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Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlkrlss 34201 The kernel of a linear functional is a subspace. (nlelshi 28889 analog.) (Contributed by NM, 16-Apr-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐾𝐺) ∈ 𝑆)

Theoremlkrssv 34202 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 34203 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 34204 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 34205* 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 34206* 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 34207 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 34208 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 34195) 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 34209 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 34210 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 34211 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 34212 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 34213 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 34214 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 34215* Lemma for lshpkrex 34224. 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 3167 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 34216* Lemma for lshpkrex 34224. 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 34217* Lemma for lshpkrex 34224. 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 34218* Lemma for lshpkrex 34224. 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 34219* Lemma for lshpkrex 34224. 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 34220* Lemma for lshpkrex 34224. 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 34221* Lemma for lshpkrex 34224. 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 34222* 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 34223* 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 34224* 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 34225* 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 34226* 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 34224? 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 34227* 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 34228* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 34227 may be more compatible with lspsn 18983. (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}))})

20.22.8  Opposite rings and dual vector spaces

Syntaxcld 34229 Extend class notation with left dualvector space.
class LDual

Definitiondf-ldual 34230* 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 7149. 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 34231* 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 34232 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 34233 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 34234 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 34235 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 34236 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 34237 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 34238 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 34239 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 34240 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 34241 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 34242* 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 34243 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 34244 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 34245 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 34246 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 34247 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 34248 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 34249 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 34250 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 34251 Lemma for ldualgrp 34252. (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 34252 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐷 ∈ Grp)

Theoremldual0 34253 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 34254 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 34255 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 34256 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 34257 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 34258 Lemma for lduallmod 34259. (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 34259 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 34260 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 18604; 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 34261 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 34262 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 34263 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 34261? (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 34264 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 34265 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 34266 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 34267 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 34268 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝐾𝐺) ∈ 𝐻𝐺0 ))

TheoremlkrpssN 34269 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ⊊ (𝐾𝐻) ↔ (𝐺0𝐻 = 0 )))

Theoremlkrin 34270 Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    + = (+g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ∩ (𝐾𝐻)) ⊆ (𝐾‘(𝐺 + 𝐻)))

Theoremeqlkr4 34271* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑 → (𝐾𝐺) = (𝐾𝐻))       (𝜑 → ∃𝑟𝑅 𝐻 = (𝑟 · 𝐺))

Theoremldual1dim 34272* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &   𝑁 = (LSpan‘𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)       (𝜑 → (𝑁‘{𝐺}) = {𝑔𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑔)})

Theoremldualkrsc 34273 The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝐾)    &   (𝜑𝑋0 )       (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿𝐺))

Theoremlkrss 34274 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝐾)       (𝜑 → (𝐿𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))

Theoremlkrss2N 34275* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐾𝐺) ⊆ (𝐾𝐻) ↔ ∃𝑟𝑅 𝐻 = (𝑟 · 𝐺)))

TheoremlkreqN 34276 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    · = ( ·𝑠𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴 ∈ (𝑅 ∖ { 0 }))    &   (𝜑𝐻𝐹)    &   (𝜑𝐺 = (𝐴 · 𝐻))       (𝜑 → (𝐾𝐺) = (𝐾𝐻))

TheoremlkrlspeqN 34277 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   𝐷 = (LDual‘𝑊)    &    0 = (0g𝐷)    &   𝑁 = (LSpan‘𝐷)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐻𝐹)    &   (𝜑𝐺 ∈ ((𝑁‘{𝐻}) ∖ { 0 }))       (𝜑 → (𝐿𝐺) = (𝐿𝐻))

20.22.9  Ortholattices and orthomodular lattices

Syntaxcops 34278 Extend class notation with orthoposets.
class OP

SyntaxccmtN 34279 Extend class notation with the commutes relation.
class cm

Syntaxcol 34280 Extend class notation with orthlattices.
class OL

Syntaxcoml 34281 Extend class notation with orthomodular lattices.
class OML

Definitiondf-oposet 34282* Define the class of orthoposets, which are bounded posets with an orthocomplementation operation. Note that (Base p ) e. dom ( lub 𝑝) means there is an upper bound 1., and similarly for the 0. element. (Contributed by NM, 20-Oct-2011.) (Revised by NM, 13-Sep-2018.)
OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑜(𝑜 = (oc‘𝑝) ∧ ∀𝑎 ∈ (Base‘𝑝)∀𝑏 ∈ (Base‘𝑝)(((𝑜𝑎) ∈ (Base‘𝑝) ∧ (𝑜‘(𝑜𝑎)) = 𝑎 ∧ (𝑎(le‘𝑝)𝑏 → (𝑜𝑏)(le‘𝑝)(𝑜𝑎))) ∧ (𝑎(join‘𝑝)(𝑜𝑎)) = (1.‘𝑝) ∧ (𝑎(meet‘𝑝)(𝑜𝑎)) = (0.‘𝑝))))}

Definitiondf-cmtN 34283* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
cm = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))})

Definitiondf-ol 34284 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OL = (Lat ∩ OP)

Definitiondf-oml 34285* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}

Theoremisopos 34286* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))

Theoremopposet 34287 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
(𝐾 ∈ OP → 𝐾 ∈ Poset)

Theoremoposlem 34288 Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))

Theoremop01dm 34289 Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))

Theoremop0cl 34290 An orthoposet has a zero element. (h0elch 28082 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ OP → 0𝐵)

Theoremop1cl 34291 An orthoposet has a unit element. (helch 28070 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP → 1𝐵)

Theoremop0le 34292 Orthoposet zero is less than or equal to any element. (ch0le 28270 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → 0 𝑋)

Theoremople0 34293 An element less than or equal to zero equals zero. (chle0 28272 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))

Theoremopnlen0 34294 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2805 and op0le 34292 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑋 𝑌) → 𝑋0 )

Theoremlub0N 34295 The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
1 = (lub‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Theoremopltn0 34296 A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 0 < 𝑋𝑋0 ))

Theoremople1 34297 Any element is less than the orthoposet unit. (chss 28056 analog.) (Contributed by NM, 23-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → 𝑋 1 )

Theoremop1le 34298 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 28272 analog.) (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 1 𝑋𝑋 = 1 ))

Theoremglb0N 34299 The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
𝐺 = (glb‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Theoremopoccl 34300 Closure of orthocomplement operation. (choccl 28135 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)

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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 42301-42316
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