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Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlflsc0N 36101 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)       (𝜑 → ((𝑉 × { 0 }) ∘f · (𝑉 × {𝑋})) = (𝑉 × { 0 }))
 
Theoremlfl1sc 36102 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &    1 = (1r𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺f · (𝑉 × { 1 })) = 𝐺)
 
Syntaxclk 36103 Extend class notation with the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space.
class LKer
 
Definitiondf-lkr 36104* Define the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))
 
Theoremlkrfval 36105* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
 
Theoremlkrval 36106 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))
 
Theoremellkr 36107 Membership in the kernel of a functional. (elnlfn 29633 analog.) (Contributed by NM, 16-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
 
Theoremlkrval2 36108* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
 
Theoremellkr2 36109 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊𝑌)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))
 
Theoremlkrcl 36110 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑌𝐺𝐹𝑋 ∈ (𝐾𝐺)) → 𝑋𝑉)
 
Theoremlkrf0 36111 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑌𝐺𝐹𝑋 ∈ (𝐾𝐺)) → (𝐺𝑋) = 0 )
 
Theoremlkr0f 36112 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → ((𝐾𝐺) = 𝑉𝐺 = (𝑉 × { 0 })))
 
Theoremlkrlss 36113 The kernel of a linear functional is a subspace. (nlelshi 29765 analog.) (Contributed by NM, 16-Apr-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐾𝐺) ∈ 𝑆)
 
Theoremlkrssv 36114 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐾𝐺) ⊆ 𝑉)
 
Theoremlkrsc 36115 The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑅𝐾)    &    0 = (0g𝐷)    &   (𝜑𝑅0 )       (𝜑 → (𝐿‘(𝐺f · (𝑉 × {𝑅}))) = (𝐿𝐺))
 
Theoremlkrscss 36116 The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑅𝐾)       (𝜑 → (𝐿𝐺) ⊆ (𝐿‘(𝐺f · (𝑉 × {𝑅}))))
 
Theoremeqlkr 36117* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝐺𝐹𝐻𝐹) ∧ (𝐿𝐺) = (𝐿𝐻)) → ∃𝑟𝐾𝑥𝑉 (𝐻𝑥) = ((𝐺𝑥) · 𝑟))
 
Theoremeqlkr2 36118* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝐺𝐹𝐻𝐹) ∧ (𝐿𝐺) = (𝐿𝐻)) → ∃𝑟𝐾 𝐻 = (𝐺f · (𝑉 × {𝑟})))
 
Theoremeqlkr3 36119 Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑 → (𝐾𝐺) = (𝐾𝐻))    &   (𝜑 → (𝐺𝑋) = (𝐻𝑋))    &   (𝜑 → (𝐺𝑋) ≠ 0 )       (𝜑𝐺 = 𝐻)
 
Theoremlkrlsp 36120 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 36107) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑋𝑉𝐺𝐹) ∧ (𝐺𝑋) ≠ 0 ) → ((𝐾𝐺) (𝑁‘{𝑋})) = 𝑉)
 
Theoremlkrlsp2 36121 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑋𝑉𝐺𝐹) ∧ ¬ 𝑋 ∈ (𝐾𝐺)) → ((𝐾𝐺) (𝑁‘{𝑋})) = 𝑉)
 
Theoremlkrlsp3 36122 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑋𝑉𝐺𝐹) ∧ ¬ 𝑋 ∈ (𝐾𝐺)) → (𝑁‘((𝐾𝐺) ∪ {𝑋})) = 𝑉)
 
Theoremlkrshp 36123 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ 𝐺𝐹𝐺 ≠ (𝑉 × { 0 })) → (𝐾𝐺) ∈ 𝐻)
 
Theoremlkrshp3 36124 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) ∈ 𝐻𝐺 ≠ (𝑉 × { 0 })))
 
Theoremlkrshpor 36125 The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) ∈ 𝐻 ∨ (𝐾𝐺) = 𝑉))
 
Theoremlkrshp4 36126 A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
𝑉 = (Base‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) ≠ 𝑉 ↔ (𝐾𝐺) ∈ 𝐻))
 
