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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lflf 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) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) | ||
Theorem | lflcl 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) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) | ||
Theorem | lfl0 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 = (0g‘𝐷) & ⊢ 𝑍 = (0g‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘𝑍) = 0 ) | ||
Theorem | lfladd 35204 | Property of a linear functional. (lnfnaddi 29474 analog.) (Contributed by NM, 18-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ ⨣ = (+g‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) ⨣ (𝐺‘𝑌))) | ||
Theorem | lflsub 35205 | Property of a linear functional. (lnfnaddi 29474 analog.) (Contributed by NM, 18-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝑀 = (-g‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(𝑋 − 𝑌)) = ((𝐺‘𝑋)𝑀(𝐺‘𝑌))) | ||
Theorem | lflmul 35206 | Property of a linear functional. (lnfnmuli 29475 analog.) (Contributed by NM, 16-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘(𝑅 · 𝑋)) = (𝑅 × (𝐺‘𝑋))) | ||
Theorem | lfl0f 35207 | The zero function is a functional. (Contributed by NM, 16-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ 𝐹) | ||
Theorem | lfl1 35208* | A nonzero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 1 = (1r‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → ∃𝑥 ∈ 𝑉 (𝐺‘𝑥) = 1 ) | ||
Theorem | lfladdcl 35209 | Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ + = (+g‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 + 𝐻) ∈ 𝐹) | ||
Theorem | lfladdcom 35210 | Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ + = (+g‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 + 𝐻) = (𝐻 ∘𝑓 + 𝐺)) | ||
Theorem | lfladdass 35211 | Associativity of functional addition. (Contributed by NM, 19-Oct-2014.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ + = (+g‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) & ⊢ (𝜑 → 𝐼 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐺 ∘𝑓 + 𝐻) ∘𝑓 + 𝐼) = (𝐺 ∘𝑓 + (𝐻 ∘𝑓 + 𝐼))) | ||
Theorem | lfladd0l 35212 | Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ + = (+g‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝑉 × { 0 }) ∘𝑓 + 𝐺) = 𝐺) | ||
Theorem | lflnegcl 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 = (invg‘𝑅) & ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥))) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → 𝑁 ∈ 𝐹) | ||
Theorem | lflnegl 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 = (invg‘𝑅) & ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥))) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ + = (+g‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → (𝑁 ∘𝑓 + 𝐺) = (𝑉 × { 0 })) | ||
Theorem | lflvscl 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) & ⊢ · = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × {𝑅})) ∈ 𝐹) | ||
Theorem | lflvsdi1 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) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐺 ∘𝑓 + 𝐻) ∘𝑓 · (𝑉 × {𝑋})) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐻 ∘𝑓 · (𝑉 × {𝑋})))) | ||
Theorem | lflvsdi2 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) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺 ∘𝑓 · (𝑉 × {𝑌})))) | ||
Theorem | lflvsdi2a 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) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺 ∘𝑓 · (𝑉 × {𝑌})))) | ||
Theorem | lflvsass 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) & ⊢ · = (.r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × {(𝑋 · 𝑌)})) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 · (𝑉 × {𝑌}))) | ||
Theorem | lfl0sc 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) & ⊢ · = (.r‘𝐷) & ⊢ 0 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × { 0 })) = (𝑉 × { 0 })) | ||
Theorem | lflsc0N 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) & ⊢ · = (.r‘𝐷) & ⊢ 0 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑉 × { 0 }) ∘𝑓 · (𝑉 × {𝑋})) = (𝑉 × { 0 })) | ||
Theorem | lfl1sc 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) & ⊢ · = (.r‘𝐷) & ⊢ 1 = (1r‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × { 1 })) = 𝐺) | ||
Syntax | clk 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 | ||
Definition | df-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))}))) | ||
Theorem | lkrfval 35225* | The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) | ||
