449 | Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) | ||
Theorem | supmo 9450* | 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.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | supexd 9451 | A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | supeu 9452* | A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | supval2 9453* | Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))) | ||
Theorem | eqsup 9454* | Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶)) | ||
Theorem | eqsupd 9455* | Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | supcl 9456* | A supremum belongs to its base class (closure law). See also supub 9457 and suplub 9458. (Contributed by NM, 12-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) | ||
Theorem | supub 9457* |
A supremum is an upper bound. See also supcl 9456 and suplub 9458.
This proof demonstrates how to expand an iota-based definition (df-iota 6496) using riotacl2 7385. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) | ||
Theorem | suplub 9458* | A supremum is the least upper bound. See also supcl 9456 and supub 9457. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)) | ||
Theorem | suplub2 9459* | Bidirectional form of suplub 9458. (Contributed by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)) | ||
Theorem | supnub 9460* | An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅))) | ||
Theorem | supex 9461 | A supremum is a set. (Contributed by NM, 22-May-1999.) |
⊢ 𝑅 Or 𝐴 ⇒ ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V | ||
Theorem | sup00 9462 | The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
⊢ sup(𝐵, ∅, 𝑅) = ∅ | ||
Theorem | sup0riota 9463* | 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) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) | ||
Theorem | sup0 9464* | 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 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋) | ||
Theorem | supmax 9465* | 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.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | fisup2g 9466* | A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | fisupcl 9467 | A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | ||
Theorem | supgtoreq 9468 | The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅)) ⇒ ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) | ||
Theorem | suppr 9469 | The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶)) | ||
Theorem | supsn 9470 | The supremum of a singleton. (Contributed by NM, 2-Oct-2007.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵) | ||
Theorem | supisolem 9471* | Lemma for supiso 9473. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))) | ||
Theorem | supisoex 9472* | Lemma for supiso 9473. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))) | ||
Theorem | supiso 9473* | Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅))) | ||
Theorem | infeq1 9474 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | ||
Theorem | infeq1d 9475 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | ||
Theorem | infeq1i 9476 | Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅) | ||
Theorem | infeq2 9477 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) | ||
Theorem | infeq3 9478 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) | ||
Theorem | infeq123d 9479 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) | ||
Theorem | nfinf 9480 | Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) | ||
Theorem | infexd 9481 | An infimum is a set. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | eqinf 9482* | Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶)) | ||
Theorem | eqinfd 9483* | Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | infval 9484* | Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))) | ||
Theorem | infcllem 9485* | Lemma for infcl 9486, inflb 9487, infglb 9488, etc. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) | ||
Theorem | infcl 9486* | An infimum belongs to its base class (closure law). See also inflb 9487 and infglb 9488. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) | ||
Theorem | inflb 9487* | An infimum is a lower bound. See also infcl 9486 and infglb 9488. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))) | ||
Theorem | infglb 9488* | An infimum is the greatest lower bound. See also infcl 9486 and inflb 9487. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) | ||
Theorem | infglbb 9489* | Bidirectional form of infglb 9488. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) | ||
Theorem | infnlb 9490* | A lower bound is not greater than the infimum. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶)) | ||
Theorem | infex 9491 | An infimum is a set. (Contributed by AV, 3-Sep-2020.) |
⊢ 𝑅 Or 𝐴 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V | ||
Theorem | infmin 9492* | 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.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | infmo 9493* | Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | ||
Theorem | infeu 9494* | An infimum is unique. (Contributed by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | ||
Theorem | fimin2g 9495* | A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) | ||
Theorem | fiming 9496* | A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦)) | ||
Theorem | fiinfg 9497* | Lemma showing existence and closure of infimum of a finite set. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐴 𝑧𝑅𝑦))) | ||
Theorem | fiinf2g 9498* | A finite set satisfies the conditions to have an infimum. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | ||
Theorem | fiinfcl 9499 | A nonempty finite set contains its infimum. (Contributed by AV, 3-Sep-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) | ||
Theorem | infltoreq 9500 | The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅)) ⇒ ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
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