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Type | Label | Description |
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Statement | ||
Theorem | xpsaddlem 17501* | Lemma for xpsadd 17502 and xpsmul 17503. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋) & ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌) & ⊢ · = (𝐸‘𝑅) & ⊢ × = (𝐸‘𝑆) & ⊢ ∙ = (𝐸‘𝑇) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) & ⊢ 𝑈 = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) & ⊢ ((𝜑 ∧ {〈∅, 𝐴〉, 〈1o, 𝐵〉} ∈ ran 𝐹 ∧ {〈∅, 𝐶〉, 〈1o, 𝐷〉} ∈ ran 𝐹) → ((◡𝐹‘{〈∅, 𝐴〉, 〈1o, 𝐵〉}) ∙ (◡𝐹‘{〈∅, 𝐶〉, 〈1o, 𝐷〉})) = (◡𝐹‘({〈∅, 𝐴〉, 〈1o, 𝐵〉} (𝐸‘𝑈){〈∅, 𝐶〉, 〈1o, 𝐷〉}))) & ⊢ (({〈∅, 𝑅〉, 〈1o, 𝑆〉} Fn 2o ∧ {〈∅, 𝐴〉, 〈1o, 𝐵〉} ∈ (Base‘𝑈) ∧ {〈∅, 𝐶〉, 〈1o, 𝐷〉} ∈ (Base‘𝑈)) → ({〈∅, 𝐴〉, 〈1o, 𝐵〉} (𝐸‘𝑈){〈∅, 𝐶〉, 〈1o, 𝐷〉}) = (𝑘 ∈ 2o ↦ (({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘𝑘)(𝐸‘({〈∅, 𝑅〉, 〈1o, 𝑆〉}‘𝑘))({〈∅, 𝐶〉, 〈1o, 𝐷〉}‘𝑘)))) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∙ 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | ||
Theorem | xpsadd 17502 | Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋) & ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌) & ⊢ · = (+g‘𝑅) & ⊢ × = (+g‘𝑆) & ⊢ ∙ = (+g‘𝑇) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∙ 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | ||
Theorem | xpsmul 17503 | Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋) & ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) & ⊢ ∙ = (.r‘𝑇) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∙ 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | ||
Theorem | xpssca 17504 | Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐺 = (Scalar‘𝑇)) | ||
Theorem | xpsvsca 17505 | Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ × = ( ·𝑠 ‘𝑆) & ⊢ ∙ = ( ·𝑠 ‘𝑇) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝐶 ∈ 𝑌) & ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋) & ⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝐴 ∙ 〈𝐵, 𝐶〉) = 〈(𝐴 · 𝐵), (𝐴 × 𝐶)〉) | ||
Theorem | xpsless 17506 | Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ ≤ = (le‘𝑇) ⇒ ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌))) | ||
Theorem | xpsle 17507 | Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ ≤ = (le‘𝑇) & ⊢ 𝑀 = (le‘𝑅) & ⊢ 𝑁 = (le‘𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ≤ 〈𝐶, 𝐷〉 ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷))) | ||
Syntax | cmre 17508 | The class of Moore systems. |
class Moore | ||
Syntax | cmrc 17509 | The class function generating Moore closures. |
class mrCls | ||
Syntax | cmri 17510 | mrInd is a class function which takes a Moore system to its set of independent sets. |
class mrInd | ||
Syntax | cacs 17511 | The class of algebraic closure (Moore) systems. |
class ACS | ||
Definition | df-mre 17512* |
Define a Moore collection, which is a family of subsets of a base set
which preserve arbitrary intersection. Elements of a Moore collection
are termed closed; Moore collections generalize the notion of
closedness from topologies (cldmre 22511) and vector spaces (lssmre 20526)
to the most general setting in which such concepts make sense.
Definition of Moore collection of sets in [Schechter] p. 78. A Moore
collection may also be called a closure system (Section 0.6 in
[Gratzer] p. 23.) The name Moore
collection is after Eliakim Hastings
Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.
See ismre 17516, mresspw 17518, mre1cl 17520 and mreintcl 17521 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17526); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17527. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.) |
⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) | ||
Definition | df-mrc 17513* |
Define the Moore closure of a generating set, which is the smallest
closed set containing all generating elements. Definition of Moore
closure in [Schechter] p. 79. This
generalizes topological closure
(mrccls 22512) and linear span (mrclsp 20549).
