P. 213 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ipfval 21201	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 21202	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 21203	The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·if‘𝑊)    ⇒   ⊢ , Fn (𝑉 × 𝑉)
Theorem	phlipf 21204	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 21205*	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 21206*	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 21207*	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 21208	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 21209	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 21210	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 21211	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 21212	Extend class notation with orthocomplement of a subset.
class ocv
Syntax	ccss 21213	Extend class notation with set of closed subspaces.
class ClSubSp
Syntax	cthl 21214	Extend class notation with the Hilbert lattice.
class toHL
Definition	df-ocv 21215*	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 21216*	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 21217	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 21218*	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 21219*	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 21220*	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 21221	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 21222	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 21223	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 21224	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 21225	Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))
Theorem	ocvin 21226	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 21227	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 21228	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 21229	The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘∅) = 𝑉
Theorem	ocvz 21230	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 21231	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 21232	The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))
Theorem	iunocv 21233*	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 21234*	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 21235	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 21236	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 21237	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 21238	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 21239	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 21240	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 21241	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 21242	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 21243	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 21244	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 21245	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 17532: 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 17597. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))
Theorem	mrccss 21246	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 21247	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 21248	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	thlbasOLD 21249	Obsolete proof of thlbas 21248 as of 11-Nov-2024. Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ 𝐶 = (Base‘𝐾)
Theorem	thlle 21250	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	thlleOLD 21251	Obsolete proof of thlle 21250 as of 11-Nov-2024. Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ≤ = (le‘𝐾)
Theorem	thlleval 21252	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 21253	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 21254	Extend class notation with orthogonal projection function.
class proj
Syntax	chil 21255	Extend class notation with class of all Hilbert spaces.
class Hil
Syntax	cobs 21256	Extend class notation with the set of orthonormal bases.
class OBasis
Definition	df-pj 21257*	Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 19504, 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 21258	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 21259*	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 21260*	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 21261	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 21262	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 21263*	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 21264	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 21265	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 21266	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 21267	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 21268	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 21269	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 21270	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 21271	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 21272	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 21273*	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 21274*	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 21275	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 21276	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 21277	Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
Theorem	obsss 21278	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 21279	A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 )
Theorem	obsocv 21280	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 21281	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 21282	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 21283	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 21284	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 20058), but the existence of a unity element is always assumed (our rings are unital, see df-ring 20057). 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 21286.
Syntax	cdsmm 21285	Class of module direct sum generator.
class ⊕m
Definition	df-dsmm 21286*	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 21287	The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ Rel dom ⊕m
Theorem	dsmmval 21288*	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 21289*	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 21290	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 21291*	Base set of the direct sum module using the fndmin 7046 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)))
Theorem	dsmmfi 21292	For finite products, the direct sum is just the module product. See also the observation in [Lang] p. 129. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → (𝑆 ⊕m 𝑅) = (𝑆Xs𝑅))
Theorem	dsmmelbas 21293*	Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐶 = (𝑆 ⊕m 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin)))
Theorem	dsmm0cl 21294	The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (𝜑 → 0 ∈ 𝐻)
Theorem	dsmmacl 21295	The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐻)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐻)    &   ⊢ + = (+g‘𝑃)    ⇒   ⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)
Theorem	prdsinvgd2 21296	Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑁 = (invg‘𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽)))
Theorem	dsmmsubg 21297	The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃))
Theorem	dsmmlss 21298*	The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → 𝑅:𝐼⟶LMod)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆)    &   ⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝑈 = (LSubSp‘𝑃)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    ⇒   ⊢ (𝜑 → 𝐻 ∈ 𝑈)
Theorem	dsmmlmod 21299*	The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → 𝑅:𝐼⟶LMod)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆)    &   ⊢ 𝐶 = (𝑆 ⊕m 𝑅)    ⇒   ⊢ (𝜑 → 𝐶 ∈ LMod)
11.1.2  Free modules According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist nonfree modules." The same definition is used in [Lang] p. 135: "By a free module we shall mean a module which admits a basis, or the zero module." In the following, a free module is defined as the direct sum of copies of a ring regarded as a left module over itself, see df-frlm 21301. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 21301 (see lmisfree 21396), the two definitions are essentially equivalent. The free modules as defined by df-frlm 21301 are also taken as a motivation to introduce free modules by [Lang] p. 135.
Syntax	cfrlm 21300	Class of free module generator.
class freeLMod