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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | qusin 16601 | 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 16602 | 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 16603 | 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 16604* | Value of the function in qusval 16599. (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 16605* | Lemma for ercpbl 16606. (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 16606* | 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 16607* | 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 16608* | 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 16609* | 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 16610* | 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 16611* | 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 16612* | 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 16613* | 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 | xpsc 16614 | A short expression for the pair function mapping 0 to 𝐴 and 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ◡({𝐴} +𝑐 {𝐵}) = (({∅} × {𝐴}) ∪ ({1o} × {𝐵})) | ||
Theorem | xpscg 16615 | A short expression for the pair function mapping 0 to 𝐴 and 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡({𝐴} +𝑐 {𝐵}) = {〈∅, 𝐴〉, 〈1o, 𝐵〉}) | ||
Theorem | xpscfn 16616 | The pair function is a function on 2o = {∅, 1o}. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡({𝐴} +𝑐 {𝐵}) Fn 2o) | ||
Theorem | xpsc0 16617 | The pair function maps 0 to 𝐴. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐴 ∈ 𝑉 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴) | ||
Theorem | xpsc1 16618 | The pair function maps 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐵 ∈ 𝑉 → (◡({𝐴} +𝑐 {𝐵})‘1o) = 𝐵) | ||
Theorem | xpscfv 16619 | 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} +𝑐 {𝐵})‘𝐶) = if(𝐶 = ∅, 𝐴, 𝐵)) | ||
Theorem | xpsfrnel 16620* | Elementhood in the target space of the function 𝐹 appearing in xpsval 16629. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)) | ||
Theorem | xpsfeq 16621 | A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ (𝐺 Fn 2o → ◡({(𝐺‘∅)} +𝑐 {(𝐺‘1o)}) = 𝐺) | ||
Theorem | xpsfrnel2 16622* | Elementhood in the target space of the function 𝐹 appearing in xpsval 16629. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | ||
Theorem | xpscf 16623 | Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (◡({𝑋} +𝑐 {𝑌}):2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | ||
Theorem | xpsfval 16624* | The value of the function appearing in xpsval 16629. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = ◡({𝑋} +𝑐 {𝑌})) | ||
Theorem | xpsff1o 16625* | The function appearing in xpsval 16629 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} +𝑐 {𝑦})) ⇒ ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) | ||
Theorem | xpsfrn 16626* | A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) ⇒ ⊢ ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) | ||
Theorem | xpsfrn2 16627* | A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐹 = X𝑘 ∈ 2o (◡({𝐴} +𝑐 {𝐵})‘𝑘)) | ||
Theorem | xpsff1o2 16628* | The function appearing in xpsval 16629 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} +𝑐 {𝑦})) ⇒ ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹 | ||
Theorem | xpsval 16629* | Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝑈 = (𝐺Xs◡({𝑅} +𝑐 {𝑆})) ⇒ ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈)) | ||
Theorem | xpslem 16630* | The indexed structure product that appears in xpsval 16629 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝑈 = (𝐺Xs◡({𝑅} +𝑐 {𝑆})) ⇒ ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈)) | ||
Theorem | xpsbas 16631 | 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 16632* | Lemma for xpsadd 16633 and xpsmul 16634. (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} +𝑐 {𝑦})) & ⊢ 𝑈 = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) & ⊢ ((𝜑 ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹 ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹) → ((◡𝐹‘◡({𝐴} +𝑐 {𝐵})) ∙ (◡𝐹‘◡({𝐶} +𝑐 {𝐷}))) = (◡𝐹‘(◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})))) & ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2o ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈) ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈)) → (◡({𝐴} +𝑐 {𝐵})(𝐸‘𝑈)◡({𝐶} +𝑐 {𝐷})) = (𝑘 ∈ 2o ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)))) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∙ 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | ||
Theorem | xpsadd 16633 | 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 16634 | 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 16635 | 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 16636 | 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 16637 | 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 16638 | 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 16639 | The class of Moore systems. |
class Moore | ||
Syntax | cmrc 16640 | The class function generating Moore closures. |
class mrCls | ||
Syntax | cmri 16641 | mrInd is a class function which takes a Moore system to its set of independent sets. |
class mrInd | ||
Syntax | cacs 16642 | The class of algebraic closure (Moore) systems. |
class ACS | ||
Definition | df-mre 16643* |
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 21301) and vector spaces (lssmre 19372)
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 16647, mresspw 16649, mre1cl 16651 and mreintcl 16652 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16657); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 16658. (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 16644* |
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 21302) and linear span (mrclsp 19395).
A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcssid 16674), (2) isotone, i.e., 𝑥 ⊆ 𝑦 implies that (𝑁‘𝑥) ⊆ (𝑁‘𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 16673), and (3) idempotent, i.e., (𝑁‘(𝑁‘𝑥)) = (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcidm 16676.) 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 16645* | 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 16646* | 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 8839 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 16647* | 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 16648 | The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 21147 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ Moore Fn V | ||
Theorem | mresspw 16649 | 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 16650 | A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋) | ||
Theorem | mre1cl 16651 | In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | ||
Theorem | mreintcl 16652 | 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 16653* | 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 16654 | 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 16655* | 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 16656 | 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 16657 | 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 16658 | 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 16659* | 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 16660* | 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 16661 | 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 16662 | 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 16663* | 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 16664 | Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ mrCls Fn ∪ ran Moore | ||
Theorem | mrcfval 16665* | 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 16666 | 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 16667* | 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 16668 | 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 16669 | 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 16670 | 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 16671 | 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 16672 | 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 16673 | 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 16674 | 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 16675 | 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 16676 | 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 16677 | 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 16678 | 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 16679 | 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 16680 | The Moore closure of a set is a subset of the base. Deduction form of mrcssv 16671. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) ⇒ ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋) | ||
Theorem | mrcssd 16681 | Moore closure preserves subset ordering. Deduction form of mrcss 16673. (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 16682 | A set is contained in its Moore closure. Deduction form of mrcssid 16674. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ (𝜑 → 𝑈 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) | ||
Theorem | mrcidmd 16683 | Moore closure is idempotent. Deduction form of mrcidm 16676. (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 16684 | 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 16685 | 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 16686 | 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 16695 and mrieqv2d 16696. 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 16687* | 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 16688* | 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 16689* | 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 16690* | 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 16691* | 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 16692 | 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 16693 | 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 16694 | 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 16695* | 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 16696* | In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆)))) | ||
Theorem | mrissmrcd 16697 | 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 16684, and so are equal by mrieqv2d 16696.) (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 = 𝑇) | ||
Theorem | mrissmrid 16698 | In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → 𝑇 ∈ 𝐼) | ||
Theorem | mreexd 16699* | 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.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌}))) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))) | ||
Theorem | mreexmrid 16700* | In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼) |
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