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Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlflf 35201 A linear functional is a function from vectors to scalars. (lnfnfi 29472 analog.) (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑋𝐺𝐹) → 𝐺:𝑉𝐾)

Theoremlflcl 35202 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑌𝐺𝐹𝑋𝑉) → (𝐺𝑋) ∈ 𝐾)

Theoremlfl0 35203 A linear functional is zero at the zero vector. (lnfn0i 29473 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑍 = (0g𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐺𝑍) = 0 )

Theoremlfladd 35204 Property of a linear functional. (lnfnaddi 29474 analog.) (Contributed by NM, 18-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    = (+g𝐷)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑋𝑉𝑌𝑉)) → (𝐺‘(𝑋 + 𝑌)) = ((𝐺𝑋) (𝐺𝑌)))

Theoremlflsub 35205 Property of a linear functional. (lnfnaddi 29474 analog.) (Contributed by NM, 18-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝑀 = (-g𝐷)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑋𝑉𝑌𝑉)) → (𝐺‘(𝑋 𝑌)) = ((𝐺𝑋)𝑀(𝐺𝑌)))

Theoremlflmul 35206 Property of a linear functional. (lnfnmuli 29475 analog.) (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    × = (.r𝐷)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉)) → (𝐺‘(𝑅 · 𝑋)) = (𝑅 × (𝐺𝑋)))

Theoremlfl0f 35207 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ 𝐹)

Theoremlfl1 35208* A nonzero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &    1 = (1r𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LVec ∧ 𝐺𝐹𝐺 ≠ (𝑉 × { 0 })) → ∃𝑥𝑉 (𝐺𝑥) = 1 )

Theoremlfladdcl 35209 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺𝑓 + 𝐻) ∈ 𝐹)

𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺𝑓 + 𝐻) = (𝐻𝑓 + 𝐺))

𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑𝐼𝐹)       (𝜑 → ((𝐺𝑓 + 𝐻) ∘𝑓 + 𝐼) = (𝐺𝑓 + (𝐻𝑓 + 𝐼)))

Theoremlfladd0l 35212 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &    0 = (0g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑉 × { 0 }) ∘𝑓 + 𝐺) = 𝐺)

Theoremlflnegcl 35213* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 35284, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐼 = (invg𝑅)    &   𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥)))    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑𝑁𝐹)

Theoremlflnegl 35214* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 35284, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐼 = (invg𝑅)    &   𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥)))    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &    + = (+g𝑅)    &    0 = (0g𝑅)       (𝜑 → (𝑁𝑓 + 𝐺) = (𝑉 × { 0 }))

Theoremlflvscl 35215 Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝑅𝐾)       (𝜑 → (𝐺𝑓 · (𝑉 × {𝑅})) ∈ 𝐹)

Theoremlflvsdi1 35216 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐺𝑓 + 𝐻) ∘𝑓 · (𝑉 × {𝑋})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐻𝑓 · (𝑉 × {𝑋}))))

Theoremlflvsdi2 35217 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))

Theoremlflvsdi2a 35218 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))

Theoremlflvsass 35219 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × {(𝑋 · 𝑌)})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 · (𝑉 × {𝑌})))

Theoremlfl0sc 35220 The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of (𝑉 × { 0 }) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × { 0 })) = (𝑉 × { 0 }))

Theoremlflsc0N 35221 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 35222 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 35223 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 35224* 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 35225* 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 35226 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 35227 Membership in the kernel of a functional. (elnlfn 29359 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 35228* 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 35229 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 35230 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 35231 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 35232 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 35233 The kernel of a linear functional is a subspace. (nlelshi 29491 analog.) (Contributed by NM, 16-Apr-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐾𝐺) ∈ 𝑆)

Theoremlkrssv 35234 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 35235 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 35236 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 35237* 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 35238* 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 35239 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 35240 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 35227) 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 35241 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 35242 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 35243 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 35244 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 35245 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 35246 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 35247* Lemma for lshpkrex 35256. 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 3367 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 35248* Lemma for lshpkrex 35256. 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 35249* Lemma for lshpkrex 35256. 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 35250* Lemma for lshpkrex 35256. 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 35251* Lemma for lshpkrex 35256. 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 35252* Lemma for lshpkrex 35256. 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 35253* Lemma for lshpkrex 35256. 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 35254* 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 35255* 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 35256* 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 35257* 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 35258* 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 35256? 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 35259* 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 35260* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 35259 may be more compatible with lspsn 19397. (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}))})

20.23.8  Opposite rings and dual vector spaces

Syntaxcld 35261 Extend class notation with left dualvector space.
class LDual

Definitiondf-ldual 35262* 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 7441. 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 35263* 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 35264 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 35265 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 35266 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 35267 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 35268 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 35269 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 35270 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 35271 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 35272 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 35273 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 35274* 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 35275 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 35276 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 35277 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 35278 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 35279 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 35280 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 35281 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 35282 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 35283 Lemma for ldualgrp 35284. (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 35284 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐷 ∈ Grp)

Theoremldual0 35285 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 35286 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 35287 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 35288 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 35289 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 35290 Lemma for lduallmod 35291. (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 35291 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 35292 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 19010; 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 35293 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 35294 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 35295 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 35293? (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 35296 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 35297 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 35298 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 35299 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 35300 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 ))

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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 43601-43639
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