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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lsatcvat 36201 | A nonzero subspace less than the sum of two atoms is an atom. (atcvati 30163 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑈 ≠ { 0 }) & ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝐴) | ||
Theorem | lsatcvat2 36202 | A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 30164 analog.) (Contributed by NM, 10-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝐴) | ||
Theorem | lsatcvat3 36203 | A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30173 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → ¬ 𝑅 ⊆ 𝑈) & ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) ⇒ ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) | ||
Theorem | islshpcv 36204 | Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈𝐶𝑉))) | ||
Theorem | l1cvpat 36205 | A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 36626 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑈𝐶𝑉) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = 𝑉) | ||
Theorem | l1cvat 36206 | Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 36627 analog.) (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐶 = ( ⋖L ‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → 𝑈𝐶𝑉) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) | ||
Theorem | lshpat 36207 | Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 37194 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 36205 and l1cvat 36206 to 𝑈 ∈ 𝐻, which in turn change 𝑈 ∈ 𝐻 in islshpcv 36204 to 𝑈𝐶𝑉, with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝐻) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) & ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) | ||
Syntax | clfn 36208 | Extend class notation with all linear functionals of a left module or left vector space. |
class LFnl | ||
Definition | df-lfl 36209* | Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.) |
⊢ LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑m (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦))}) | ||
Theorem | lflset 36210* | The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝐹 = {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))}) | ||
Theorem | islfl 36211* | The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))) | ||
Theorem | lfli 36212 | Property of a linear functional. (lnfnli 29817 analog.) (Contributed by NM, 16-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ 𝐹 = (LFnl‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌))) | ||
Theorem | islfld 36213* | Properties that determine a linear functional. TODO: use this in place of islfl 36211 when it shortens the proof. (Contributed by NM, 19-Oct-2014.) |
⊢ (𝜑 → 𝑉 = (Base‘𝑊)) & ⊢ (𝜑 → + = (+g‘𝑊)) & ⊢ (𝜑 → 𝐷 = (Scalar‘𝑊)) & ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊)) & ⊢ (𝜑 → 𝐾 = (Base‘𝐷)) & ⊢ (𝜑 → ⨣ = (+g‘𝐷)) & ⊢ (𝜑 → × = (.r‘𝐷)) & ⊢ (𝜑 → 𝐹 = (LFnl‘𝑊)) & ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐹) | ||
Theorem | lflf 36214 | A linear functional is a function from vectors to scalars. (lnfnfi 29818 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 36215 | 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 36216 | A linear functional is zero at the zero vector. (lnfn0i 29819 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 36217 | Property of a linear functional. (lnfnaddi 29820 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 36218 | Property of a linear functional. (lnfnaddi 29820 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 36219 | Property of a linear functional. (lnfnmuli 29821 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 36220 | 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 36221* | 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 36222 | 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 36223 | 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 36224 | 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 36225 | 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 36226* | Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 36297, 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 36227* | A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 36297, 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 36228 | 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 36229 | 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 36230 | 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 36231 | 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 36232 | 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 36233 | 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 36234 | 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 36235 | 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 36236 | 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 36237* | 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 36238* | 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 36239 | 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 36240 | Membership in the kernel of a functional. (elnlfn 29705 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 36241* | 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 36242 | 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 36243 | 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 36244 | 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 36245 | 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 36246 | The kernel of a linear functional is a subspace. (nlelshi 29837 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 36247 | 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 36248 | 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 36249 | 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 36250* | 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 36251* | 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 36252 | 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 36253 | The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 36240) 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 36254 | 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 36255 | 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 36256 | 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 36257 | 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 36258 | 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 36259 | 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 36260* | Lemma for lshpkrex 36269. 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 3453 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 36261* | Lemma for lshpkrex 36269. 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 36262* | Lemma for lshpkrex 36269. 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 36263* | Lemma for lshpkrex 36269. 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 36264* | Lemma for lshpkrex 36269. 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 36265* | Lemma for lshpkrex 36269. 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 36266* | Lemma for lshpkrex 36269. 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 36267* | 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 36268* | 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 36269* | 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 36270* | 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 36271* | 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 36269? 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 36272* | 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 36273* | Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 36272 may be more compatible with lspsn 19774. (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 36274 | Extend class notation with left dualvector space. |
class LDual | ||
Definition | df-ldual 36275* | 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 7685. 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 36276* | 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 36277 | 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 36278 | 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 36279 | 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 36280 | 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 36281 | 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 36282 | 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 36283 | 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 36284 | 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 36285 | 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 36286 | 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 36287* | 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 36288 | 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 36289 | 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 36290 | 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 36291 | 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 36292 | 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 36293 | 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 36294 | 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 36295 | 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 36296 | Lemma for ldualgrp 36297. (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 36297 | 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 36298 | 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 36299 | 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 36300 | 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) |
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