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	eqlkr3 39101	Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻))    &   ⊢ (𝜑 → (𝐺‘𝑋) = (𝐻‘𝑋))    &   ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 )    ⇒   ⊢ (𝜑 → 𝐺 = 𝐻)
Theorem	lkrlsp 39102	The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 39089) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ (𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐾‘𝐺) ⊕ (𝑁‘{𝑋})) = 𝑉)
Theorem	lkrlsp2 39103	The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ (𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹) ∧ ¬ 𝑋 ∈ (𝐾‘𝐺)) → ((𝐾‘𝐺) ⊕ (𝑁‘{𝑋})) = 𝑉)
Theorem	lkrlsp3 39104	The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ (𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹) ∧ ¬ 𝑋 ∈ (𝐾‘𝐺)) → (𝑁‘((𝐾‘𝐺) ∪ {𝑋})) = 𝑉)
Theorem	lkrshp 39105	The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻)
Theorem	lkrshp3 39106	The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ (𝑉 × { 0 })))
Theorem	lkrshpor 39107	The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉))
Theorem	lkrshp4 39108	A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐾‘𝐺) ≠ 𝑉 ↔ (𝐾‘𝐺) ∈ 𝐻))
Theorem	lshpsmreu 39109*	Lemma for lshpkrex 39118. 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 3341 for 𝑎 to 𝑐? (Contributed by NM, 4-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ (𝜑 → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)))
Theorem	lshpkrlem1 39110*	Lemma for lshpkrex 39118. The value of tentative functional 𝐺 is zero iff its argument belongs to hyperplane 𝑈. (Contributed by NM, 14-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝐺‘𝑋) = 0 ))
Theorem	lshpkrlem2 39111*	Lemma for lshpkrex 39118. The value of tentative functional 𝐺 is a scalar. (Contributed by NM, 16-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐾)
Theorem	lshpkrlem3 39112*	Lemma for lshpkrex 39118. Defining property of 𝐺‘𝑋. (Contributed by NM, 15-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝑋 = (𝑧 + ((𝐺‘𝑋) · 𝑍)))
Theorem	lshpkrlem4 39113*	Lemma for lshpkrex 39118. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    ⇒   ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍)))
Theorem	lshpkrlem5 39114*	Lemma for lshpkrex 39118. Part of showing linearity of 𝐺. (Contributed by NM, 16-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    ⇒   ⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑈 ∧ (𝑠 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)) ∧ ((𝑙 · 𝑢) + 𝑣) = (𝑧 + ((𝐺‘((𝑙 · 𝑢) + 𝑣)) · 𝑍)))) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)))
Theorem	lshpkrlem6 39115*	Lemma for lshpkrex 39118. Show linearlity of 𝐺. (Contributed by NM, 17-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    ⇒   ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝐺‘((𝑙 · 𝑢) + 𝑣)) = ((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)))
Theorem	lshpkrcl 39116*	The set 𝐺 defined by hyperplane 𝑈 is a linear functional. (Contributed by NM, 17-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝐹)
Theorem	lshpkr 39117*	The kernel of functional 𝐺 is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))))    &   ⊢ 𝐿 = (LKer‘𝑊)    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) = 𝑈)
Theorem	lshpkrex 39118*	There exists a functional whose kernel equals a given hyperplane. Part of Th. 1.27 of Barbu and Precupanu, Convexity and Optimization in Banach Spaces. (Contributed by NM, 17-Jul-2014.)
⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝐻) → ∃𝑔 ∈ 𝐹 (𝐾‘𝑔) = 𝑈)
Theorem	lshpset2N 39119*	The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → 𝐻 = {𝑠 ∣ ∃𝑔 ∈ 𝐹 (𝑔 ≠ (𝑉 × { 0 }) ∧ 𝑠 = (𝐾‘𝑔))})
Theorem	islshpkrN 39120*	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 39118? 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 39121*	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 39122*	Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 39121 may be more compatible with lspsn 20915. (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 39123	Extend class notation with left dualvector space.
class LDual
Definition	df-ldual 39124*	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 7966. 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 39125*	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 39126	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 39127	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 39128	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 39129	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 39130	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 39131	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 39132	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 39133	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 39134	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 39135	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 39136*	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 39137	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 39138	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 39139	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 39140	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 39141	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 39142	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 39143	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 39144	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 39145	Lemma for ldualgrp 39146. (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 39146	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 39147	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 39148	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 39149	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 39150	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 39151	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 39152	Lemma for lduallmod 39153. (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 39153	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 39154	The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 20253; 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 39155	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 39156	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 39157	The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 39155? (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 39158	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 39159	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 39160	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 39161	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 39162	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 39163	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 39164	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 39165*	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 39166*	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 39167	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 39168	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 39169*	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 39170	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 39171	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 39172	Extend class notation with orthoposets.
class OP
Syntax	ccmtN 39173	Extend class notation with the commutes relation.
class cm
Syntax	col 39174	Extend class notation with orthlattices.
class OL
Syntax	coml 39175	Extend class notation with orthomodular lattices.
class OML
Definition	df-oposet 39176*	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 39177*	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 39178	Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
⊢ OL = (Lat ∩ OP)
Definition	df-oml 39179*	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 39180*	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 39181	Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
Theorem	oposlem 39182	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 39183	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 39184	An orthoposet has a zero element. (h0elch 31191 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵)
Theorem	op1cl 39185	An orthoposet has a unity element. (helch 31179 analog.) (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵)
Theorem	op0le 39186	Orthoposet zero is less than or equal to any element. (ch0le 31377 analog.) (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋)
Theorem	ople0 39187	An element less than or equal to zero equals zero. (chle0 31379 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 39188	An element not less than another is nonzero. TODO: Look for uses of necon3bd 2940 and op0le 39186 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 39189	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 39190	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 39191	Any element is less than the orthoposet unity. (chss 31165 analog.) (Contributed by NM, 23-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 )
Theorem	op1le 39192	If the orthoposet unity is less than or equal to an element, the element equals the unit. (chle0 31379 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 39193	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 39194	Closure of orthocomplement operation. (choccl 31242 analog.) (Contributed by NM, 20-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
Theorem	opococ 39195	Double negative law for orthoposets. (ococ 31342 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
Theorem	opcon3b 39196	Contraposition law for orthoposets. (chcon3i 31402 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)))
Theorem	opcon2b 39197	Orthocomplement contraposition law. (negcon2 11482 analog.) (Contributed by NM, 16-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋)))
Theorem	opcon1b 39198	Orthocomplement contraposition law. (negcon1 11481 analog.) (Contributed by NM, 24-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = 𝑌 ↔ ( ⊥ ‘𝑌) = 𝑋))
Theorem	oplecon3 39199	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 39200	Contraposition law for orthoposets. (chsscon3 31436 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))