A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcssid 17543), (2) isotone, i.e., 𝑥 ⊆ 𝑦 implies that (𝑁‘𝑥) ⊆ (𝑁‘𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 17542), and (3) idempotent, i.e., (𝑁‘(𝑁‘𝑥)) = (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcidm 17545.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation 𝑁 on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.) |
⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠})) | ||
Definition | df-mri 17514* | In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.) |
⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))}) | ||
Definition | df-acs 17515* | An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 9620 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}) | ||
Theorem | ismre 17516* | Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶))) | ||
Theorem | fnmre 17517 | The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22355 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ Moore Fn V | ||
Theorem | mresspw 17518 | A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) | ||
Theorem | mress 17519 | A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋) | ||
Theorem | mre1cl 17520 | In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | ||
Theorem | mreintcl 17521 | A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) | ||
Theorem | mreiincl 17522* | A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) | ||
Theorem | mrerintcl 17523 | The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶) | ||
Theorem | mreriincl 17524* | The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶) | ||
Theorem | mreincl 17525 | Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶) | ||
Theorem | mreuni 17526 | Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) | ||
Theorem | mreunirn 17527 | Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ ∪ ran Moore ↔ 𝐶 ∈ (Moore‘∪ 𝐶)) | ||
Theorem | ismred 17528* | Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋) & ⊢ (𝜑 → 𝑋 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) | ||
Theorem | ismred2 17529* | Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋)) | ||
Theorem | mremre 17530 | The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋)) | ||
Theorem | submre 17531 | The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴)) | ||
Theorem | mrcflem 17532* | The domain and codomain of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶) | ||
Theorem | fnmrc 17533 | Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ mrCls Fn ∪ ran Moore | ||
Theorem | mrcfval 17534* | Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})) | ||
Theorem | mrcf 17535 | The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶) | ||
Theorem | mrcval 17536* | Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) | ||
Theorem | mrccl 17537 | The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶) | ||
Theorem | mrcsncl 17538 | The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝑋) → (𝐹‘{𝑈}) ∈ 𝐶) | ||
Theorem | mrcid 17539 | The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈) | ||
Theorem | mrcssv 17540 | The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋) | ||
Theorem | mrcidb 17541 | A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈)) | ||
Theorem | mrcss 17542 | Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉)) | ||
Theorem | mrcssid 17543 | The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝐹‘𝑈)) | ||
Theorem | mrcidb2 17544 | A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈)) | ||
Theorem | mrcidm 17545 | The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘(𝐹‘𝑈)) = (𝐹‘𝑈)) | ||
Theorem | mrcsscl 17546 | The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉) | ||
Theorem | mrcuni 17547 | Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑈) = (𝐹‘∪ (𝐹 “ 𝑈))) | ||
Theorem | mrcun 17548 | Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉)))) | ||
Theorem | mrcssvd 17549 | The Moore closure of a set is a subset of the base. Deduction form of mrcssv 17540. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) ⇒ ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋) | ||
Theorem | mrcssd 17550 | Moore closure preserves subset ordering. Deduction form of mrcss 17542. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑈 ⊆ 𝑉) & ⊢ (𝜑 → 𝑉 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉)) | ||
Theorem | mrcssidd 17551 | A set is contained in its Moore closure. Deduction form of mrcssid 17543. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑈 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) | ||
Theorem | mrcidmd 17552 | Moore closure is idempotent. Deduction form of mrcidm 17545. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑈 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈)) | ||
Theorem | mressmrcd 17553 | In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | ||
Theorem | submrc 17554 | In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) & ⊢ 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷)) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈)) | ||
Theorem | mrieqvlemd 17555 | In a Moore system, if 𝑌 is a member of 𝑆, (𝑆 ∖ {𝑌}) and 𝑆 have the same closure if and only if 𝑌 is in the closure of (𝑆 ∖ {𝑌}). Used in the proof of mrieqvd 17564 and mrieqv2d 17565. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))) | ||
Theorem | mrisval 17556* | Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}) | ||
Theorem | ismri 17557* | Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) | ||
Theorem | ismri2 17558* | Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) | ||
Theorem | ismri2d 17559* | Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) | ||
Theorem | ismri2dd 17560* | Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ⇒ ⊢ (𝜑 → 𝑆 ∈ 𝐼) | ||
Theorem | mriss 17561 | An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋) | ||
