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Type | Label | Description |
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Statement | ||
Theorem | imasaddfnlem 16801* | The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) ⇒ ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵)) | ||
Theorem | imasaddvallem 16802* | The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) | ||
Theorem | imasaddflem 16803* | The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) ⇒ ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵) | ||
Theorem | imasaddfn 16804* | The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ · = (+g‘𝑅) & ⊢ ∙ = (+g‘𝑈) ⇒ ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵)) | ||
Theorem | imasaddval 16805* | The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ · = (+g‘𝑅) & ⊢ ∙ = (+g‘𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) | ||
Theorem | imasaddf 16806* | The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ · = (+g‘𝑅) & ⊢ ∙ = (+g‘𝑈) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) ⇒ ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵) | ||
Theorem | imasmulfn 16807* | The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ · = (.r‘𝑅) & ⊢ ∙ = (.r‘𝑈) ⇒ ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵)) | ||
Theorem | imasmulval 16808* | The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ · = (.r‘𝑅) & ⊢ ∙ = (.r‘𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) | ||
Theorem | imasmulf 16809* | The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ · = (.r‘𝑅) & ⊢ ∙ = (.r‘𝑈) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) ⇒ ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵) | ||
Theorem | imasvscafn 16810* | The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ ∙ = ( ·𝑠 ‘𝑈) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞)))) ⇒ ⊢ (𝜑 → ∙ Fn (𝐾 × 𝐵)) | ||
Theorem | imasvscaval 16811* | The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ ∙ = ( ·𝑠 ‘𝑈) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞)))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) | ||
Theorem | imasvscaf 16812* | The image structure's scalar multiplication is closed in the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ ∙ = ( ·𝑠 ‘𝑈) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) ⇒ ⊢ (𝜑 → ∙ :(𝐾 × 𝐵)⟶𝐵) | ||
Theorem | imasless 16813 | The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ ≤ = (le‘𝑈) ⇒ ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) | ||
Theorem | imasleval 16814* | The value of the image structure's ordering when the order is compatible with the mapping function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ ≤ = (le‘𝑈) & ⊢ 𝑁 = (le‘𝑅) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝑎𝑁𝑏 ↔ 𝑐𝑁𝑑))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ 𝑋𝑁𝑌)) | ||
Theorem | qusval 16815* | Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ (𝜑 → ∼ ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | ||
Theorem | quslem 16816* | The function in qusval 16815 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ (𝜑 → ∼ ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ )) | ||
Theorem | qusin 16817 | Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉) ⇒ ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) | ||
Theorem | qusbas 16818 | Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → (𝑉 / ∼ ) = (Base‘𝑈)) | ||
Theorem | quss 16819 | The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐾 = (Scalar‘𝑅) ⇒ ⊢ (𝜑 → 𝐾 = (Scalar‘𝑈)) | ||
Theorem | divsfval 16820* | Value of the function in qusval 16815. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑉 ∈ V) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) | ||
Theorem | ercpbllem 16821* | Lemma for ercpbl 16822. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑉 ∈ V) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵)) | ||
Theorem | ercpbl 16822* | Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑉 ∈ V) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉) & ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)))) | ||
Theorem | erlecpbl 16823* | Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑉 ∈ V) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷))) | ||
Theorem | qusaddvallem 16824* | Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) | ||
Theorem | qusaddflem 16825* | The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) & ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) ⇒ ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ )) | ||
Theorem | qusaddval 16826* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) & ⊢ · = (+g‘𝑅) & ⊢ ∙ = (+g‘𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) | ||
Theorem | qusaddf 16827* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) & ⊢ · = (+g‘𝑅) & ⊢ ∙ = (+g‘𝑈) ⇒ ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ )) | ||
Theorem | qusmulval 16828* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) & ⊢ · = (.r‘𝑅) & ⊢ ∙ = (.r‘𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) | ||
Theorem | qusmulf 16829* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) & ⊢ · = (.r‘𝑅) & ⊢ ∙ = (.r‘𝑈) ⇒ ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ )) | ||
Theorem | fnpr2o 16830 | Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) | ||
Theorem | fnpr2ob 16831 | Biconditional version of fnpr2o 16830. (Contributed by Jim Kingdon, 27-Sep-2023.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) | ||
Theorem | fvpr0o 16832 | The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) |
⊢ (𝐴 ∈ 𝑉 → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘∅) = 𝐴) | ||
Theorem | fvpr1o 16833 | The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) |
⊢ (𝐵 ∈ 𝑉 → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘1o) = 𝐵) | ||
Theorem | fvprif 16834 | The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 2o) → ({〈∅, 𝐴〉, 〈1o, 𝐵〉}‘𝐶) = if(𝐶 = ∅, 𝐴, 𝐵)) | ||
Theorem | xpsfrnel 16835* | Elementhood in the target space of the function 𝐹 appearing in xpsval 16843. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)) | ||
Theorem | xpsfeq 16836 | A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ (𝐺 Fn 2o → {〈∅, (𝐺‘∅)〉, 〈1o, (𝐺‘1o)〉} = 𝐺) | ||
Theorem | xpsfrnel2 16837* | Elementhood in the target space of the function 𝐹 appearing in xpsval 16843. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | ||
Theorem | xpscf 16838 | Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | ||
