 Home Metamath Proof ExplorerTheorem List (p. 161 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26941) Hilbert Space Explorer (26942-28466) Users' Mathboxes (28467-42420)

Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmrcssvd 16001 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 15992. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)       (𝜑 → (𝑁𝐵) ⊆ 𝑋)

Theoremmrcssd 16002 Moore closure preserves subset ordering. Deduction form of mrcss 15994. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑉𝑋)       (𝜑 → (𝑁𝑈) ⊆ (𝑁𝑉))

Theoremmrcssidd 16003 A set is contained in its Moore closure. Deduction form of mrcssid 15995. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑𝑈 ⊆ (𝑁𝑈))

Theoremmrcidmd 16004 Moore closure is idempotent. Deduction form of mrcidm 15997. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑 → (𝑁‘(𝑁𝑈)) = (𝑁𝑈))

Theoremmressmrcd 16005 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.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑁𝑆) = (𝑁𝑇))

Theoremsubmrc 16006 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) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))

Theoremmrieqvlemd 16007 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 16016 and mrieqv2d 16017. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆𝑋)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))

7.2.2  Independent sets in a Moore system

Theoremmrisval 16008* 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 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Theoremismri 16009* 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) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))

Theoremismri2 16010* 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) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Theoremismri2d 16011* 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)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Theoremismri2dd 16012* 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)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝑋)    &   (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))       (𝜑𝑆𝐼)

Theoremmriss 16013 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) ∧ 𝑆𝐼) → 𝑆𝑋)

Theoremmrissd 16014 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)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝐼)       (𝜑𝑆𝑋)

Theoremismri2dad 16015 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)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝐼)    &   (𝜑𝑌𝑆)       (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Theoremmrieqvd 16016* 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.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))

Theoremmrieqv2d 16017* 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.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))

Theoremmrissmrcd 16018 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 16005, and so are equal by mrieqv2d 16017.) (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)    &   (𝜑𝑆𝐼)       (𝜑𝑆 = 𝑇)

Theoremmrissmrid 16019 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝑆)       (𝜑𝑇𝐼)

Theoremmreexd 16020* 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 us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
(𝜑𝑋𝑉)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝑋)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁𝑆))       (𝜑𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))

Theoremmreexmrid 16021* 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.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑌𝑋)    &   (𝜑 → ¬ 𝑌 ∈ (𝑁𝑆))       (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)

Theoremmreexexlemd 16022* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16026. (Contributed by David Moews, 1-May-2017.)
(𝜑𝑋𝐽)    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹𝐾𝐺𝐾))    &   (𝜑 → ∀𝑡𝑢 ∈ 𝒫 (𝑋𝑡)∀𝑣 ∈ 𝒫 (𝑋𝑡)(((𝑢𝐾𝑣𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣𝑡)) ∧ (𝑢𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢𝑖 ∧ (𝑖𝑡) ∈ 𝐼)))       (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))

Theoremmreexexlem2d 16023* Used in mreexexlem4d 16025 to prove the induction step in mreexexd 16026. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑𝑌𝐹)       (𝜑 → ∃𝑔𝐺𝑔 ∈ (𝐹 ∖ {𝑌}) ∧ ((𝐹 ∖ {𝑌}) ∪ (𝐻 ∪ {𝑔})) ∈ 𝐼))

Theoremmreexexlem3d 16024* Base case of the induction in mreexexd 16026. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))       (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹𝑖 ∧ (𝑖𝐻) ∈ 𝐼))

Theoremmreexexlem4d 16025* Induction step of the induction in mreexexd 16026. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑𝐿 ∈ ω)    &   (𝜑 → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))    &   (𝜑 → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))       (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))

Theoremmreexexd 16026* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹𝐻) is independent, 𝐹 is contained in the closure of (𝐺𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴𝐵) or (𝐵𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16024 for the base case and mreexexlem4d 16025 for the induction step. (Contributed by David Moews, 1-May-2017.) Removed dependencies on ax-rep 4597 and ax-ac2 9048. (Revised by Brendan Leahy, 2-Jun-2021.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))       (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹𝑞 ∧ (𝑞𝐻) ∈ 𝐼))

TheoremmreexexdOLD 16027* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹𝐻) is independent, 𝐹 is contained in the closure of (𝐺𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴𝐵) or (𝐵𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16024 for the base case and mreexexlem4d 16025 for the induction step. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))       (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹𝑞 ∧ (𝑞𝐻) ∈ 𝐼))

Theoremmreexdomd 16028* 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 16026. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑋)    &   (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))    &   (𝜑𝑆𝐼)       (𝜑𝑆𝑇)

Theoremmreexfidimd 16029* 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 16028 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝐼)    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))       (𝜑𝑆𝑇)

7.2.3  Algebraic closure systems

Theoremisacs 16030* 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(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))

Theoremacsmre 16031 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))

Theoremisacs2 16032* 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(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))

Theoremacsfiel 16033* 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) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))

Theoremacsfiel2 16034* 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) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))

Theoremacsmred 16035 An algebraic closure system is also a Moore system. Deduction form of acsmre 16031. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))       (𝜑𝐴 ∈ (Moore‘𝑋))

Theoremisacs1i 16036* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))

Theoremmreacs 16037 Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

Theoremacsfn 16038* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} ∈ (ACS‘𝑋))

Theoremacsfn0 16039* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉𝐾𝑋) → {𝑎 ∈ 𝒫 𝑋𝐾𝑎} ∈ (ACS‘𝑋))