Theorem | lkrval 35226 | Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) | ||
Theorem | ellkr 35227 | Membership in the kernel of a functional. (elnlfn 29359 analog.) (Contributed by NM, 16-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | ||
Theorem | lkrval2 35228* | Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 }) | ||
Theorem | ellkr2 35229 | Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑌) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) | ||
Theorem | lkrcl 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) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → 𝑋 ∈ 𝑉) | ||
Theorem | lkrf0 35231 | The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) | ||
Theorem | lkr0f 35232 | The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.) |
⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) | ||
Theorem | lkrlss 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 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ∈ 𝑆) | ||
Theorem | lkrssv 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) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐾‘𝐺) ⊆ 𝑉) | ||
Theorem | lkrsc 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) & ⊢ · = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐿 = (LKer‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝐾) & ⊢ 0 = (0g‘𝐷) & ⊢ (𝜑 → 𝑅 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) | ||
Theorem | lkrscss 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) & ⊢ · = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐿 = (LKer‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) | ||
Theorem | eqlkr 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) & ⊢ · = (.r‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐿 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)) | ||
Theorem | eqlkr2 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) & ⊢ · = (.r‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐿 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘𝑓 · (𝑉 × {𝑟}))) | ||
Theorem | eqlkr3 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 = (0g‘𝑆) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) & ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻)) & ⊢ (𝜑 → (𝐺‘𝑋) = (𝐻‘𝑋)) & ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) ⇒ ⊢ (𝜑 → 𝐺 = 𝐻) | ||
Theorem | lkrlsp 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 = (0g‘𝐷) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ (𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐾‘𝐺) ⊕ (𝑁‘{𝑋})) = 𝑉) | ||
Theorem | lkrlsp2 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 ∧ (𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹) ∧ ¬ 𝑋 ∈ (𝐾‘𝐺)) → ((𝐾‘𝐺) ⊕ (𝑁‘{𝑋})) = 𝑉) | ||
Theorem | lkrlsp3 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 ∧ (𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹) ∧ ¬ 𝑋 ∈ (𝐾‘𝐺)) → (𝑁‘((𝐾‘𝐺) ∪ {𝑋})) = 𝑉) | ||
Theorem | lkrshp 35243 | The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) | ||
Theorem | lkrshp3 35244 | The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ (𝑉 × { 0 }))) | ||
Theorem | lkrshpor 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) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) | ||
Theorem | lkrshp4 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) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐾‘𝐺) ≠ 𝑉 ↔ (𝐾‘𝐺) ∈ 𝐻)) | ||
Theorem | lshpsmreu 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) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ (𝜑 → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍))) | ||
Theorem | lshpkrlem1 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) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝐺‘𝑋) = 0 )) | ||
Theorem | lshpkrlem2 35249* | Lemma for lshpkrex 35256. The value of tentative functional 𝐺 is a scalar. (Contributed by NM, 16-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐾) | ||
Theorem | lshpkrlem3 35250* | Lemma for lshpkrex 35256. Defining property of 𝐺‘𝑋. (Contributed by NM, 15-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍))) | ||
Theorem | lshpkrlem4 35251* | Lemma for lshpkrex 35256. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) ⇒ ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍))) | ||
Theorem | lshpkrlem5 35252* | Lemma for lshpkrex 35256. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) ⇒ ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) | ||
Theorem | lshpkrlem6 35253* | Lemma for lshpkrex 35256. Show linearlity of 𝐺. (Contributed by NM, 17-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) ⇒ ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣))) | ||
