HomeTheorem List (p. 505 of 515)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | lubeldm2 50401* | Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) & ⊢ (𝜑 → 𝐾 ∈ Poset) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓))) | ||
| Theorem | glbeldm2 50402* | Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ Poset) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓))) | ||
| Theorem | lubeldm2d 50403* | Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → ≤ = (le‘𝐾)) & ⊢ (𝜑 → 𝑈 = (lub‘𝐾)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) & ⊢ (𝜑 → 𝐾 ∈ Poset) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓))) | ||
| Theorem | glbeldm2d 50404* | Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → ≤ = (le‘𝐾)) & ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) & ⊢ (𝜑 → 𝐾 ∈ Poset) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓))) | ||
| Theorem | lubsscl 50405 | If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) & ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇) ⇒ ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆))) | ||
| Theorem | glbsscl 50406 | If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) & ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇) ⇒ ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆))) | ||
| Theorem | lubprlem 50407 | Lemma for lubprdm 50408 and lubpr 50409. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌)) | ||
| Theorem | lubprdm 50408 | The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) | ||
| Theorem | lubpr 50409 | The LUB of the set of two comparable elements in a poset is the greater one of the two. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) & ⊢ 𝑈 = (lub‘𝐾) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = 𝑌) | ||
| Theorem | glbprlem 50410 | Lemma for glbprdm 50411 and glbpr 50412. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ∧ (𝐺‘𝑆) = 𝑋)) | ||
| Theorem | glbprdm 50411 | The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | ||
| Theorem | glbpr 50412 | The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ≤ = (le‘𝐾) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) & ⊢ 𝐺 = (glb‘𝐾) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = 𝑋) | ||
| Theorem | joindm2 50413* | The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈)) | ||
| Theorem | joindm3 50414* | The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ≤ = (le‘𝐾) ⇒ ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤)))) | ||
| Theorem | meetdm2 50415* | The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺)) | ||
| Theorem | meetdm3 50416* | The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ≤ = (le‘𝐾) ⇒ ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧)))) | ||
| Theorem | posjidm 50417 | Poset join is idempotent. latjidm 18686 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋) | ||
| Theorem | posmidm 50418 | Poset meet is idempotent. latmidm 18698 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋) | ||
| Theorem | resiposbas 50419 | Construct a poset (resipos 50420) for any base set. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾)) | ||
| Theorem | resipos 50420 | A set equipped with an order where no distinct elements are comparable is a poset. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐾 ∈ Poset) | ||
| Theorem | exbaspos 50421* | There exists a poset for any base set. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Poset 𝐵 = (Base‘𝑘)) | ||
| Theorem | exbasprs 50422* | There exists a preordered set for any base set. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Proset 𝐵 = (Base‘𝑘)) | ||
| Theorem | basresposfo 50423 | The base function restricted to the class of posets maps the class of posets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (Base ↾ Poset):Poset⟶ontoV | ||
| Theorem | basresprsfo 50424 | The base function restricted to the class of preordered sets maps the class of preordered sets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (Base ↾ Proset ): Proset ⟶ontoV | ||
| Theorem | posnex 50425 | The class of posets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ Poset ∉ V | ||
| Theorem | prsnex 50426 | The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ Proset ∉ V | ||
| Theorem | toslat 50427 | A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Lat) | ||
| Theorem | isclatd 50428* | The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝑈 = (lub‘𝐾)) & ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈) & ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺) ⇒ ⊢ (𝜑 → 𝐾 ∈ CLat) | ||
| Theorem | intubeu 50429* | Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})) | ||
| Theorem | unilbeu 50430* | Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})) | ||
| Theorem | ipolublem 50431* | Lemma for ipolubdm 50432 and ipolub 50433. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑧 → 𝑋 ⊆ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑋 ≤ 𝑧)))) | ||
| Theorem | ipolubdm 50432* | The domain of the LUB of the inclusion poset. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑇 ∈ 𝐹)) | ||
| Theorem | ipolub 50433* | The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 18636 is in quantified form. mrelatlub 18786 could potentially be shortened using this. See mrelatlubALT 50440. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) & ⊢ (𝜑 → 𝑇 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) | ||
| Theorem | ipoglblem 50434* | Lemma for ipoglbdm 50435 and ipoglb 50436. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋)))) | ||
| Theorem | ipoglbdm 50435* | The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆}) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹)) | ||
