P. 219 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ip0l 21801	Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (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) → ( 0 , 𝐴) = 𝑍)
Theorem	ip0r 21802	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 21803	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 21804	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 21805	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 21806	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 21807	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 21808	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 21809	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 21810	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 21811	"Associative" law for second argument of inner product (compare ipass 21810). (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 21812	"Associative" law for inner product. Conjugate version of ipassr 21811. (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 21813*	The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ · = (·if‘𝑊)    ⇒   ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
Theorem	ipfval 21814	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 21815	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 21816	The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·if‘𝑊)    ⇒   ⊢ , Fn (𝑉 × 𝑉)
Theorem	phlipf 21817	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 21818*	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 21819*	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 21820*	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 21821	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 21822	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 21823	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)))
Theorem	phlssphl 21824	A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)
10.9.2  Orthocomplements and closed subspaces
Syntax	cocv 21825	Extend class notation with orthocomplement of a subset.
class ocv
Syntax	ccss 21826	Extend class notation with set of closed subspaces.
class ClSubSp
Syntax	cthl 21827	Extend class notation with the Hilbert lattice.
class toHL
Definition	df-ocv 21828*	Define the orthocomplement function in a given set (which usually is a pre-Hilbert space): it associates with a subset its orthogonal subset (which in the case of a closed linear subspace is its orthocomplement). (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 21829*	Define the set of closed (linear) subspaces of a given pre-Hilbert space. (Contributed by NM, 7-Oct-2011.)
⊢ ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
Definition	df-thl 21830	Define the Hilbert lattice of closed subspaces of a given pre-Hilbert space. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
Theorem	ocvfval 21831*	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 21832*	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 21833*	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 21834	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 21835	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 21836	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 21837	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 21838	Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))
Theorem	ocvin 21839	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 21840	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 21841	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 21842	The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘∅) = 𝑉
Theorem	ocvz 21843	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 21844	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 21845	The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))
Theorem	iunocv 21846*	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 21847*	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 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})
Theorem	iscss 21848	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 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))))
Theorem	cssi 21849	Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))
Theorem	cssss 21850	A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑉)
Theorem	iscss2 21851	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 = (ClSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
Theorem	ocvcss 21852	The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐶)
Theorem	cssincl 21853	The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)
Theorem	css0 21854	The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)
Theorem	css1 21855	The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝑉 ∈ 𝐶)
Theorem	csslss 21856	A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ 𝐿)
Theorem	lsmcss 21857	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 = (ClSubSp‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ PreHil)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑉)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝑆 ⊕ ( ⊥ ‘𝑆)))    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)
Theorem	cssmre 21858	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 17651: 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 17720. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))
Theorem	mrccss 21859	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 = (ClSubSp‘𝑊)    &   ⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘𝑆)))
Theorem	thlval 21860	Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⊥ ⟩))
Theorem	thlbas 21861	Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ 𝐶 = (Base‘𝐾)
Theorem	thlle 21862	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ≤ = (le‘𝐾)
Theorem	thlleval 21863	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐶) → (𝑆 ≤ 𝑇 ↔ 𝑆 ⊆ 𝑇))
Theorem	thloc 21864	Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ⊥ = (oc‘𝐾)
10.9.3  Orthogonal projection and orthonormal bases
Syntax	cpj 21865	Extend class notation with orthogonal projection function.
class proj
Syntax	chil 21866	Extend class notation with class of all Hilbert spaces.
class Hil
Syntax	cobs 21867	Extend class notation with the set of orthonormal bases.
class OBasis
Definition	df-pj 21868*	Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 19717, 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 21869	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) = (ClSubSp‘ℎ)}
Definition	df-obs 21870*	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 21871*	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 21872	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 21873	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 21874*	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 21875	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 21876	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 21877	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 21878	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 21879	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 21880	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 21881	A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ 𝐶)
Theorem	ocvpj 21882	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 21883	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 = (ClSubSp‘𝐻)    ⇒   ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
Theorem	ishil2 21884*	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 = (ClSubSp‘𝐻)    ⇒   ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ ∀𝑠 ∈ 𝐶 (𝑠 ⊕ ( ⊥ ‘𝑠)) = 𝑉))
Theorem	isobs 21885*	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 21886	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 21887	A basis element has length one. (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 21888	Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
Theorem	obsss 21889	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 21890	A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 )
Theorem	obsocv 21891	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 21892	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 21893	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 21894	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 21895	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 = (ClSubSp‘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." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part. Unless otherwise stated, the rings of scalars need not be commutative (see df-cring 20328), but the existence of a unity element is always assumed (our rings are unital, see df-ring 20327). For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space.
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 componentwise 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 Definition df-dsmm 21897.
Syntax	cdsmm 21896	Class of module direct sum generator.
class ⊕m
Definition	df-dsmm 21897*	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 21898	The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ Rel dom ⊕m
Theorem	dsmmval 21899*	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 21900*	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)))