Theorem | lshpkrcl 35254* | The set 𝐺 defined by hyperplane 𝑈 is a linear functional. (Contributed by NM, 17-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐹) | ||
Theorem | lshpkr 35255* | The kernel of functional 𝐺 is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) & ⊢ 𝐿 = (LKer‘𝑊) ⇒ ⊢ (𝜑 → (𝐿‘𝐺) = 𝑈) | ||
Theorem | lshpkrex 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 ∧ 𝑈 ∈ 𝐻) → ∃𝑔 ∈ 𝐹 (𝐾‘𝑔) = 𝑈) | ||
Theorem | lshpset2N 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 = (0g‘𝐷) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → 𝐻 = {𝑠 ∣ ∃𝑔 ∈ 𝐹 (𝑔 ≠ (𝑉 × { 0 }) ∧ 𝑠 = (𝐾‘𝑔))}) | ||
Theorem | islshpkrN 35258* | The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be 𝑈 = (𝐾‘𝑔) or (𝐾‘𝑔) = 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 = (0g‘𝐷) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐾 = (LKer‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝑈 ∈ 𝐻 ↔ ∃𝑔 ∈ 𝐹 (𝑔 ≠ (𝑉 × { 0 }) ∧ 𝑈 = (𝐾‘𝑔)))) | ||
Theorem | lfl1dim 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) & ⊢ · = (.r‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘𝑓 · (𝑉 × {𝑘}))}) | ||
Theorem | lfl1dim2N 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) & ⊢ · = (.r‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∈ 𝐹 ∣ ∃𝑘 ∈ 𝐾 𝑔 = (𝐺 ∘𝑓 · (𝑉 × {𝑘}))}) | ||
Syntax | cld 35261 | Extend class notation with left dualvector space. |
class LDual | ||
Definition | df-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 (+g‘𝑣) 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) ↦ (𝑓 ∘𝑓 (.r‘(Scalar‘𝑣))((Base‘𝑣) × {𝑘})))〉})) | ||
Theorem | ldualset 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) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = ( ∘𝑓 + ↾ (𝐹 × 𝐹)) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘𝑓 · (𝑉 × {𝑘}))) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐷 = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ✚ 〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), ∙ 〉})) | ||
Theorem | ldualvbase 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) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑉 = 𝐹) | ||
Theorem | ldualelvbase 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) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝑉) | ||
Theorem | ldualfvadd 35266 | Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ + = (+g‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ✚ = (+g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ ⨣ = ( ∘𝑓 + ↾ (𝐹 × 𝐹)) ⇒ ⊢ (𝜑 → ✚ = ⨣ ) | ||
Theorem | ldualvadd 35267 | Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ + = (+g‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ✚ = (+g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ✚ 𝐻) = (𝐺 ∘𝑓 + 𝐻)) | ||
Theorem | ldualvaddcl 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) & ⊢ + = (+g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 + 𝐻) ∈ 𝐹) | ||
Theorem | ldualvaddval 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) & ⊢ + = (+g‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ✚ = (+g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐺 ✚ 𝐻)‘𝑋) = ((𝐺‘𝑋) + (𝐻‘𝑋))) | ||
Theorem | ldualsca 35270 | The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝑂 = (oppr‘𝐹) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑅 = (Scalar‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑅 = 𝑂) | ||
Theorem | ldualsbase 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) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐾 = 𝐿) | ||
Theorem | ldualsaddN 35272 | Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ + = (+g‘𝐹) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑅 = (Scalar‘𝐷) & ⊢ ✚ = (+g‘𝑅) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) ⇒ ⊢ (𝜑 → ✚ = + ) | ||
Theorem | ldualsmul 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) & ⊢ · = (.r‘𝐹) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑅 = (Scalar‘𝐷) & ⊢ ∙ = (.r‘𝑅) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)) | ||
Theorem | ldualfvs 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) & ⊢ × = (.r‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ 𝑌) & ⊢ · = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘𝑓 × (𝑉 × {𝑘}))) ⇒ ⊢ (𝜑 → ∙ = · ) | ||
Theorem | ldualvs 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) & ⊢ × = (.r‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑋 ∙ 𝐺) = (𝐺 ∘𝑓 × (𝑉 × {𝑋}))) | ||
Theorem | ldualvsval 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) & ⊢ × = (.r‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑋 ∙ 𝐺)‘𝐴) = ((𝐺‘𝐴) × 𝑋)) | ||
