P. 392 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	islshpkrN 39101*	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 39099? 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 39102*	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 39103*	Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 39102 may be more compatible with lspsn 21017. (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}))})
21.28.8  Opposite rings and dual vector spaces
Syntax	cld 39104	Extend class notation with left dualvector space.
class LDual
Definition	df-ldual 39105*	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 to reuse our existing collection of left vector space theorems. The restriction on ∘f (+g‘𝑣) allows it to be a set; see ofmres 8007. Note the operation reversal in the scalar product to allow 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 39106*	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 to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow 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 39107	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 39108	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 39109	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 39110	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 39111	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 39112	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 39113	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 39114	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 39115	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 39116	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 39117*	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 39118	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 39119	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 39120	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 39121	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 39122	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 39123	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 39124	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 39125	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 39126	Lemma for ldualgrp 39127. (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 39127	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 39128	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 39129	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 39130	The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝑀 = (invg‘𝑅)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐷)    &   ⊢ 𝑁 = (invg‘𝑆)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 𝑁 = 𝑀)
Theorem	ldual0v 39131	The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ 𝑂 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 }))
Theorem	ldual0vcl 39132	The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 0 ∈ 𝐹)
Theorem	lduallmodlem 39133	Lemma for lduallmod 39134. (Contributed by NM, 22-Oct-2014.)
⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = ∘f (+g‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐷)    ⇒   ⊢ (𝜑 → 𝐷 ∈ LMod)
Theorem	lduallmod 39134	The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 𝐷 ∈ LMod)
Theorem	lduallvec 39135	The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 20350; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → 𝐷 ∈ LVec)
Theorem	ldualvsub 39136	The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ + = (+g‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ − = (-g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻)))
Theorem	ldualvsubcl 39137	Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ − = (-g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 − 𝐻) ∈ 𝐹)
Theorem	ldualvsubval 39138	The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 39136? (Requires 𝐷 to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝑆 = (-g‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ − = (-g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐺 − 𝐻)‘𝑋) = ((𝐺‘𝑋)𝑆(𝐻‘𝑋)))
Theorem	ldualssvscl 39139	Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ 𝑆 = (LSubSp‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑈)
Theorem	ldualssvsubcl 39140	Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝑆 = (LSubSp‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑈)
Theorem	ldual0vs 39141	Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ 𝑂 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ( 0 · 𝐺) = 𝑂)
Theorem	lkr0f2 39142	The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = 0 ))
Theorem	lduallkr3 39143	The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ 0 ))
Theorem	lkrpssN 39144	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 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))
Theorem	lkrin 39145	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)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ∩ (𝐾‘𝐻)) ⊆ (𝐾‘(𝐺 + 𝐻)))
Theorem	eqlkr4 39146*	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)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))
Theorem	ldual1dim 39147*	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)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)})
Theorem	ldualkrsc 39148	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 = (0g‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐿 = (LKer‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺))
Theorem	lkrss 39149	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)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))
Theorem	lkrss2N 39150*	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)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)))
Theorem	lkreqN 39151	Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ 𝐷 = (LDual‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑅 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (𝐴 · 𝐻))    ⇒   ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻))
Theorem	lkrlspeqN 39152	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 = (0g‘𝐷)    &   ⊢ 𝑁 = (LSpan‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ ((𝑁‘{𝐻}) ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) = (𝐿‘𝐻))
21.28.9  Ortholattices and orthomodular lattices
Syntax	cops 39153	Extend class notation with orthoposets.
class OP
Syntax	ccmtN 39154	Extend class notation with the commutes relation.
class cm
Syntax	col 39155	Extend class notation with orthlattices.
class OL
Syntax	coml 39156	Extend class notation with orthomodular lattices.
