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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | renegclALT 36101 | Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 10951. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | ||
Theorem | elimhyps2 36102 | Generalization of elimhyps 36099 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.) |
⊢ [𝐵 / 𝑥]𝜑 ⇒ ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑 | ||
Theorem | dedths2 36103 | Generalization of dedths 36100 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.) |
⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) | ||
Theorem | nfcxfrdf 36104 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐴) | ||
Theorem | nfded 36105 | A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3696. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2988. (Contributed by NM, 19-Nov-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶) & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐶) | ||
Theorem | nfded2 36106 | A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g., ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) for nfopd 4822) that starts from abidnf 3696. The last is assigned to the inference form (e.g., Ⅎ𝑥〈{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}〉 for nfop 4821) whose hypotheses are satisfied using nfaba1 2988. (Contributed by NM, 19-Nov-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) & ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷) & ⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝜑 → Ⅎ𝑥𝐷) | ||
Theorem | nfunidALT2 36107 | Deduction version of nfuni 4847. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | ||
Theorem | nfunidALT 36108 | Deduction version of nfuni 4847. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | ||
Theorem | nfopdALT 36109 | Deduction version of bound-variable hypothesis builder nfop 4821. This shows how the deduction version of a not-free theorem such as nfop 4821 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | ||
Theorem | cnaddcom 36110 | Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | toycom 36111* | Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on ℂ. This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.) |
⊢ 𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ} & ⊢ + = (+g‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Syntax | clsa 36112 | Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space. |
class LSAtoms | ||
Syntax | clsh 36113 | Extend class notation with all subspaces of a left module or left vector space that are hyperplanes. |
class LSHyp | ||
Definition | df-lsatoms 36114* | Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) |
⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))) | ||
Definition | df-lshyp 36115* | Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.) |
⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))}) | ||
Theorem | lshpset 36116* | The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}) | ||
Theorem | islshp 36117* | The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) | ||
Theorem | islshpsm 36118* | Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑈 ⊕ (𝑁‘{𝑣})) = 𝑉))) | ||
Theorem | lshplss 36119 | A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑆) | ||
Theorem | lshpne 36120 | A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → 𝑈 ≠ 𝑉) | ||
Theorem | lshpnel 36121 | A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | ||
Theorem | lshpnelb 36122 | The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) | ||
Theorem | lshpnel2N 36123 | Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) | ||
Theorem | lshpne0 36124 | The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.) (Proof shortened by AV, 19-Jul-2022.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) ⇒ ⊢ (𝜑 → 𝑋 ≠ 0 ) | ||
Theorem | lshpdisj 36125 | A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) ⇒ ⊢ (𝜑 → (𝑈 ∩ (𝑁‘{𝑋})) = { 0 }) | ||
Theorem | lshpcmp 36126 | If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.) |
⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐻) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) | ||
Theorem | lshpinN 36127 | The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.) |
⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐻) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) ⇒ ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ∈ 𝐻 ↔ 𝑇 = 𝑈)) | ||
Theorem | lsatset 36128* | The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))) | ||
Theorem | islsat 36129* | The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) | ||
Theorem | lsatlspsn2 36130 | The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 36131 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) | ||
Theorem | lsatlspsn 36131 | The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) | ||
Theorem | islsati 36132* | A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣})) | ||
Theorem | lsateln0 36133* | A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) | ||
Theorem | lsatlss 36134 | The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 𝐴 ⊆ 𝑆) | ||
Theorem | lsatlssel 36135 | An atom is a subspace. (Contributed by NM, 25-Aug-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑆) | ||
Theorem | lsatssv 36136 | An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑄 ⊆ 𝑉) | ||
Theorem | lsatn0 36137 | A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 30124 analog.) (Contributed by NM, 25-Aug-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑈 ≠ { 0 }) | ||
Theorem | lsatspn0 36138 | The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) | ||
Theorem | lsator0sp 36139 | The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ∨ (𝑁‘{𝑋}) = { 0 })) | ||
