P. 199 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ip0r 19801	Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑍 = (0g‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 , 0 ) = 𝑍)
Theorem	ipeq0 19802	The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑍 = (0g‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 𝑍 ↔ 𝐴 = 0 ))
Theorem	ipdir 19803	Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ ⨣ = (+g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) ⨣ (𝐵 , 𝐶)))
Theorem	ipdi 19804	Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ ⨣ = (+g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) ⨣ (𝐴 , 𝐶)))
Theorem	ip2di 19805	Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ PreHil)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶))))
Theorem	ipsubdir 19806	Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑆 = (-g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) , 𝐶) = ((𝐴 , 𝐶)𝑆(𝐵 , 𝐶)))
Theorem	ipsubdi 19807	Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑆 = (-g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)))
Theorem	ip2subdi 19808	Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑆 = (-g‘𝐹)    &   ⊢ + = (+g‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ PreHil)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))
Theorem	ipass 19809	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶)))
Theorem	ipassr 19810	"Associative" law for second argument of inner product (compare ipass 19809). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝐹)    &   ⊢ ∗ = (*𝑟‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))
Theorem	ipassr2 19811	"Associative" law for inner product. Conjugate version of ipassr 19810. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝐹)    &   ⊢ ∗ = (*𝑟‘𝐹)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( ∗ ‘𝐶) · 𝐵)))
Theorem	ipffval 19812*	The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ · = (·if‘𝑊)    ⇒   ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
Theorem	ipfval 19813	The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ · = (·if‘𝑊)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))
Theorem	ipfeq 19814	If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ · = (·if‘𝑊)    ⇒   ⊢ ( , Fn (𝑉 × 𝑉) → · = , )
Theorem	ipffn 19815	The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·if‘𝑊)    ⇒   ⊢ , Fn (𝑉 × 𝑉)
Theorem	phlipf 19816	The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·if‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾)
Theorem	ip2eq 19817*	Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵)))
Theorem	isphld 19818*	Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ (𝜑 → 𝑉 = (Base‘𝑊))    &   ⊢ (𝜑 → + = (+g‘𝑊))    &   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))    &   ⊢ (𝜑 → 𝐼 = (·𝑖‘𝑊))    &   ⊢ (𝜑 → 0 = (0g‘𝑊))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝐹))    &   ⊢ (𝜑 → ⨣ = (+g‘𝐹))    &   ⊢ (𝜑 → × = (.r‘𝐹))    &   ⊢ (𝜑 → ∗ = (*𝑟‘𝐹))    &   ⊢ (𝜑 → 𝑂 = (0g‘𝐹))    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝐹 ∈ *-Ring)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐾 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) ⨣ (𝑦𝐼𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ( ∗ ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))    ⇒   ⊢ (𝜑 → 𝑊 ∈ PreHil)
Theorem	phlpropd 19819*	If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))    &   ⊢ 𝑃 = (Base‘𝐹)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(·𝑖‘𝐾)𝑦) = (𝑥(·𝑖‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Theorem	ssipeq 19820	The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑃 = (·𝑖‘𝑋)    ⇒   ⊢ (𝑈 ∈ 𝑆 → 𝑃 = , )
Theorem	phssipval 19821	The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑃 = (·𝑖‘𝑋)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝑃𝐵) = (𝐴 , 𝐵))
Theorem	phssip 19822	The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ · = (·if‘𝑊)    &   ⊢ 𝑃 = (·if‘𝑋)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))
10.12.2  Orthocomplements and closed subspaces
Syntax	cocv 19823	Extend class notation with orthocomplement of a subspace.
class ocv
Syntax	ccss 19824	Extend class notation with set of closed subspaces.
class CSubSp
Syntax	cthl 19825	Extend class notation with the Hilbert lattice.
class toHL
Definition	df-ocv 19826*	Define orthocomplement of a subspace. (Contributed by NM, 7-Oct-2011.)
⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
Definition	df-css 19827*	Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)
⊢ CSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
Definition	df-thl 19828	Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(CSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
Theorem	ocvfval 19829*	The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
Theorem	ocvval 19830*	Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
Theorem	elocv 19831*	Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
Theorem	ocvi 19832	Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 )
Theorem	ocvss 19833	The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
Theorem	ocvocv 19834	A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)))
Theorem	ocvlss 19835	The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐿)
Theorem	ocv2ss 19836	Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))
Theorem	ocvin 19837	An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 })
Theorem	ocvsscon 19838	Two ways to say that 𝑆 and 𝑇 are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑉) → (𝑆 ⊆ ( ⊥ ‘𝑇) ↔ 𝑇 ⊆ ( ⊥ ‘𝑆)))
Theorem	ocvlsp 19839	The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘(𝑁‘𝑆)) = ( ⊥ ‘𝑆))
Theorem	ocv0 19840	The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘∅) = 𝑉
Theorem	ocvz 19841	The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → ( ⊥ ‘{ 0 }) = 𝑉)
Theorem	ocv1 19842	The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → ( ⊥ ‘𝑉) = { 0 })
Theorem	unocv 19843	The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))
Theorem	iunocv 19844*	The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵))
Theorem	cssval 19845*	The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
Theorem	iscss 19846	The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))
Theorem	cssi 19847	Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
Theorem	cssss 19848	A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑉)
Theorem	iscss2 19849	It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
Theorem	ocvcss 19850	The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐶)
Theorem	cssincl 19851	The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)
Theorem	css0 19852	The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)
Theorem	css1 19853	The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝑉 ∈ 𝐶)
Theorem	csslss 19854	A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ 𝐿)
Theorem	lsmcss 19855	A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ PreHil)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑉)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝑆 ⊕ ( ⊥ ‘𝑆)))    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)
Theorem	cssmre 19856	The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 16072: consider the Hilbert space of sequences ℕ⟶ℝ with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 16138. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))
Theorem	mrccss 19857	The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘𝑆)))
Theorem	thlval 19858	Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
Theorem	thlbas 19859	Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ 𝐶 = (Base‘𝐾)
Theorem	thlle 19860	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ≤ = (le‘𝐾)
Theorem	thlleval 19861	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐶) → (𝑆 ≤ 𝑇 ↔ 𝑆 ⊆ 𝑇))
Theorem	thloc 19862	Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ⊥ = (oc‘𝐾)
10.12.3  Orthogonal projection and orthonormal bases
Syntax	cpj 19863	Extend class notation with orthogonal projection function.
class proj
Syntax	chs 19864	Extend class notation with class of all Hilbert spaces.
class Hil
Syntax	cobs 19865	Extend class notation with the set of orthonormal bases.
class OBasis
Definition	df-pj 19866*	Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 17875, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj1‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑𝑚 (Base‘ℎ)))))
Definition	df-hil 19867	Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.)
⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (CSubSp‘ℎ)}
Definition	df-obs 19868*	Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})
Theorem	pjfval 19869*	The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉)))
Theorem	pjdm 19870	A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))
Theorem	pjpm 19871	The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ 𝐾 ∈ ((𝑉 ↑𝑚 𝑉) ↑pm 𝐿)
Theorem	pjfval2 19872*	Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
Theorem	pjval 19873	Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇)))
Theorem	pjdm2 19874	A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)))
Theorem	pjff 19875	A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝐾:dom 𝐾⟶(𝑊 LMHom 𝑊))
Theorem	pjf 19876	A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉)
Theorem	pjf2 19877	A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)
Theorem	pjfo 19878	A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇)
Theorem	pjcss 19879	A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ 𝐶)