Theoremlshpsmreu 36127* Lemma for lshpkrex 36136. Show uniqueness of ring multiplier 𝑘 when a vector 𝑋 is broken down into components, one in a hyperplane and the other outside of it . TODO: do we need the cbvrexv 3454 for 𝑎 to 𝑐? (Contributed by NM, 4-Jan-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)       (𝜑 → ∃!𝑘𝐾𝑦𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)))
 
Theoremlshpkrlem1 36128* Lemma for lshpkrex 36136. The value of tentative functional 𝐺 is zero iff its argument belongs to hyperplane 𝑈. (Contributed by NM, 14-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐷)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))       (𝜑 → (𝑋𝑈 ↔ (𝐺𝑋) = 0 ))
 
Theoremlshpkrlem2 36129* Lemma for lshpkrex 36136. The value of tentative functional 𝐺 is a scalar. (Contributed by NM, 16-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐷)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))       (𝜑 → (𝐺𝑋) ∈ 𝐾)
 
Theoremlshpkrlem3 36130* Lemma for lshpkrex 36136. Defining property of 𝐺𝑋. (Contributed by NM, 15-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐷)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))       (𝜑 → ∃𝑧𝑈 𝑋 = (𝑧 + ((𝐺𝑋) · 𝑍)))
 
Theoremlshpkrlem4 36131* Lemma for lshpkrex 36136. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐷)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))       (((𝜑𝑙𝐾𝑢𝑉) ∧ (𝑣𝑉𝑟𝑉𝑠𝑉) ∧ (𝑢 = (𝑟 + ((𝐺𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺𝑣) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r𝐷)(𝐺𝑢))(+g𝐷)(𝐺𝑣)) · 𝑍)))
 
Theoremlshpkrlem5 36132* Lemma for lshpkrex 36136. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐷)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))       (((𝜑𝑙𝐾𝑢𝑉) ∧ (𝑣𝑉𝑟𝑈 ∧ (𝑠𝑈𝑧𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r𝐷)(𝐺𝑢))(+g𝐷)(𝐺𝑣)))
 
Theoremlshpkrlem6 36133* Lemma for lshpkrex 36136. Show linearlity of 𝐺. (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝐷)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))       ((𝜑 ∧ (𝑙𝐾𝑢𝑉𝑣𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r𝐷)(𝐺𝑢))(+g𝐷)(𝐺𝑣)))
 
Theoremlshpkrcl 36134* The set 𝐺 defined by hyperplane 𝑈 is a linear functional. (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    &   𝐹 = (LFnl‘𝑊)       (𝜑𝐺𝐹)
 
Theoremlshpkr 36135* The kernel of functional 𝐺 is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑍})) = 𝑉)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = ( ·𝑠𝑊)    &   𝐺 = (𝑥𝑉 ↦ (𝑘𝐾𝑦𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    &   𝐿 = (LKer‘𝑊)       (𝜑 → (𝐿𝐺) = 𝑈)
 
Theoremlshpkrex 36136* There exists a functional whose kernel equals a given hyperplane. Part of Th. 1.27 of Barbu and Precupanu, Convexity and Optimization in Banach Spaces. (Contributed by NM, 17-Jul-2014.)
𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ 𝑈𝐻) → ∃𝑔𝐹 (𝐾𝑔) = 𝑈)
 
Theoremlshpset2N 36137* The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       (𝑊 ∈ LVec → 𝐻 = {𝑠 ∣ ∃𝑔𝐹 (𝑔 ≠ (𝑉 × { 0 }) ∧ 𝑠 = (𝐾𝑔))})
 
TheoremislshpkrN 36138* The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be 𝑈 = (𝐾𝑔) or (𝐾𝑔) = 𝑈 as in lshpkrex 36136? Both standards seem to be used randomly throughout set.mm; we should decide on a preferred one. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       (𝑊 ∈ LVec → (𝑈𝐻 ↔ ∃𝑔𝐹 (𝑔 ≠ (𝑉 × { 0 }) ∧ 𝑈 = (𝐾𝑔))))
 