Theoremacsfn1 16040* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑋 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏𝑎 𝐸𝑎} ∈ (ACS‘𝑋))

Theoremacsfn1c 16041* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉 ∧ ∀𝑏𝐾𝑐𝑋 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏𝐾𝑐𝑎 𝐸𝑎} ∈ (ACS‘𝑋))

Theoremacsfn2 16042* Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑋𝑐𝑋 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏𝑎𝑐𝑎 𝐸𝑎} ∈ (ACS‘𝑋))

PART 8  BASIC CATEGORY THEORY

8.1  Categories

8.1.1  Categories

Syntaxccat 16043 Extend class notation with the class of categories.
class Cat

Syntaxccid 16044 Extend class notation with the identity arrow of a category.
class Id

Syntaxchomf 16045 Extend class notation to include functionalized Hom-set extractor.
class Homf

Syntaxccomf 16046 Extend class notation to include functionalized composition operation.
class compf

Definitiondf-cat 16047* 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. 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 16048. (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) / ][(comp‘𝑐) / 𝑜]𝑥𝑏 (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))}

Definitiondf-cid 16048* 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) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))

Definitiondf-homf 16049* 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)))

Definitiondf-comf 16050* 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 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))

Theoremiscat 16051* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       (𝐶𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))

Theoremiscatd 16052* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐻 = (Hom ‘𝐶))    &   (𝜑· = (comp‘𝐶))    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))       (𝜑𝐶 ∈ Cat)

Theoremcatidex 16053* 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))

Theoremcatideu 16054* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))

Theoremcidfval 16055* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)       (𝜑1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))

Theoremcidval 16056* 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)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) = (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))

Theoremcidffn 16057 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
Id Fn Cat

Theoremcidfn 16058 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)       (𝐶 ∈ Cat → 1 Fn 𝐵)

Theoremcatidd 16059* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐻 = (Hom ‘𝐶))    &   (𝜑· = (comp‘𝐶))    &   (𝜑𝐶 ∈ Cat)    &   ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)       (𝜑 → (Id‘𝐶) = (𝑥𝐵1 ))

Theoremiscatd2 16060* Version of iscatd 16052 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐻 = (Hom ‘𝐶))    &   (𝜑· = (comp‘𝐶))    &   (𝜑𝐶𝑉)    &   (𝜓 ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))    &   ((𝜑𝑦𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)    &   ((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)    &   ((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    &   ((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))       (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦𝐵1 )))

Theoremcatidcl 16061 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))

Theoremcatlid 16062 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)

Theoremcatrid 16063 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)

Theoremcatcocl 16064 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍))

Theoremcatass 16065 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑𝑊𝐵)    &   (𝜑𝐾 ∈ (𝑍𝐻𝑊))       (𝜑 → ((𝐾(⟨𝑌, 𝑍· 𝑊)𝐺)(⟨𝑋, 𝑌· 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍· 𝑊)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)))

Theorem0catg 16066 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

Theorem0cat 16067 The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
∅ ∈ Cat

Theoremhomffval 16068* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))

Theoremfnhomeqhomf 16069 If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Theoremhomfval 16070 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Theoremhomffn 16071 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)       𝐹 Fn (𝐵 × 𝐵)

Theoremhomfeq 16072* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐵 = (Base‘𝐷))       (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))

Theoremhomfeqd 16073 If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (Base‘𝐶) = (Base‘𝐷))    &   (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))       (𝜑 → (Homf𝐶) = (Homf𝐷))

Theoremhomfeqbas 16074 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Theoremhomfeqval 16075 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))

Theoremcomfffval 16076* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Theoremcomffval 16077* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))

Theoremcomfval 16078 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))

Theoremcomfffval2 16079* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Theoremcomffval2 16080* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))

Theoremcomfval2 16081 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))

Theoremcomfffn 16082 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)       𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Theoremcomffn 16083 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))

Theoremcomfeq 16084* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
· = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐵 = (Base‘𝐷))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → ((compf𝐶) = (compf𝐷) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))

Theoremcomfeqd 16085 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (comp‘𝐶) = (comp‘𝐷))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (compf𝐶) = (compf𝐷))

Theoremcomfeqval 16086 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))

Theoremcatpropd 16087 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))

Theoremcidpropd 16088 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (Id‘𝐶) = (Id‘𝐷))

8.1.2  Opposite category

Syntaxcoppc 16089 The opposite category operation.
class oppCat

Definitiondf-oppc 16090* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑓)(1st𝑢)))⟩))

Theoremoppcval 16091* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)       (𝐶𝑉𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))

Theoremoppchomfval 16092 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)       tpos 𝐻 = (Hom ‘𝑂)

Theoremoppchom 16093 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)       (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋)

Theoremoppccofval 16094 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌· 𝑋))

Theoremoppcco 16095 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍)𝐹) = (𝐹(⟨𝑍, 𝑌· 𝑋)𝐺))

Theoremoppcbas 16096 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Base‘𝐶)       𝐵 = (Base‘𝑂)

Theoremoppccatid 16097 Lemma for oppccat 16100. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))

Theoremoppchomf 16098 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐻 = (Homf𝐶)       tpos 𝐻 = (Homf𝑂)

Theoremoppcid 16099 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Id‘𝐶)       (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵)

Theoremoppccat 16100 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (𝐶 ∈ Cat → 𝑂 ∈ Cat)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42420
 Copyright terms: Public domain < Previous  Next >