Theorem | ldualvscl 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) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹) | ||
Theorem | ldualvaddcom 35278 | Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ + = (+g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐹) & ⊢ (𝜑 → 𝑌 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | ldualvsass 35279 | Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝑌 · 𝐺))) | ||
Theorem | ldualvsass2 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) & ⊢ × = (.r‘𝑄) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑌) · 𝐺) = (𝑋 · (𝑌 · 𝐺))) | ||
Theorem | ldualvsdi1 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) & ⊢ + = (+g‘𝐷) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑋 · (𝐺 + 𝐻)) = ((𝑋 · 𝐺) + (𝑋 · 𝐻))) | ||
Theorem | ldualvsdi2 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) & ⊢ + = (+g‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ ✚ = (+g‘𝐷) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝐺) = ((𝑋 · 𝐺) ✚ (𝑌 · 𝐺))) | ||
Theorem | ldualgrplem 35283 | Lemma for ldualgrp 35284. (Contributed by NM, 22-Oct-2014.) |
⊢ 𝐷 = (LDual‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = ∘𝑓 (+g‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐷) ⇒ ⊢ (𝜑 → 𝐷 ∈ Grp) | ||
Theorem | ldualgrp 35284 | The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐷 = (LDual‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝐷 ∈ Grp) | ||
Theorem | ldual0 35285 | The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐷) & ⊢ 𝑂 = (0g‘𝑆) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝑂 = 0 ) | ||
Theorem | ldual1 35286 | The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐷) & ⊢ 𝐼 = (1r‘𝑆) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝐼 = 1 ) | ||
Theorem | ldualneg 35287 | The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝑀 = (invg‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐷) & ⊢ 𝑁 = (invg‘𝑆) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝑁 = 𝑀) | ||
Theorem | ldual0v 35288 | The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 𝑂 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) | ||
Theorem | ldual0vcl 35289 | The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ 0 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 0 ∈ 𝐹) | ||
Theorem | lduallmodlem 35290 | Lemma for lduallmod 35291. (Contributed by NM, 22-Oct-2014.) |
⊢ 𝐷 = (LDual‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = ∘𝑓 (+g‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐷) ⇒ ⊢ (𝜑 → 𝐷 ∈ LMod) | ||
Theorem | lduallmod 35291 | The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.) |
⊢ 𝐷 = (LDual‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → 𝐷 ∈ LMod) | ||
Theorem | lduallvec 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) & ⊢ (𝜑 → 𝑊 ∈ LVec) ⇒ ⊢ (𝜑 → 𝐷 ∈ LVec) | ||
Theorem | ldualvsub 35293 | The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ + = (+g‘𝐷) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ − = (-g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻))) | ||
Theorem | ldualvsubcl 35294 | Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ − = (-g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 − 𝐻) ∈ 𝐹) | ||
Theorem | ldualvsubval 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 = (-g‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ − = (-g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝐻 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐺 − 𝐻)‘𝑋) = ((𝐺‘𝑋)𝑆(𝐻‘𝑋))) | ||
Theorem | ldualssvscl 35296 | Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.) |
⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ 𝑆 = (LSubSp‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑈) | ||
Theorem | ldualssvsubcl 35297 | Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.) |
⊢ 𝐷 = (LDual‘𝑊) & ⊢ − = (-g‘𝐷) & ⊢ 𝑆 = (LSubSp‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑈) | ||
Theorem | ldual0vs 35298 | Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.) |
⊢ 𝐹 = (LFnl‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (LDual‘𝑊) & ⊢ · = ( ·𝑠 ‘𝐷) & ⊢ 𝑂 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ( 0 · 𝐺) = 𝑂) | ||
Theorem | lkr0f2 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 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = 0 )) | ||
Theorem | lduallkr3 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 = (0g‘𝐷) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ 0 )) |
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