Theorem | mrissd 17562 | An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | ||
Theorem | ismri2dad 17563 | Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) | ||
Theorem | mrieqvd 17564* | In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆))) | ||
Theorem | mrieqv2d 17565* | In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)))) | ||
Theorem | mrissmrcd 17566 | In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 17553, and so are equal by mrieqv2d 17565.) (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 = 𝑇) | ||
Theorem | mrissmrid 17567 | In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → 𝑇 ∈ 𝐼) | ||
Theorem | mreexd 17568* | In a Moore system, the closure operator is said to have the exchange property if, for all elements 𝑦 and 𝑧 of the base set and subsets 𝑆 of the base set such that 𝑧 is in the closure of (𝑆 ∪ {𝑦}) but not in the closure of 𝑆, 𝑦 is in the closure of (𝑆 ∪ {𝑧}) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌}))) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))) | ||
Theorem | mreexmrid 17569* | In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼) | ||
Theorem | mreexexlemd 17570* | This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 17574. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾)) & ⊢ (𝜑 → ∀𝑡∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexexlem2d 17571* | Used in mreexexlem4d 17573 to prove the induction step in mreexexd 17574. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝐹) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (¬ 𝑔 ∈ (𝐹 ∖ {𝑌}) ∧ ((𝐹 ∖ {𝑌}) ∪ (𝐻 ∪ {𝑔})) ∈ 𝐼)) | ||
Theorem | mreexexlem3d 17572* | Base case of the induction in mreexexd 17574. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅)) ⇒ ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexexlem4d 17573* | Induction step of the induction in mreexexd 17574. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ ω) & ⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))) & ⊢ (𝜑 → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿)) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexexd 17574* | Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹 ∪ 𝐻) is independent, 𝐹 is contained in the closure of (𝐺 ∪ 𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞 ∪ 𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴 ∖ 𝐵) or (𝐵 ∖ 𝐴) is finite. The theorem is proven by induction using mreexexlem3d 17572 for the base case and mreexexlem4d 17573 for the induction step. (Contributed by David Moews, 1-May-2017.) Remove dependencies on ax-rep 5278 and ax-ac2 10440. (Revised by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin)) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹 ≈ 𝑞 ∧ (𝑞 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexdomd 17575* | In a Moore system whose closure operator has the exchange property, if 𝑆 is independent and contained in the closure of 𝑇, and either 𝑆 or 𝑇 is finite, then 𝑇 dominates 𝑆. This is an immediate consequence of mreexexd 17574. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑋) & ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 ≼ 𝑇) | ||
Theorem | mreexfidimd 17576* | In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 17575 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐼) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
Theorem | isacs 17577* | A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) | ||
Theorem | acsmre 17578 | Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) | ||
Theorem | isacs2 17579* | In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))) | ||
Theorem | acsfiel 17580* | A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))) | ||
Theorem | acsfiel2 17581* | A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)) | ||
Theorem | acsmred 17582 | An algebraic closure system is also a Moore system. Deduction form of acsmre 17578. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | ||
Theorem | isacs1i 17583* | A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋)) | ||
Theorem | mreacs 17584 | Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)) | ||
Theorem | acsfn 17585* | Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn0 17586* | Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ 𝐾 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn1 17587* | Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn1c 17588* | Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn2 17589* | Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Syntax | ccat 17590 | Extend class notation with the class of categories. |
class Cat | ||
Syntax | ccid 17591 | Extend class notation with the identity arrow of a category. |
class Id | ||
Syntax | chomf 17592 | Extend class notation to include functionalized Hom-set extractor. |
class Homf | ||
Syntax | ccomf 17593 | Extend class notation to include functionalized composition operation. |
class compf | ||
Definition | df-cat 17594* | A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53, without the axiom CAT 1, i.e., pairwise disjointness of hom-sets (cat1 18029). See setc2obas 18026 and setc2ohom 18027 for a counterexample. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ((Base‘𝑐)), the morphisms "hom" ((Hom ‘𝑐)) and the composition law "o" ((comp‘𝑐)). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 17595. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} | ||
Definition | df-cid 17595* | Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓)))) | ||
Definition | df-homf 17596* | Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | ||
Definition | df-comf 17597* | Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)))) | ||
Theorem | iscat 17598* | The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) | ||
Theorem | iscatd 17599* | Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | catidex 17600* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) |
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