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Type | Label | Description |
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Statement | ||
Theorem | fh1i 29401 | Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 𝐶ℋ 𝐵 & ⊢ 𝐴 𝐶ℋ 𝐶 ⇒ ⊢ (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) | ||
Theorem | fh2i 29402 | Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 𝐶ℋ 𝐵 & ⊢ 𝐴 𝐶ℋ 𝐶 ⇒ ⊢ (𝐵 ∩ (𝐴 ∨ℋ 𝐶)) = ((𝐵 ∩ 𝐴) ∨ℋ (𝐵 ∩ 𝐶)) | ||
Theorem | fh3i 29403 | Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 𝐶ℋ 𝐵 & ⊢ 𝐴 𝐶ℋ 𝐶 ⇒ ⊢ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)) = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶)) | ||
Theorem | fh4i 29404 | Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 𝐶ℋ 𝐵 & ⊢ 𝐴 𝐶ℋ 𝐶 ⇒ ⊢ (𝐵 ∨ℋ (𝐴 ∩ 𝐶)) = ((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶)) | ||
Theorem | cm2ji 29405 | A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 𝐶ℋ 𝐵 & ⊢ 𝐴 𝐶ℋ 𝐶 ⇒ ⊢ 𝐴 𝐶ℋ (𝐵 ∨ℋ 𝐶) | ||
Theorem | cm2mi 29406 | A lattice element that commutes with two others also commutes with their meet. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 𝐶ℋ 𝐵 & ⊢ 𝐴 𝐶ℋ 𝐶 ⇒ ⊢ 𝐴 𝐶ℋ (𝐵 ∩ 𝐶) | ||
Theorem | qlax1i 29407 | One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ ⇒ ⊢ 𝐴 = (⊥‘(⊥‘𝐴)) | ||
Theorem | qlax2i 29408 | One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) | ||
Theorem | qlax3i 29409 | One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ ⇒ ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) | ||
Theorem | qlax4i 29410 | One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (𝐵 ∨ℋ (⊥‘𝐵))) = (𝐵 ∨ℋ (⊥‘𝐵)) | ||
Theorem | qlax5i 29411 | One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ∨ℋ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵))) = 𝐴 | ||
Theorem | qlaxr1i 29412 | One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝐵 = 𝐴 | ||
Theorem | qlaxr2i 29413 | One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | qlaxr4i 29414 | One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐴 = 𝐵 ⇒ ⊢ (⊥‘𝐴) = (⊥‘𝐵) | ||
Theorem | qlaxr5i 29415 | One of the conditions showing Cℋ is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∨ℋ 𝐶) = (𝐵 ∨ℋ 𝐶) | ||
Theorem | qlaxr3i 29416 | A variation of the orthomodular law, showing Cℋ is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.) (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ (𝐶 ∨ℋ (⊥‘𝐶)) = ((⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ∨ℋ (⊥‘(𝐴 ∨ℋ 𝐵))) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | chscllem1 29417* | Lemma for chscl 29421. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) & ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) & ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) ⇒ ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) | ||
Theorem | chscllem2 29418* | Lemma for chscl 29421. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) & ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) & ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑣 ) | ||
Theorem | chscllem3 29419* | Lemma for chscl 29421. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) & ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) & ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ 𝐵) & ⊢ (𝜑 → (𝐻‘𝑁) = (𝐶 +ℎ 𝐷)) ⇒ ⊢ (𝜑 → 𝐶 = (𝐹‘𝑁)) | ||
Theorem | chscllem4 29420* | Lemma for chscl 29421. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) & ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) & ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐵)‘(𝐻‘𝑛))) ⇒ ⊢ (𝜑 → 𝑢 ∈ (𝐴 +ℋ 𝐵)) | ||
Theorem | chscl 29421 | The subspace sum of two closed orthogonal spaces is closed. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ∈ Cℋ ) & ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) ⇒ ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ∈ Cℋ ) | ||
Theorem | osumi 29422 | If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the (countable) Axiom of Choice is used for this proof via pjhth 29173, although "the hard part" of this proof, chscl 29421, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) | ||
Theorem | osumcori 29423 | Corollary of osumi 29422. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((𝐴 ∩ 𝐵) +ℋ (𝐴 ∩ (⊥‘𝐵))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) | ||
Theorem | osumcor2i 29424 | Corollary of osumi 29422, showing it holds under the weaker hypothesis that 𝐴 and 𝐵 commute. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ (𝐴 𝐶ℋ 𝐵 → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) | ||
Theorem | osum 29425 | If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) | ||
