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Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshsel 28301* Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦)))

Theoremshsel3 28302* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 𝑦)))

Theoremshseli 28303* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremshscli 28304 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) ∈ S

Theoremshscl 28305 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ∈ S )

Theoremshscom 28306 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremshsva 28307 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → ((𝐶𝐴𝐷𝐵) → (𝐶 + 𝐷) ∈ (𝐴 + 𝐵)))

Theoremshsel1 28308 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶𝐴𝐶 ∈ (𝐴 + 𝐵)))

Theoremshsel2 28309 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐶𝐵𝐶 ∈ (𝐴 + 𝐵)))

Theoremshsvs 28310 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → ((𝐶𝐴𝐷𝐵) → (𝐶 𝐷) ∈ (𝐴 + 𝐵)))

Theoremshsub1 28311 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐴 + 𝐵))

Theoremshsub2 28312 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐵 + 𝐴))

Theoremchoc0 28313 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
(⊥‘0) = ℋ

Theoremchoc1 28314 The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(⊥‘ ℋ) = 0

Theoremchocnul 28315 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
(⊥‘∅) = ℋ

Theoremshintcli 28316 Closure of intersection of a nonempty subset of S. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
(𝐴S𝐴 ≠ ∅)        𝐴S

Theoremshintcl 28317 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐴 ≠ ∅) → 𝐴S )

Theoremchintcli 28318 The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
(𝐴C𝐴 ≠ ∅)        𝐴C

Theoremchintcl 28319 The intersection (infimum) of a nonempty subset of C belongs to C. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
((𝐴C𝐴 ≠ ∅) → 𝐴C )

Theoremspanval 28320* Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (span‘𝐴) = {𝑥S𝐴𝑥})

Theoremhsupval 28321 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 28396. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Theoremchsupval 28322 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 28397. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Theoremspancl 28323 The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (span‘𝐴) ∈ S )

Theoremelspancl 28324 A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ∈ (span‘𝐴)) → 𝐵 ∈ ℋ)

Theoremshsupcl 28325 Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → (span‘ 𝐴) ∈ S )

Theoremhsupcl 28326 Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to C even if the subsets do not. (Contributed by NM, 10-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) ∈ C )

Theoremchsupcl 28327 Closure of supremum of subset of C. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that C is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999.) (New usage is discouraged.)
(𝐴C → ( 𝐴) ∈ C )

Theoremhsupss 28328 Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ⊆ 𝒫 ℋ ∧ 𝐵 ⊆ 𝒫 ℋ) → (𝐴𝐵 → ( 𝐴) ⊆ ( 𝐵)))

Theoremchsupss 28329 Subset relation for supremum of subset of C. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 → ( 𝐴) ⊆ ( 𝐵)))

Theoremhsupunss 28330 The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → 𝐴 ⊆ ( 𝐴))

Theoremchsupunss 28331 The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C 𝐴 ⊆ ( 𝐴))

Theoremspanss2 28332 A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → 𝐴 ⊆ (span‘𝐴))

Theoremshsupunss 28333 The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
(𝐴S 𝐴 ⊆ (span‘ 𝐴))

Theoremspanid 28334 A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
(𝐴S → (span‘𝐴) = 𝐴)

Theoremspanss 28335 Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
((𝐵 ⊆ ℋ ∧ 𝐴𝐵) → (span‘𝐴) ⊆ (span‘𝐵))

Theoremspanssoc 28336 The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (span‘𝐴) ⊆ (⊥‘(⊥‘𝐴)))

Theoremsshjval 28337 Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremshjval 28338 Value of join in S. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremchjval 28339 Value of join in C. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Theoremchjvali 28340 Value of join in C. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵)))

Theoremsshjval3 28341 Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice C. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))

Theoremsshjcl 28342 Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) ∈ C )

Theoremshjcl 28343 Closure of join in S. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) ∈ C )

Theoremchjcl 28344 Closure of join in C. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) ∈ C )

Theoremshjcom 28345 Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))

Theoremshless 28346 Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐴 + 𝐶) ⊆ (𝐵 + 𝐶))

Theoremshlej1 28347 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremshlej2 28348 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴S𝐵S𝐶S ) ∧ 𝐴𝐵) → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremshincli 28349 Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ∈ S

Theoremshscomi 28350 Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (𝐵 + 𝐴)

Theoremshsvai 28351 Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶𝐴𝐷𝐵) → (𝐶 + 𝐷) ∈ (𝐴 + 𝐵))

Theoremshsel1i 28352 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶𝐴𝐶 ∈ (𝐴 + 𝐵))

Theoremshsel2i 28353 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐶𝐵𝐶 ∈ (𝐴 + 𝐵))

Theoremshsvsi 28354 Vector subtraction belongs to subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶𝐴𝐷𝐵) → (𝐶 𝐷) ∈ (𝐴 + 𝐵))

Theoremshunssi 28355 Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ⊆ (𝐴 + 𝐵)

Theoremshunssji 28356 Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵) ⊆ (𝐴 𝐵)

Theoremshsleji 28357 Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) ⊆ (𝐴 𝐵)

Theoremshjcomi 28358 Commutative law for join in S. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) = (𝐵 𝐴)

Theoremshsub1i 28359 Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐴 + 𝐵)

Theoremshsub2i 28360 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐵 + 𝐴)

Theoremshub1i 28361 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       𝐴 ⊆ (𝐴 𝐵)

Theoremshjcli 28362 Closure of C join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ C

Theoremshjshcli 28363 S closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) ∈ S

Theoremshlessi 28364 Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐴 + 𝐶) ⊆ (𝐵 + 𝐶))