class OML
Definition	df-oposet 39157*	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)(((𝑜‘𝑎) ∈ (Base‘𝑝) ∧ (𝑜‘(𝑜‘𝑎)) = 𝑎 ∧ (𝑎(le‘𝑝)𝑏 → (𝑜‘𝑏)(le‘𝑝)(𝑜‘𝑎))) ∧ (𝑎(join‘𝑝)(𝑜‘𝑎)) = (1.‘𝑝) ∧ (𝑎(meet‘𝑝)(𝑜‘𝑎)) = (0.‘𝑝))))}
Definition	df-cmtN 39158*	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))))})
Definition	df-ol 39159	Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
⊢ OL = (Lat ∩ OP)
Definition	df-oml 39160*	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)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
Theorem	isopos 39161*	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 𝑈 ∧ 𝐵 ∈ dom 𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))
Theorem	opposet 39162	Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
Theorem	oposlem 39163	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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
Theorem	op01dm 39164	Conditions necessary for zero and unity elements to exist. (Contributed by NM, 14-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺))
Theorem	op0cl 39165	An orthoposet has a zero element. (h0elch 31283 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵)
Theorem	op1cl 39166	An orthoposet has a unity element. (helch 31271 analog.) (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵)
Theorem	op0le 39167	Orthoposet zero is less than or equal to any element. (ch0le 31469 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋)
Theorem	ople0 39168	An element less than or equal to zero equals zero. (chle0 31471 analog.) (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 ))
Theorem	opnlen0 39169	An element not less than another is nonzero. TODO: Look for uses of necon3bd 2951 and op0le 39167 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑋 ≤ 𝑌) → 𝑋 ≠ 0 )
Theorem	lub0N 39170	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 )
Theorem	opltn0 39171	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 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 ))
Theorem	ople1 39172	Any element is less than the orthoposet unity. (chss 31257 analog.) (Contributed by NM, 23-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 )
Theorem	op1le 39173	If the orthoposet unity is less than or equal to an element, the element equals the unit. (chle0 31471 analog.) (Contributed by NM, 5-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ 𝑋 = 1 ))
Theorem	glb0N 39174	The greatest lower bound of the empty set is the unity element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → (𝐺‘∅) = 1 )
Theorem	opoccl 39175	Closure of orthocomplement operation. (choccl 31334 analog.) (Contributed by NM, 20-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
Theorem	opococ 39176	Double negative law for orthoposets. (ococ 31434 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
Theorem	opcon3b 39177	Contraposition law for orthoposets. (chcon3i 31494 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)))
Theorem	opcon2b 39178	Orthocomplement contraposition law. (negcon2 11559 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋)))
Theorem	opcon1b 39179	Orthocomplement contraposition law. (negcon1 11558 analog.) (Contributed by NM, 24-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋))
Theorem	oplecon3 39180	Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
Theorem	oplecon3b 39181	Contraposition law for orthoposets. (chsscon3 31528 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
Theorem	oplecon1b 39182	Contraposition law for strict ordering in orthoposets. (chsscon1 31529 analog.) (Contributed by NM, 6-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋))
Theorem	opoc1 39183	Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 )
Theorem	opoc0 39184	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 )
Theorem	opltcon3b 39185	Contraposition law for strict ordering in orthoposets. (chpsscon3 31531 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ( ⊥ ‘𝑌) < ( ⊥ ‘𝑋)))
Theorem	opltcon1b 39186	Contraposition law for strict ordering in orthoposets. (chpsscon1 31532 analog.) (Contributed by NM, 5-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < 𝑋))
Theorem	opltcon2b 39187	Contraposition law for strict ordering in orthoposets. (chsscon2 31530 analog.) (Contributed by NM, 5-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋)))
Theorem	opexmid 39188	Law of excluded middle for orthoposets. (chjo 31543 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )
Theorem	opnoncon 39189	Law of contradiction for orthoposets. (chocin 31523 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )
Theorem	riotaocN 39190*	The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12244 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))
Theorem	cmtfvalN 39191*	Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
Theorem	cmtvalN 39192	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 31612 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
Theorem	isolat 39193	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
Theorem	ollat 39194	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
Theorem	olop 39195	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
Theorem	olposN 39196	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)
Theorem	isolatiN 39197	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
⊢ 𝐾 ∈ Lat    &   ⊢ 𝐾 ∈ OP    ⇒   ⊢ 𝐾 ∈ OL
Theorem	oldmm1 39198	De Morgan's law for meet in an ortholattice. (chdmm1 31553 analog.) (Contributed by NM, 6-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)))
Theorem	oldmm2 39199	De Morgan's law for meet in an ortholattice. (chdmm2 31554 analog.) (Contributed by NM, 6-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) = (𝑋 ∨ ( ⊥ ‘𝑌)))
Theorem	oldmm3N 39200	De Morgan's law for meet in an ortholattice. (chdmm3 31555 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))