Theorem | lsatssn0 36140 | A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝑈 ≠ { 0 }) | ||
Theorem | lsatcmp 36141 | If two atoms are comparable, they are equal. (atsseq 30126 analog.) TODO: can lspsncmp 19890 shorten this? (Contributed by NM, 25-Aug-2014.) |
⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐴) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) | ||
Theorem | lsatcmp2 36142 | If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 36141. TODO: can lspsncmp 19890 shorten this? (Contributed by NM, 3-Feb-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝐴) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∨ 𝑈 = { 0 })) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) | ||
Theorem | lsatel 36143 | A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋})) | ||
Theorem | lsatelbN 36144 | A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋}))) | ||
Theorem | lsat2el 36145 | Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑄) ⇒ ⊢ (𝜑 → 𝑃 = 𝑄) | ||
Theorem | lsmsat 36146* | Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 36943 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑇 ≠ { 0 }) & ⊢ (𝜑 → 𝑄 ⊆ (𝑇 ⊕ 𝑈)) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈))) | ||
Theorem | lsatfixedN 36147* | Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 19902. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋})) & ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)})) | ||
Theorem | lsmsatcv 36148 | Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29431 analog.) Explicit atom version of lsmcv 19915. (Contributed by NM, 29-Oct-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) | ||
Theorem | lssatomic 36149* | The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 30137 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ≠ { 0 }) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) | ||
Theorem | lssats 36150* | The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 30140 analog.) (Contributed by NM, 9-Apr-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 = (𝑁‘∪ {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝑈})) | ||
Theorem | lpssat 36151* | Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 30142 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ⊊ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇)) | ||
Theorem | lrelat 36152* | Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 30143 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ⊊ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) | ||
Theorem | lssatle 36153* | The ordering of two subspaces is determined by the atoms under them. (chrelat3 30150 analog.) (Contributed by NM, 29-Oct-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈))) | ||
Theorem | lssat 36154* | Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 30142 analog.) (Contributed by NM, 9-Apr-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑉 ∈ 𝑆) ∧ 𝑈 ⊊ 𝑉) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ 𝑈)) | ||
Theorem | islshpat 36155* | Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 36118. (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑞 ∈ 𝐴 (𝑈 ⊕ 𝑞) = 𝑉))) | ||
Syntax | clcv 36156 | Extend class notation with the covering relation for a left module or left vector space. |
class ⋖L | ||
Definition | df-lcv 36157* | Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation 𝐴( ⋖L ‘𝑊)𝐵 is read "𝐵 covers 𝐴 " or "𝐴 is covered by 𝐵 " , and it means that 𝐵 is larger than 𝐴 and there is nothing in between. See lcvbr 36159 for binary relation. (df-cv 30058 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ ⋖L = (𝑤 ∈ V ↦ {〈𝑡, 𝑢〉 ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}) | ||
Theorem | lcvfbr 36158* | The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐶 = {〈𝑡, 𝑢〉 ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}) | ||
Theorem | lcvbr 36159* | The covers relation for a left vector space (or a left module). (cvbr 30061 analog.) (Contributed by NM, 9-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))) | ||
Theorem | lcvbr2 36160* | The covers relation for a left vector space (or a left module). (cvbr2 30062 analog.) (Contributed by NM, 9-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈)))) | ||
Theorem | lcvbr3 36161* | The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = 𝑇 ∨ 𝑠 = 𝑈))))) | ||
Theorem | lcvpss 36162 | The covers relation implies proper subset. (cvpss 30064 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇𝐶𝑈) ⇒ ⊢ (𝜑 → 𝑇 ⊊ 𝑈) | ||
Theorem | lcvnbtwn 36163 | The covers relation implies no in-betweenness. (cvnbtwn 30065 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑅𝐶𝑇) ⇒ ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) | ||
Theorem | lcvntr 36164 | The covers relation is not transitive. (cvntr 30071 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑅𝐶𝑇) & ⊢ (𝜑 → 𝑇𝐶𝑈) ⇒ ⊢ (𝜑 → ¬ 𝑅𝐶𝑈) | ||
Theorem | lcvnbtwn2 36165 | The covers relation implies no in-betweenness. (cvnbtwn2 30066 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑅𝐶𝑇) & ⊢ (𝜑 → 𝑅 ⊊ 𝑈) & ⊢ (𝜑 → 𝑈 ⊆ 𝑇) ⇒ ⊢ (𝜑 → 𝑈 = 𝑇) | ||
Theorem | lcvnbtwn3 36166 | The covers relation implies no in-betweenness. (cvnbtwn3 30067 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑅𝐶𝑇) & ⊢ (𝜑 → 𝑅 ⊆ 𝑈) & ⊢ (𝜑 → 𝑈 ⊊ 𝑇) ⇒ ⊢ (𝜑 → 𝑈 = 𝑅) | ||
Theorem | lsmcv2 36167 | Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 30072 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝑈𝐶(𝑈 ⊕ (𝑁‘{𝑋}))) | ||
Theorem | lcvat 36168* | If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 30145 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇𝐶𝑈) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊕ 𝑞) = 𝑈) | ||
Theorem | lsatcv0 36169 | An atom covers the zero subspace. (atcv0 30121 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → { 0 }𝐶𝑄) | ||