Theorem	ocvpj 19880	The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( ⊥ ‘𝑇) ∈ dom 𝐾)
Theorem	ishil 19881	The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐾 = (proj‘𝐻)    &   ⊢ 𝐶 = (CSubSp‘𝐻)    ⇒   ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
Theorem	ishil2 19882*	The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝑉 = (Base‘𝐻)    &   ⊢ ⊕ = (LSSum‘𝐻)    &   ⊢ ⊥ = (ocv‘𝐻)    &   ⊢ 𝐶 = (CSubSp‘𝐻)    ⇒   ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ ∀𝑠 ∈ 𝐶 (𝑠 ⊕ ( ⊥ ‘𝑠)) = 𝑉))
Theorem	isobs 19883*	The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑌 = (0g‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))
Theorem	obsip 19884	The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Theorem	obsipid 19885	A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 )
Theorem	obsrcl 19886	Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
Theorem	obsss 19887	An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ 𝑉)
Theorem	obsne0 19888	A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 )
Theorem	obsocv 19889	An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘𝐵) = { 0 })
Theorem	obs2ocv 19890	The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝑉)
Theorem	obselocv 19891	A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ¬ 𝐴 ∈ 𝐶))
Theorem	obs2ss 19892	A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)
Theorem	obslbs 19893	An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐶 = (CSubSp‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → (𝐵 ∈ 𝐽 ↔ (𝑁‘𝐵) ∈ 𝐶))
PART 11  BASIC LINEAR ALGEBRA According to Wikipedia ("Linear algebra", 03-Mar-2019, https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." However, dealing with modules (over rings) instead of vector spaces (over fields) allows for a more general approach. Therefore, "vectors" are regarded as members (elements of the base set) of a (free) module over a ring (see df-frlm 19910) in the following. By this, the number of entries in a vector is determined by the size of the index set of the direct sum building the free module the vector is belonging to. Since every vector space is isomorphic to a free module (see lvecisfrlm 20001), the theorems stated for free modules are also valid for vector spaces. Until not explicitly stated, the underlying ring needs not to be commutative (see df-cring 18373), but the existence of a multiplicative neutral element is always presumed (the ring is a unital ring, see also df-ring 18372). In this sense, linear equations, matrices and determinants are usually regarded as "over a ring" in this part.
11.1  Vectors and free modules
11.1.1  Direct sum of left modules According to Wikipedia ("Direct sum of modules", 28-Mar-2019, https://en.wikipedia.org/wiki/Direct_sum_of_modules) "Let R be a ring, and { Mi: i ∈ I } a family of left R-modules indexed by the set I. The direct sum of {Mi} is then defined to be the set of all sequences (αi) where αi ∈ Mi and αi = 0 for cofinitely many indices i. (The direct product is analogous but the indices do not need to cofinitely vanish.)". In this definition, "cofinitely many" means "almost all" or "for all but finitely many". Furthemore, "This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences α and β can be added by writing (α + β)i = αi + βi for all i (note that this is again zero for all but finitely many indices), and such a sequence can be multiplied with an element r from R by defining r(α)i = (rα)i for all i.". In [Lang] p. 128, the definition of the direct sum of left modules is based on direct sums of abelian groups ("We define on [the direct sum of abelian groups Mi] M a structure of A-module: If (xi)i ∈ I is an element of M, i.e. a familiy of elements xi ∈ Mi such that xi = 0 for almost all i, and if a ∈ A, then we define a(xi)i ∈ I = (axi)i ∈ I, that is we define multiplication by a componentwise.") which itself is based on the direct product of abelian groups ([Lang] p. 36: "Let {Ai}i ∈ I be a family of abelian groups. We define their direct sum A ... to be the subset of the direct product ... consisting of all families (xi)i ∈ I with xi ∈ Ai such that xi = 0 for all but a finite number of indices i"). In short, the direct sum of a familiy of (left) modules {Mi}i ∈ I is the restriction of the direct product of {Mi}i ∈ I to the elements with index function having finite support, as formalized by the definition df-dsmm 19895.
Syntax	cdsmm 19894	Class of module direct sum generator.
class ⊕m
Definition	df-dsmm 19895*	The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin}))
Theorem	reldmdsmm 19896	The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ Rel dom ⊕m
Theorem	dsmmval 19897*	Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵))
Theorem	dsmmbase 19898*	Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))
Theorem	dsmmval2 19899	Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐵 = (Base‘(𝑆 ⊕m 𝑅))    ⇒   ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)
Theorem	dsmmbas2 19900*	Base set of the direct sum module using the fndmin 6232 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}    ⇒   ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))