Theoremshlej1i 28365 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremshlej2i 28366 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐵 → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremshslej 28367 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ⊆ (𝐴 𝐵))

Theoremshincl 28368 Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴𝐵) ∈ S )

Theoremshub1 28369 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐴 𝐵))

Theoremshub2 28370 A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → 𝐴 ⊆ (𝐵 𝐴))

Theoremshsidmi 28371 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
𝐴S       (𝐴 + 𝐴) = 𝐴

Theoremshslubi 28372 The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 + 𝐵) ⊆ 𝐶)

Theoremshlesb1i 28373 Hilbert lattice ordering in terms of subspace sum. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴𝐵 ↔ (𝐴 + 𝐵) = 𝐵)

Theoremshsval2i 28374* An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = {𝑥S ∣ (𝐴𝐵) ⊆ 𝑥}

Theoremshsval3i 28375 An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 + 𝐵) = (span‘(𝐴𝐵))

Theoremshmodsi 28376 The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (𝐴𝐶 → ((𝐴 + 𝐵) ∩ 𝐶) ⊆ (𝐴 + (𝐵𝐶)))

Theoremshmodi 28377 The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S       (((𝐴 + 𝐵) = (𝐴 𝐵) ∧ 𝐴𝐶) → ((𝐴 𝐵) ∩ 𝐶) ⊆ (𝐴 (𝐵𝐶)))

19.4.5  Projection theorem

Theorempjhthlem1 28378* Lemma for pjhth 28380. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   (𝜑𝐴 ∈ ℋ)    &   (𝜑𝐵𝐻)    &   (𝜑𝐶𝐻)    &   (𝜑 → ∀𝑥𝐻 (norm‘(𝐴 𝐵)) ≤ (norm‘(𝐴 𝑥)))    &   𝑇 = (((𝐴 𝐵) ·ih 𝐶) / ((𝐶 ·ih 𝐶) + 1))       (𝜑 → ((𝐴 𝐵) ·ih 𝐶) = 0)

Theorempjhthlem2 28379* Lemma for pjhth 28380. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   (𝜑𝐴 ∈ ℋ)       (𝜑 → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))

Theorempjhth 28380 Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐻C → (𝐻 + (⊥‘𝐻)) = ℋ)

Theorempjhtheu 28381* Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102. See pjhtheu2 28403 for the uniqueness of 𝑦. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃!𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))

19.4.6  Projectors

Definitiondf-pjh 28382* Define the projection function on a Hilbert space, as a mapping from the Hilbert lattice to a function on Hilbert space. Every closed subspace is associated with a unique projection function. Remark in [Kalmbach] p. 66, adopted as a definition. (proj𝐻)‘𝐴 is the projection of vector 𝐴 onto closed subspace 𝐻. Note that the range of proj is the set of all projection operators, so 𝑇 ∈ ran proj means that 𝑇 is a projection operator. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))

Theorempjhfval 28383* The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
(𝐻C → (proj𝐻) = (𝑥 ∈ ℋ ↦ (𝑧𝐻𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 + 𝑦))))

Theorempjhval 28384* Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))

Theorempjpreeq 28385* Equality with a projection. This version of pjeq 28386 does not assume the Axiom of Choice via pjhth 28380. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ (𝐻 + (⊥‘𝐻))) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))

Theorempjeq 28386* Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (((proj𝐻)‘𝐴) = 𝐵 ↔ (𝐵𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (𝐵 + 𝑥))))

Theoremaxpjcl 28387 Closure of a projection in its subspace. If we consider this together with axpjpj 28407 to be axioms, the need for the ax-hcompl 28187 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjomli 28422.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ 𝐻)

Theorempjhcl 28388 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) ∈ ℋ)

19.5  Properties of Hilbert subspaces

19.5.1  Orthomodular law

Theoremomlsilem 28389 Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐺S    &   𝐻S    &   𝐺𝐻    &   (𝐻 ∩ (⊥‘𝐺)) = 0    &   𝐴𝐻    &   𝐵𝐺    &   𝐶 ∈ (⊥‘𝐺)       (𝐴 = (𝐵 + 𝐶) → 𝐴𝐺)

Theoremomlsii 28390 Subspace inference form of orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴C    &   𝐵S    &   𝐴𝐵    &   (𝐵 ∩ (⊥‘𝐴)) = 0       𝐴 = 𝐵

Theoremomlsi 28391 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)

Theoremococi 28392 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
𝐴C       (⊥‘(⊥‘𝐴)) = 𝐴

Theoremococ 28393 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴C → (⊥‘(⊥‘𝐴)) = 𝐴)

Theoremdfch2 28394 Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
C = {𝑥 ∈ 𝒫 ℋ ∣ (⊥‘(⊥‘𝑥)) = 𝑥}

Theoremococin 28395* The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = {𝑥C𝐴𝑥})

Theoremhsupval2 28396* Alternate definition of supremum of a subset of the Hilbert lattice. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. We actually define it on any collection of Hilbert space subsets, not just the Hilbert lattice C, to allow more general theorems. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = {𝑥C 𝐴𝑥})

Theoremchsupval2 28397* The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( 𝐴) = {𝑥C 𝐴𝑥})

Theoremsshjval2 28398* Value of join in the set of closed subspaces of Hilbert space C. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = {𝑥C ∣ (𝐴𝐵) ⊆ 𝑥})

Theoremchsupid 28399* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝑥C𝑥𝐴}) = 𝐴)

Theoremchsupsn 28400 Value of supremum of subset of C on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴C → ( ‘{𝐴}) = 𝐴)

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