| Theorem | ipoglb 50436* | The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 18637 is in quantified form. mrelatglb 18784 could potentially be shortened using this. See mrelatglbALT 50441. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆}) & ⊢ (𝜑 → 𝑇 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) | ||
| Theorem | ipolub0 50437 | The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹) | ||
| Theorem | ipolub00 50438 | The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → ∅ ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑈‘∅) = ∅) | ||
| Theorem | ipoglb0 50439 | The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → ∪ 𝐹 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐹) | ||
| Theorem | mrelatlubALT 50440 | Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐹 = (mrCls‘𝐶) & ⊢ 𝐿 = (lub‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈)) | ||
| Theorem | mrelatglbALT 50441 | Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) | ||
| Theorem | mreclat 50442 | A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat) | ||
| Theorem | topclat 50443 | A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat) | ||
| Theorem | toplatglb0 50444 | The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐽) | ||
| Theorem | toplatlub 50445 | Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ 𝐽) & ⊢ 𝑈 = (lub‘𝐼) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = ∪ 𝑆) | ||
| Theorem | toplatglb 50446 | Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ 𝐽) & ⊢ 𝐺 = (glb‘𝐼) & ⊢ (𝜑 → 𝑆 ≠ ∅) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆)) | ||
| Theorem | toplatjoin 50447 | Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ ∨ = (join‘𝐼) ⇒ ⊢ (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵)) | ||
| Theorem | toplatmeet 50448 | Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ ∧ = (meet‘𝐼) ⇒ ⊢ (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵)) | ||
| Theorem | topdlat 50449 | A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat) | ||
| Theorem | elmgpcntrd 50450* | The center of a ring. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntr‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(.r‘𝑅)𝑦) = (𝑦(.r‘𝑅)𝑋)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑍) | ||
| Theorem | asclelbasALT 50451 | Alternate proof of asclelbas 22246. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑊)) | ||
| Theorem | asclcntr 50452 | The algebra scalar lifting function maps into the center of the algebra. Equivalently, a lifted scalar is a center of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ 𝑀 = (mulGrp‘𝑊) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Cntr‘𝑀)) | ||
| Theorem | asclcom 50453 |
Scalars are commutative after being lifted.
However, the scalars themselves are not necessarily commutative if the algebra is not a faithful module. For example, Let 𝐹 be the 2 by 2 upper triangular matrix algebra over a commutative ring 𝑊. It is provable that 𝐹 is in general non-commutative. Define scalar multiplication 𝐶 · 𝑋 as multipying the top-left entry, which is a "vector" element of 𝑊, of the "scalar" 𝐶, which is now an upper triangular matrix, with the "vector" 𝑋 ∈ (Base‘𝑊). Equivalently, the algebra scalar lifting function is not necessarily injective unless the algebra is faithful. Therefore, all "scalar injection" was renamed. Alternate proof involves assa2ass 22226, assa2ass2 22227, and asclval 22242, by setting 𝑋 and 𝑌 the multiplicative identity of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ∗ = (.r‘𝐹) & ⊢ (𝜑 → 𝐷 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶))) | ||
| Theorem | homf0 50454 | The base is empty iff the functionalized Hom-set operation is empty. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅) | ||
| Theorem | catprslem 50455* | Lemma for catprs 50456. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
| Theorem | catprs 50456* | A preorder can be extracted from a category. See catprs2 50457 for more details. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))) | ||
| Theorem | catprs2 50457* | A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 50458 and catprsc2 50459 for constructions satisfying the hypothesis "catprs.1". See catprs 50456 for a more primitive version. See prsthinc 50930 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
| Theorem | catprsc 50458* | A construction of the preorder induced by a category. See catprs2 50457 for details. See also catprsc2 50459 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
| Theorem | catprsc2 50459* | An alternate construction of the preorder induced by a category. See catprs2 50457 for details. See also catprsc 50458 for a different construction. The two constructions are different because df-cat 17892 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.) |
| ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
| Theorem | endmndlem 50460 | A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 51047 for converting a monoid to a category. Lemma for bj-endmnd 38507. (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀)) & ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀)) ⇒ ⊢ (𝜑 → 𝑀 ∈ Mnd) | ||
| Theorem | oppccatb 50461 | An opposite category is a category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝑂 ∈ Cat)) | ||
| Theorem | oppcmndclem 50462 | Lemma for oppcmndc 50464. Everything is true for two distinct elements in a singleton or an empty set (since it is impossible). Note that if this theorem and oppcendc 50463 are in ¬ 𝑥 = 𝑦 form, then both proofs should be one step shorter. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ (𝜑 → 𝐵 = {𝐴}) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → 𝜓)) | ||