Theoremlfl1dim 36139* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → {𝑔𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑔)} = {𝑔 ∣ ∃𝑘𝐾 𝑔 = (𝐺f · (𝑉 × {𝑘}))})
 
Theoremlfl1dim2N 36140* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 36139 may be more compatible with lspsn 19705. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → {𝑔𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑔)} = {𝑔𝐹 ∣ ∃𝑘𝐾 𝑔 = (𝐺f · (𝑉 × {𝑘}))})
 
20.24.8  Opposite rings and dual vector spaces
 
Syntaxcld 36141 Extend class notation with left dualvector space.
class LDual
 
Definitiondf-ldual 36142* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. The restriction on f (+g𝑣) allows it to be a set; see ofmres 7676. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
LDual = (𝑣 ∈ V ↦ ({⟨(Base‘ndx), (LFnl‘𝑣)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑣)) ↾ ((LFnl‘𝑣) × (LFnl‘𝑣)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑣))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑣)), 𝑓 ∈ (LFnl‘𝑣) ↦ (𝑓f (.r‘(Scalar‘𝑣))((Base‘𝑣) × {𝑘})))⟩}))
 
Theoremldualset 36143* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑅)    &    = ( ∘f + ↾ (𝐹 × 𝐹))    &   𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓f · (𝑉 × {𝑘})))    &   (𝜑𝑊𝑋)       (𝜑𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
 
Theoremldualvbase 36144 The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &   𝑉 = (Base‘𝐷)    &   (𝜑𝑊𝑋)       (𝜑𝑉 = 𝐹)
 
Theoremldualelvbase 36145 Utility theorem for converting a functional to a vector of the dual space in order to use standard vector theorems. (Contributed by NM, 6-Jan-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &   𝑉 = (Base‘𝐷)    &   (𝜑𝑊𝑋)    &   (𝜑𝐺𝐹)       (𝜑𝐺𝑉)
 
Theoremldualfvadd 36146 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐷 = (LDual‘𝑊)    &    = (+g𝐷)    &   (𝜑𝑊𝑋)    &    = ( ∘f + ↾ (𝐹 × 𝐹))       (𝜑 = )
 
Theoremldualvadd 36147 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐷 = (LDual‘𝑊)    &    = (+g𝐷)    &   (𝜑𝑊𝑋)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺 𝐻) = (𝐺f + 𝐻))
 
Theoremldualvaddcl 36148 The value of vector addition in the dual of a vector space is a functional. (Contributed by NM, 21-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺 + 𝐻) ∈ 𝐹)
 
Theoremldualvaddval 36149 The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    = (+g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐺 𝐻)‘𝑋) = ((𝐺𝑋) + (𝐻𝑋)))
 
Theoremldualsca 36150 The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
𝐹 = (Scalar‘𝑊)    &   𝑂 = (oppr𝐹)    &   𝐷 = (LDual‘𝑊)    &   𝑅 = (Scalar‘𝐷)    &   (𝜑𝑊𝑋)       (𝜑𝑅 = 𝑂)
 
Theoremldualsbase 36151 Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐿 = (Base‘𝐹)    &   𝐷 = (LDual‘𝑊)    &   𝑅 = (Scalar‘𝐷)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝑊𝑉)       (𝜑𝐾 = 𝐿)
 
TheoremldualsaddN 36152 Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
𝐹 = (Scalar‘𝑊)    &    + = (+g𝐹)    &   𝐷 = (LDual‘𝑊)    &   𝑅 = (Scalar‘𝐷)    &    = (+g𝑅)    &   (𝜑𝑊𝑉)       (𝜑 = + )
 
Theoremldualsmul 36153 Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = (.r𝐹)    &   𝐷 = (LDual‘𝑊)    &   𝑅 = (Scalar‘𝐷)    &    = (.r𝑅)    &   (𝜑𝑊𝑉)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)       (𝜑 → (𝑋 𝑌) = (𝑌 · 𝑋))
 
Theoremldualfvs 36154* Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐷 = (LDual‘𝑊)    &    = ( ·𝑠𝐷)    &   (𝜑𝑊𝑌)    &    · = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓f × (𝑉 × {𝑘})))       (𝜑 = · )
 
