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Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlkrshp3 33301 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 33302 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 33303 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 33304* Lemma for lshpkrex 33313. 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 3052 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 33305* Lemma for lshpkrex 33313. 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 33306* Lemma for lshpkrex 33313. 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 33307* Lemma for lshpkrex 33313. 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 33308* Lemma for lshpkrex 33313. 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 33309* Lemma for lshpkrex 33313. 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 33310* Lemma for lshpkrex 33313. 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 33311* 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 33312* 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 33313* 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 33314* 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 33315* 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 33313? 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 33316* 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 33317* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 33316 may be more compatible with lspsn 18727. (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 33318 Extend class notation with left dualvector space.
class LDual

Definitiondf-ldual 33319* 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 6930. 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 33320* 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 33321 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 33322 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 33323 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 33324 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 33325 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 33326 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 33327 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 33328 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 33329 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 33330 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 33331* 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 33332 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 33333 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 33334 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 33335 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 33336 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 33337 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 33338 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 33339 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 33340 Lemma for ldualgrp 33341. (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 33341 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
𝐷 = (LDual‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐷 ∈ Grp)

Theoremldual0 33342 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 33343 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 33344 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 33345 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 33346 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 33347 Lemma for lduallmod 33348. (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 33348 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 33349 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 18353; 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 33350 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 33351 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 33352 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 33350? (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 33353 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 33354 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 33355 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 33356 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 33357 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 33358 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 33359 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 33360* 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 33361* 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 33362 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 33363 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 33364* 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 33365 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 33366 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 33367 Extend class notation with orthoposets.
class OP

SyntaxccmtN 33368 Extend class notation with the commutes relation.
class cm

Syntaxcol 33369 Extend class notation with orthlattices.
class OL

Syntaxcoml 33370 Extend class notation with orthomodular lattices.
class OML

Definitiondf-oposet 33371* 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 33372* 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 33373 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
OL = (Lat ∩ OP)

Definitiondf-oml 33374* 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 33375* 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 33376 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
(𝐾 ∈ OP → 𝐾 ∈ Poset)

Theoremoposlem 33377 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 33378 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 33379 An orthoposet has a zero element. (h0elch 27284 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ OP → 0𝐵)

Theoremop1cl 33380 An orthoposet has a unit element. (helch 27272 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ OP → 1𝐵)

Theoremop0le 33381 Orthoposet zero is less than or equal to any element. (ch0le 27472 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → 0 𝑋)

Theoremople0 33382 An element less than or equal to zero equals zero. (chle0 27474 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))

Theoremopnlen0 33383 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2700 and op0le 33381 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 33384 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 33385 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 33386 Any element is less than the orthoposet unit. (chss 27258 analog.) (Contributed by NM, 23-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → 𝑋 1 )

Theoremop1le 33387 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 27474 analog.) (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 1 𝑋𝑋 = 1 ))

Theoremglb0N 33388 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 33389 Closure of orthocomplement operation. (choccl 27337 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)

Theoremopococ 33390 Double negative law for orthoposets. (ococ 27437 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)

Theoremopcon3b 33391 Contraposition law for orthoposets. (chcon3i 27497 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))

Theoremopcon2b 33392 Orthocomplement contraposition law. (negcon2 10085 analog.) (Contributed by NM, 16-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))

Theoremopcon1b 33393 Orthocomplement contraposition law. (negcon1 10084 analog.) (Contributed by NM, 24-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = 𝑌 ↔ ( 𝑌) = 𝑋))

Theoremoplecon3 33394 Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))

Theoremoplecon3b 33395 Contraposition law for orthoposets. (chsscon3 27531 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))

Theoremoplecon1b 33396 Contraposition law for strict ordering in orthoposets. (chsscon1 27532 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))

Theoremopoc1 33397 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OP → ( 1 ) = 0 )

Theoremopoc0 33398 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OP → ( 0 ) = 1 )

Theoremopltcon3b 33399 Contraposition law for strict ordering in orthoposets. (chpsscon3 27534 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ( 𝑌) < ( 𝑋)))

Theoremopltcon1b 33400 Contraposition law for strict ordering in orthoposets. (chpsscon1 27535 analog.) (Contributed by NM, 5-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) < 𝑌 ↔ ( 𝑌) < 𝑋))

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