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Type | Label | Description |
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Statement | ||
Theorem | imasvscaval 16801* | 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 16802* | 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 16803 | 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 16804* | 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 16805* | 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 16806* | The function in qusval 16805 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 16807 | 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 16808 | 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 16809 | 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 16810* | Value of the function in qusval 16805. (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 16811* | Lemma for ercpbl 16812. (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 16812* | 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 16813* | 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 16814* | 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 16815* | 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 16816* | 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 16817* | 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 16818* | 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 16819* | 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 16820 | Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) | ||
Theorem | fnpr2ob 16821 | Biconditional version of fnpr2o 16820. (Contributed by Jim Kingdon, 27-Sep-2023.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) | ||
Theorem | fvpr0o 16822 | 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 16823 | 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 16824 | 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 16825* | Elementhood in the target space of the function 𝐹 appearing in xpsval 16833. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)) | ||
Theorem | xpsfeq 16826 | 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 16827* | Elementhood in the target space of the function 𝐹 appearing in xpsval 16833. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | ||
Theorem | xpscf 16828 | 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 16829* | The value of the function appearing in xpsval 16833. (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 16830* | The function appearing in xpsval 16833 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 16831* | 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 16832* | The function appearing in xpsval 16833 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 16833* | 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 16834* | The indexed structure product that appears in xpsval 16833 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 16835 | 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 16836* | Lemma for xpsadd 16837 and xpsmul 16838. (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 16837 | 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 16838 | 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 16839 | 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 16840 | 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 16841 | 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 16842 | 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 16843 | The class of Moore systems. |
class Moore | ||
Syntax | cmrc 16844 | The class function generating Moore closures. |
class mrCls | ||
Syntax | cmri 16845 | mrInd is a class function which takes a Moore system to its set of independent sets. |
class mrInd | ||
Syntax | cacs 16846 | The class of algebraic closure (Moore) systems. |
class ACS | ||
Definition | df-mre 16847* |
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 21616) and vector spaces (lssmre 19669)
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 16851, mresspw 16853, mre1cl 16855 and mreintcl 16856 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16861); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 16862. (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 16848* |
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 21617) and linear span (mrclsp 19692).
A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcssid 16878), (2) isotone, i.e., 𝑥 ⊆ 𝑦 implies that (𝑁‘𝑥) ⊆ (𝑁‘𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 16877), and (3) idempotent, i.e., (𝑁‘(𝑁‘𝑥)) = (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcidm 16880.) 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 16849* | 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 16850* | 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 9095 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 16851* | 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 16852 | The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 21462 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ Moore Fn V | ||
Theorem | mresspw 16853 | 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 16854 | A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋) | ||
Theorem | mre1cl 16855 | In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | ||
Theorem | mreintcl 16856 | 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 16857* | 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 16858 | 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 16859* | 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 16860 | 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 16861 | 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 16862 | 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 16863* | 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 16864* | 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 16865 | 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 16866 | 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 16867* | 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 16868 | Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ mrCls Fn ∪ ran Moore | ||
Theorem | mrcfval 16869* | 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 16870 | 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 16871* | 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 16872 | 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 16873 | 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 16874 | 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 16875 | 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 16876 | 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 16877 | 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 16878 | 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 16879 | 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 16880 | 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 16881 | 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 16882 | 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 16883 | 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 16884 | The Moore closure of a set is a subset of the base. Deduction form of mrcssv 16875. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) ⇒ ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋) | ||
Theorem | mrcssd 16885 | Moore closure preserves subset ordering. Deduction form of mrcss 16877. (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 16886 | A set is contained in its Moore closure. Deduction form of mrcssid 16878. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑈 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) | ||
Theorem | mrcidmd 16887 | Moore closure is idempotent. Deduction form of mrcidm 16880. (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 16888 | 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 16889 | 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 16890 | 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 16899 and mrieqv2d 16900. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))) | ||
Theorem | mrisval 16891* | Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}) | ||
Theorem | ismri 16892* | Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) | ||
Theorem | ismri2 16893* | Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) | ||
Theorem | ismri2d 16894* | Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) | ||
Theorem | ismri2dd 16895* | Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ⇒ ⊢ (𝜑 → 𝑆 ∈ 𝐼) | ||
Theorem | mriss 16896 | An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐼 = (mrInd‘𝐴) ⇒ ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋) | ||
Theorem | mrissd 16897 | An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | ||
Theorem | ismri2dad 16898 | Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) | ||
Theorem | mrieqvd 16899* | In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆))) | ||
Theorem | mrieqv2d 16900* | In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)))) |
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