Theoremldualvs 36155 Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐷 = (LDual‘𝑊)    &    = ( ·𝑠𝐷)    &   (𝜑𝑊𝑌)    &   (𝜑𝑋𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝑋 𝐺) = (𝐺f × (𝑉 × {𝑋})))
 
Theoremldualvsval 36156 Value of scalar product operation value for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐷 = (LDual‘𝑊)    &    = ( ·𝑠𝐷)    &   (𝜑𝑊𝑌)    &   (𝜑𝑋𝐾)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑋 𝐺)‘𝐴) = ((𝐺𝐴) × 𝑋))
 
Theoremldualvscl 36157 The scalar product operation value is a functional. (Contributed by NM, 18-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
 
Theoremldualvaddcom 36158 Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐹)    &   (𝜑𝑌𝐹)       (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theoremldualvsass 36159 Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝑌 · 𝐺)))
 
Theoremldualvsass2 36160 Associative law for scalar product operation, using operations from the dual space. (Contributed by NM, 20-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐷 = (LDual‘𝑊)    &   𝑄 = (Scalar‘𝐷)    &    × = (.r𝑄)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑋 × 𝑌) · 𝐺) = (𝑋 · (𝑌 · 𝐺)))
 
Theoremldualvsdi1 36161 Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝑋 · 𝐺) + (𝑋 · 𝐻)))
 
Theoremldualvsdi2 36162 Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐷 = (LDual‘𝑊)    &    = (+g𝐷)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑋 + 𝑌) · 𝐺) = ((𝑋 · 𝐺) (𝑌 · 𝐺)))
 
Theoremldualgrplem 36163 Lemma for ldualgrp 36164. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   𝑉 = (Base‘𝑊)    &    + = ∘f (+g𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    · = ( ·𝑠𝐷)       (𝜑𝐷 ∈ Grp)
 
Theoremldualgrp 36164 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐷 ∈ Grp)
 
Theoremldual0 36165 The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.)
𝑅 = (Scalar‘𝑊)    &    0 = (0g𝑅)    &   𝐷 = (LDual‘𝑊)    &   𝑆 = (Scalar‘𝐷)    &   𝑂 = (0g𝑆)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝑂 = 0 )
 
Theoremldual1 36166 The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
𝑅 = (Scalar‘𝑊)    &    1 = (1r𝑅)    &   𝐷 = (LDual‘𝑊)    &   𝑆 = (Scalar‘𝐷)    &   𝐼 = (1r𝑆)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐼 = 1 )
 
Theoremldualneg 36167 The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
𝑅 = (Scalar‘𝑊)    &   𝑀 = (invg𝑅)    &   𝐷 = (LDual‘𝑊)    &   𝑆 = (Scalar‘𝐷)    &   𝑁 = (invg𝑆)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝑁 = 𝑀)
 
Theoremldual0v 36168 The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    0 = (0g𝑅)    &   𝐷 = (LDual‘𝑊)    &   𝑂 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝑂 = (𝑉 × { 0 }))
 
Theoremldual0vcl 36169 The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)       (𝜑0𝐹)
 
Theoremlduallmodlem 36170 Lemma for lduallmod 36171. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   𝑉 = (Base‘𝑊)    &    + = ∘f (+g𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    · = ( ·𝑠𝐷)       (𝜑𝐷 ∈ LMod)
 
Theoremlduallmod 36171 The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐷 ∈ LMod)
 
Theoremlduallvec 36172 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 19304; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LVec)       (𝜑𝐷 ∈ LVec)
 
Theoremldualvsub 36173 The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
𝑅 = (Scalar‘𝑊)    &   𝑁 = (invg𝑅)    &    1 = (1r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &    · = ( ·𝑠𝐷)    &    = (-g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺 𝐻) = (𝐺 + ((𝑁1 ) · 𝐻)))
 
Theoremldualvsubcl 36174 Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    = (-g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺 𝐻) ∈ 𝐹)
 
Theoremldualvsubval 36175 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 36173? (Requires 𝐷 to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝑆 = (-g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    = (-g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐺 𝐻)‘𝑋) = ((𝐺𝑋)𝑆(𝐻𝑋)))
 