Theorem | lsatcveq0 36170 | A subspace covered by an atom must be the zero subspace. (atcveq0 30127 analog.) (Contributed by NM, 7-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑈𝐶𝑄 ↔ 𝑈 = { 0 })) | ||
Theorem | lsat0cv 36171 | A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ { 0 }𝐶𝑈)) | ||
Theorem | lcvexchlem1 36172 | Lemma for lcvexch 36177. (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈)) | ||
Theorem | lcvexchlem2 36173 | Lemma for lcvexch 36177. (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑅) & ⊢ (𝜑 → 𝑅 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ∩ 𝑈) = 𝑅) | ||
Theorem | lcvexchlem3 36174 | Lemma for lcvexch 36177. (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ⊆ 𝑅) & ⊢ (𝜑 → 𝑅 ⊆ (𝑇 ⊕ 𝑈)) ⇒ ⊢ (𝜑 → ((𝑅 ∩ 𝑈) ⊕ 𝑇) = 𝑅) | ||
Theorem | lcvexchlem4 36175 | Lemma for lcvexch 36177. (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈)) ⇒ ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈) | ||
Theorem | lcvexchlem5 36176 | Lemma for lcvexch 36177. (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈) ⇒ ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈)) | ||
Theorem | lcvexch 36177 | Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 30148 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ 𝑇𝐶(𝑇 ⊕ 𝑈))) | ||
Theorem | lcvp 36178 | Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30154 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑈 ∩ 𝑄) = { 0 } ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) | ||
Theorem | lcv1 36179 | Covering property of a subspace plus an atom. (chcv1 30134 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) | ||
Theorem | lcv2 36180 | Covering property of a subspace plus an atom. (chcv2 30135 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑈 ⊊ (𝑈 ⊕ 𝑄) ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) | ||
Theorem | lsatexch 36181 | The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 30160 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) & ⊢ (𝜑 → (𝑈 ∩ 𝑄) = { 0 }) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄)) | ||
Theorem | lsatnle 36182 | The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 30155 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ (𝑈 ∩ 𝑄) = { 0 })) | ||
Theorem | lsatnem0 36183 | The meet of distinct atoms is the zero subspace. (atnemeq0 30156 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) | ||
Theorem | lsatexch1 36184 | The atom exch1ange property. (hlatexch1 36533 analog.) (Contributed by NM, 14-Jan-2015.) |
⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ⊆ (𝑆 ⊕ 𝑅)) & ⊢ (𝜑 → 𝑄 ≠ 𝑆) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (𝑆 ⊕ 𝑄)) | ||
Theorem | lsatcv0eq 36185 | If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 30158 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) ⇒ ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅)) | ||
Theorem | lsatcv1 36186 | Two atoms covering the zero subspace are equal. (atcv1 30159 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅)) | ||
Theorem | lsatcvatlem 36187 | Lemma for lsatcvat 36188. (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑈 ≠ { 0 }) & ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝐴) | ||
Theorem | lsatcvat 36188 | A nonzero subspace less than the sum of two atoms is an atom. (atcvati 30165 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑈 ≠ { 0 }) & ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝐴) | ||
Theorem | lsatcvat2 36189 | A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 30166 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝐴) | ||
Theorem | lsatcvat3 36190 | A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30175 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → ¬ 𝑅 ⊆ 𝑈) & ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) | ||
Theorem | islshpcv 36191 | Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈𝐶𝑉))) | ||
Theorem | l1cvpat 36192 | A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 36613 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑈𝐶𝑉) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = 𝑉) | ||
Theorem | l1cvat 36193 | Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 36614 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → 𝑈𝐶𝑉) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) | ||
Theorem | lshpat 36194 | Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 37181 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 36192 and l1cvat 36193 to 𝑈 ∈ 𝐻, which in turn change 𝑈 ∈ 𝐻 in islshpcv 36191 to 𝑈𝐶𝑉, with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) | ||
Syntax | clfn 36195 | Extend class notation with all linear functionals of a left module or left vector space. |
class LFnl | ||
Definition | df-lfl 36196* | Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.) |
⊢ LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑m (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦))}) | ||
Theorem | lflset 36197* | The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐹 = {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))}) | ||
Theorem | islfl 36198* | The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))) | ||
Theorem | lfli 36199 | Property of a linear functional. (lnfnli 29819 analog.) (Contributed by NM, 16-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌))) | ||
Theorem | islfld 36200* | Properties that determine a linear functional. TODO: use this in place of islfl 36198 when it shortens the proof. (Contributed by NM, 19-Oct-2014.) |
⊢ (𝜑 → 𝑉 = (Base‘𝑊)) & ⊢ (𝜑 → + = (+g‘𝑊)) & ⊢ (𝜑 → 𝐷 = (Scalar‘𝑊)) & ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊)) & ⊢ (𝜑 → 𝐾 = (Base‘𝐷)) & ⊢ (𝜑 → ⨣ = (+g‘𝐷)) & ⊢ (𝜑 → × = (.r‘𝐷)) & ⊢ (𝜑 → 𝐹 = (LFnl‘𝑊)) & ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
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