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
21.28.8  Opposite rings and dual vector spaces
Syntax	cld 39101	Extend class notation with left dualvector space.
class LDual
Definition	df-ldual 39102*	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 7926. 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 39103*	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 39104	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 39105	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 39106	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 39107	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 39108	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 39109	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 39110	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 39111	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 39112	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 39113	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 39114*	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 39115	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 39116	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 39117	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 39118	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 39119	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 39120	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 39121	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 39122	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 39123	Lemma for ldualgrp 39124. (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 39124	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 39125	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 39126	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 39127	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 39128	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 39129	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 39130	Lemma for lduallmod 39131. (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 39131	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 39132	The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 20240; 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 39133	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 39134	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 39135	The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 39133? (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 39136	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 39137	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 39138	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 39139	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 39140	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 39141	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 39142	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 39143*	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 39144*	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 39145	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 39146	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 39147*	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 39148	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 39149	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 39150	Extend class notation with orthoposets.
class OP
Syntax	ccmtN 39151	Extend class notation with the commutes relation.
class cm
Syntax	col 39152	Extend class notation with orthlattices.
class OL
Syntax	coml 39153	Extend class notation with orthomodular lattices.
class OML
Definition	df-oposet 39154*	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 39155*	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 39156	Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
⊢ OL = (Lat ∩ OP)
Definition	df-oml 39157*	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 39158*	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 39159	Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
Theorem	oposlem 39160	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 39161	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 39162	An orthoposet has a zero element. (h0elch 31217 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵)
Theorem	op1cl 39163	An orthoposet has a unity element. (helch 31205 analog.) (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵)
Theorem	op0le 39164	Orthoposet zero is less than or equal to any element. (ch0le 31403 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋)
Theorem	ople0 39165	An element less than or equal to zero equals zero. (chle0 31405 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 39166	An element not less than another is nonzero. TODO: Look for uses of necon3bd 2939 and op0le 39164 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 39167	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 39168	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 39169	Any element is less than the orthoposet unity. (chss 31191 analog.) (Contributed by NM, 23-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 )
Theorem	op1le 39170	If the orthoposet unity is less than or equal to an element, the element equals the unit. (chle0 31405 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 39171	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 39172	Closure of orthocomplement operation. (choccl 31268 analog.) (Contributed by NM, 20-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
Theorem	opococ 39173	Double negative law for orthoposets. (ococ 31368 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
Theorem	opcon3b 39174	Contraposition law for orthoposets. (chcon3i 31428 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)))
Theorem	opcon2b 39175	Orthocomplement contraposition law. (negcon2 11435 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋)))
Theorem	opcon1b 39176	Orthocomplement contraposition law. (negcon1 11434 analog.) (Contributed by NM, 24-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋))
Theorem	oplecon3 39177	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 39178	Contraposition law for orthoposets. (chsscon3 31462 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 39179	Contraposition law for strict ordering in orthoposets. (chsscon1 31463 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 39180	Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 )
Theorem	opoc0 39181	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 )
Theorem	opltcon3b 39182	Contraposition law for strict ordering in orthoposets. (chpsscon3 31465 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 39183	Contraposition law for strict ordering in orthoposets. (chpsscon1 31466 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 39184	Contraposition law for strict ordering in orthoposets. (chsscon2 31464 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 39185	Law of excluded middle for orthoposets. (chjo 31477 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 39186	Law of contradiction for orthoposets. (chocin 31457 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 39187*	The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12122 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 39188*	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 39189	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 31546 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 39190	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
Theorem	ollat 39191	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
Theorem	olop 39192	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
Theorem	olposN 39193	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)
Theorem	isolatiN 39194	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
⊢ 𝐾 ∈ Lat    &   ⊢ 𝐾 ∈ OP    ⇒   ⊢ 𝐾 ∈ OL
Theorem	oldmm1 39195	De Morgan's law for meet in an ortholattice. (chdmm1 31487 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 39196	De Morgan's law for meet in an ortholattice. (chdmm2 31488 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 39197	De Morgan's law for meet in an ortholattice. (chdmm3 31489 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))
Theorem	oldmm4 39198	De Morgan's law for meet in an ortholattice. (chdmm4 31490 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	oldmj1 39199	De Morgan's law for join in an ortholattice. (chdmj1 31491 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	oldmj2 39200	De Morgan's law for join in an ortholattice. (chdmj2 31492 analog.) (Contributed by NM, 7-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ ( ⊥ ‘𝑌)))