Theoremldualssvscl 36176 Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   𝑆 = (LSubSp‘𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 · 𝑌) ∈ 𝑈)
 
Theoremldualssvsubcl 36177 Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
𝐷 = (LDual‘𝑊)    &    = (-g𝐷)    &   𝑆 = (LSubSp‘𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 𝑌) ∈ 𝑈)
 
Theoremldual0vs 36178 Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
𝐹 = (LFnl‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    0 = (0g𝑅)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   𝑂 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → ( 0 · 𝐺) = 𝑂)
 
Theoremlkr0f2 36179 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) = 𝑉𝐺 = 0 ))
 
Theoremlduallkr3 36180 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) ∈ 𝐻𝐺0 ))
 
TheoremlkrpssN 36181 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ⊊ (𝐾𝐻) ↔ (𝐺0𝐻 = 0 )))
 
Theoremlkrin 36182 Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ∩ (𝐾𝐻)) ⊆ (𝐾‘(𝐺 + 𝐻)))
 
Theoremeqlkr4 36183* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑 → (𝐾𝐺) = (𝐾𝐻))       (𝜑 → ∃𝑟𝑅 𝐻 = (𝑟 · 𝐺))
 
Theoremldual1dim 36184* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &   𝑁 = (LSpan‘𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → (𝑁‘{𝐺}) = {𝑔𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑔)})
 
Theoremldualkrsc 36185 The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝐾)    &   (𝜑𝑋0 )       (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿𝐺))
 
Theoremlkrss 36186 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝐾)       (𝜑 → (𝐿𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))
 
Theoremlkrss2N 36187* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ⊆ (𝐾𝐻) ↔ ∃𝑟𝑅 𝐻 = (𝑟 · 𝐺)))
 
TheoremlkreqN 36188 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴 ∈ (𝑅 ∖ { 0 }))    &   (𝜑𝐻𝐹)    &   (𝜑𝐺 = (𝐴 · 𝐻))       (𝜑 → (𝐾𝐺) = (𝐾𝐻))
 
TheoremlkrlspeqN 36189 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   𝑁 = (LSpan‘𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐻𝐹)    &   (𝜑𝐺 ∈ ((𝑁‘{𝐻}) ∖ { 0 }))       (𝜑 → (𝐿𝐺) = (𝐿𝐻))
 
20.24.9  Ortholattices and orthomodular lattices
 
Syntaxcops 36190 Extend class notation with orthoposets.
class OP
 
SyntaxccmtN 36191 Extend class notation with the commutes relation.
class cm
 
Syntaxcol 36192 Extend class notation with orthlattices.
class OL
 
Syntaxcoml 36193 Extend class notation with orthomodular lattices.
class OML
 
Definitiondf-oposet 36194* Define the class of orthoposets, which are bounded posets with an orthocomplementation operation. Note that (Base p ) e. dom ( lub 𝑝) means there is an upper bound 1., and similarly for the 0. element. (Contributed by NM, 20-Oct-2011.) (Revised by NM, 13-Sep-2018.)
OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑜(𝑜 = (oc‘𝑝) ∧ ∀𝑎 ∈ (Base‘𝑝)∀𝑏 ∈ (Base‘𝑝)(((𝑜𝑎) ∈ (Base‘𝑝) ∧ (𝑜‘(𝑜𝑎)) = 𝑎 ∧ (𝑎(le‘𝑝)𝑏 → (𝑜𝑏)(le‘𝑝)(𝑜𝑎))) ∧ (𝑎(join‘𝑝)(𝑜𝑎)) = (1.‘𝑝) ∧ (𝑎(meet‘𝑝)(𝑜𝑎)) = (0.‘𝑝))))}
 
Definitiondf-cmtN 36195* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
cm = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))})
 
Definitiondf-ol 36196 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OL = (Lat ∩ OP)
 
Definitiondf-oml 36197* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
 
Theoremisopos 36198* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
 
Theoremopposet 36199 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
(𝐾 ∈ OP → 𝐾 ∈ Poset)
 
Theoremoposlem 36200 Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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