| Theorem | oppcendc 50463* | The opposite category of a category whose morphisms are all endomorphisms has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅)) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂)) | ||
| Theorem | oppcmndc 50464 | The opposite category of a category whose base set is a singleton or an empty set has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐵 = {𝑋}) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂)) | ||
| Theorem | idmon 50465 | An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) | ||
| Theorem | idepi 50466 | An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋)) | ||
| Theorem | sectrcl 50467 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | sectrcl2 50468 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | invrcl 50469 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | invrcl2 50470 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | isinv2 50471 | The property "𝐹 is an inverse of 𝐺". (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)) | ||
| Theorem | isisod 50472 | The predicate "is an isomorphism" (deduction form). (Contributed by Zhi Wang, 16-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) & ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)) & ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | ||
| Theorem | upeu2lem 50473* | Lemma for upeu2 50619. There exists a unique morphism from 𝑌 to 𝑍 that commutes if 𝐹:𝑋⟶𝑌 is an isomorphism. (Contributed by Zhi Wang, 20-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑍)) ⇒ ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) | ||
| Theorem | sectfn 50474 | The function value of the function returning the sections of a category is a function over the Cartesian square of the base set of the category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | invfn 50475 | The function value of the function returning the inverses of a category is a function over the Cartesian square of the base set of the category. Simplifies isofn 18000 (see isofnALT 50476). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofnALT 50476 | The function value of the function returning the isomorphisms of a category is a function over the Cartesian square of the base set of the category. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofval2 50477* | Function value of the function returning the isomorphisms of a category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦))) | ||
| Theorem | isorcl 50478 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | isorcl2 50479 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | isoval2 50480 | The isomorphisms are the domain of the inverse relation. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌) | ||
| Theorem | sectpropdlem 50481 | Lemma for sectpropd 50482. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Sect‘𝐶)) → 𝑃 ∈ (Sect‘𝐷)) | ||
| Theorem | sectpropd 50482 | Two structures with the same base, hom-sets and composition operation have the same sections. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Sect‘𝐶) = (Sect‘𝐷)) | ||
| Theorem | invpropdlem 50483 | Lemma for invpropd 50484. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐷)) | ||
| Theorem | invpropd 50484 | Two structures with the same base, hom-sets and composition operation have the same inverses. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Inv‘𝐶) = (Inv‘𝐷)) | ||
| Theorem | isopropdlem 50485 | Lemma for isopropd 50486. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷)) | ||
| Theorem | isopropd 50486 | Two structures with the same base, hom-sets and composition operation have the same isomorphisms. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Iso‘𝐶) = (Iso‘𝐷)) | ||
| Theorem | cicfn 50487 | ≃𝑐 is a function on Cat. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ ≃𝑐 Fn Cat | ||
| Theorem | cicrcl2 50488 | Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝐶 ∈ Cat) | ||
| Theorem | oppccic 50489 | Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑅( ≃𝑐 ‘𝐶)𝑆) ⇒ ⊢ (𝜑 → 𝑅( ≃𝑐 ‘𝑂)𝑆) | ||
| Theorem | relcic 50490 | The set of isomorphic objects is a relation. Simplifies cicer 18031 (see cicerALT 50491). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶)) | ||
| Theorem | cicerALT 50491 | Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶)) | ||
| Theorem | cic1st2nd 50492 | Reconstruction of a pair of isomorphic objects in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩) | ||
| Theorem | cic1st2ndbr 50493 | Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) | ||
| Theorem | cicpropdlem 50494 | Lemma for cicpropd 50495. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷)) | ||
| Theorem | cicpropd 50495 | Two structures with the same base, hom-sets and composition operation have the same isomorphic objects. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝐷)) | ||
| Theorem | oppccicb 50496 | Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅( ≃𝑐 ‘𝑂)𝑆) | ||
| Theorem | oppcciceq 50497 | The opposite category has the same isomorphic objects as the original category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝑂) | ||
| Theorem | dmdm 50498 | The double domain of a function on a Cartesian square. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴) | ||
| Theorem | iinfssclem1 50499* | Lemma for iinfssc 50502. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝐾 = (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆, 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 (𝑧𝐻𝑤))) | ||
| Theorem | iinfssclem2 50500* | Lemma for iinfssc 50502. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝐾 Fn (∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆)) | ||
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