Theorem | xpsfval 16839* | The value of the function appearing in xpsval 16843. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) | ||
Theorem | xpsff1o 16840* | The function appearing in xpsval 16843 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) ⇒ ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) | ||
Theorem | xpsfrn 16841* | A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) ⇒ ⊢ ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) | ||
Theorem | xpsff1o2 16842* | The function appearing in xpsval 16843 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) ⇒ ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹 | ||
Theorem | xpsval 16843* | Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝑈 = (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ⇒ ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈)) | ||
Theorem | xpsrnbas 16844* | The indexed structure product that appears in xpsval 16843 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝑈 = (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ⇒ ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈)) | ||
Theorem | xpsbas 16845 | The base set of the 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 → (𝑋 × 𝑌) = (Base‘𝑇)) | ||
Theorem | xpsaddlem 16846* | Lemma for xpsadd 16847 and xpsmul 16848. (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 16847 | 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 16848 | 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 16849 | 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 16850 | 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 16851 | 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 16852 | 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 16853 | The class of Moore systems. |
class Moore | ||
Syntax | cmrc 16854 | The class function generating Moore closures. |
class mrCls | ||
Syntax | cmri 16855 | mrInd is a class function which takes a Moore system to its set of independent sets. |
class mrInd | ||
Syntax | cacs 16856 | The class of algebraic closure (Moore) systems. |
class ACS | ||
Definition | df-mre 16857* |
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 21686) and vector spaces (lssmre 19738)
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 16861, mresspw 16863, mre1cl 16865 and mreintcl 16866 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16871); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 16872. (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 16858* |
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 21687) and linear span (mrclsp 19761).
A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcssid 16888), (2) isotone, i.e., 𝑥 ⊆ 𝑦 implies that (𝑁‘𝑥) ⊆ (𝑁‘𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 16887), and (3) idempotent, i.e., (𝑁‘(𝑁‘𝑥)) = (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcidm 16890.) 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 16859* | 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 16860* | 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 9106 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 16861* | 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 16862 | The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 21532 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ Moore Fn V | ||
Theorem | mresspw 16863 | 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 16864 | A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋) | ||
Theorem | mre1cl 16865 | In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | ||
Theorem | mreintcl 16866 | 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 16867* | 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 16868 | 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 16869* | 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 16870 | 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 16871 | 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 16872 | 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 16873* | 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 16874* | 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 16875 | 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 16876 | 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 16877* | The domain and range 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 16878 | Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ mrCls Fn ∪ ran Moore | ||
Theorem | mrcfval 16879* | 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 16880 | 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 16881* | 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 16882 | 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 16883 | 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 16884 | 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 16885 | 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 16886 | 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 16887 | 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 16888 | 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 16889 | 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 16890 | 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 16891 | 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 16892 | 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 16893 | 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 16894 | The Moore closure of a set is a subset of the base. Deduction form of mrcssv 16885. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) ⇒ ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋) | ||
Theorem | mrcssd 16895 | Moore closure preserves subset ordering. Deduction form of mrcss 16887. (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 16896 | A set is contained in its Moore closure. Deduction form of mrcssid 16888. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑈 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) | ||
Theorem | mrcidmd 16897 | Moore closure is idempotent. Deduction form of mrcidm 16890. (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 16898 | 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 16899 | 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 16900 | 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 16909 and mrieqv2d 16910. 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))) |
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