Theorem | spansnji 29426 | The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Proof suggested by Eric Schechter 1-Jun-2004.) (Contributed by NM, 1-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 +ℋ (span‘{𝐵})) = (𝐴 ∨ℋ (span‘{𝐵})) | ||
Theorem | spansnj 29427 | The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℋ (span‘{𝐵})) = (𝐴 ∨ℋ (span‘{𝐵}))) | ||
Theorem | spansnscl 29428 | The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℋ (span‘{𝐵})) ∈ Cℋ ) | ||
Theorem | sumspansn 29429 | The sum of two vectors belong to the span of one of them iff the other vector also belongs. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴}))) | ||
Theorem | spansnm0i 29430 | The meet of different one-dimensional subspaces is the zero subspace. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (¬ 𝐴 ∈ (span‘{𝐵}) → ((span‘{𝐴}) ∩ (span‘{𝐵})) = 0ℋ) | ||
Theorem | nonbooli 29431 | A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ but (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ 0ℋ. The antecedent specifies that the vectors 𝐴 and 𝐵 are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to 𝐹, 𝐺, and 𝐻. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐹 = (span‘{𝐴}) & ⊢ 𝐺 = (span‘{𝐵}) & ⊢ 𝐻 = (span‘{(𝐴 +ℎ 𝐵)}) ⇒ ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) → (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) | ||
Theorem | spansncvi 29432 | Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨ℋ (span‘{𝐶}))) → 𝐵 = (𝐴 ∨ℋ (span‘{𝐶}))) | ||
Theorem | spansncv 29433 | Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨ℋ (span‘{𝐶}))) → 𝐵 = (𝐴 ∨ℋ (span‘{𝐶})))) | ||
Theorem | 5oalem1 29434 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝑅 ∈ Sℋ ⇒ ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅)))) | ||
Theorem | 5oalem2 29435 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝐷 ∈ Sℋ ⇒ ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑤)) → (𝑥 −ℎ 𝑧) ∈ ((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷))) | ||
Theorem | 5oalem3 29436 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝐷 ∈ Sℋ & ⊢ 𝐹 ∈ Sℋ & ⊢ 𝐺 ∈ Sℋ ⇒ ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺)) ∧ ((𝑥 +ℎ 𝑦) = (𝑓 +ℎ 𝑔) ∧ (𝑧 +ℎ 𝑤) = (𝑓 +ℎ 𝑔))) → (𝑥 −ℎ 𝑧) ∈ (((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) +ℋ ((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)))) | ||
Theorem | 5oalem4 29437 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝐷 ∈ Sℋ & ⊢ 𝐹 ∈ Sℋ & ⊢ 𝐺 ∈ Sℋ ⇒ ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ (𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺)) ∧ ((𝑥 +ℎ 𝑦) = (𝑓 +ℎ 𝑔) ∧ (𝑧 +ℎ 𝑤) = (𝑓 +ℎ 𝑔))) → (𝑥 −ℎ 𝑧) ∈ (((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) +ℋ ((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺))))) | ||
Theorem | 5oalem5 29438 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝐷 ∈ Sℋ & ⊢ 𝐹 ∈ Sℋ & ⊢ 𝐺 ∈ Sℋ & ⊢ 𝑅 ∈ Sℋ & ⊢ 𝑆 ∈ Sℋ ⇒ ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) ∧ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ (𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆))) ∧ (((𝑥 +ℎ 𝑦) = (𝑣 +ℎ 𝑢) ∧ (𝑧 +ℎ 𝑤) = (𝑣 +ℎ 𝑢)) ∧ (𝑓 +ℎ 𝑔) = (𝑣 +ℎ 𝑢))) → (𝑥 −ℎ 𝑧) ∈ ((((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)))) ∩ ((((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))) +ℋ (((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)) ∩ (((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆))))))) | ||
Theorem | 5oalem6 29439 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝐷 ∈ Sℋ & ⊢ 𝐹 ∈ Sℋ & ⊢ 𝐺 ∈ Sℋ & ⊢ 𝑅 ∈ Sℋ & ⊢ 𝑆 ∈ Sℋ ⇒ ⊢ (((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ℎ = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) ∧ ℎ = (𝑧 +ℎ 𝑤))) ∧ (((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐺) ∧ ℎ = (𝑓 +ℎ 𝑔)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆) ∧ ℎ = (𝑣 +ℎ 𝑢)))) → ℎ ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ ((((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)))) ∩ ((((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))) +ℋ (((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)) ∩ (((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))))))))) | ||
Theorem | 5oalem7 29440 | Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) TODO: replace uses of ee4anv 2371 with 4exdistrv 1956 as in 3oalem3 29444. (New usage is discouraged.) |
⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Sℋ & ⊢ 𝐶 ∈ Sℋ & ⊢ 𝐷 ∈ Sℋ & ⊢ 𝐹 ∈ Sℋ & ⊢ 𝐺 ∈ Sℋ & ⊢ 𝑅 ∈ Sℋ & ⊢ 𝑆 ∈ Sℋ ⇒ ⊢ (((𝐴 +ℋ 𝐵) ∩ (𝐶 +ℋ 𝐷)) ∩ ((𝐹 +ℋ 𝐺) ∩ (𝑅 +ℋ 𝑆))) ⊆ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ ((((𝐴 +ℋ 𝐶) ∩ (𝐵 +ℋ 𝐷)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)))) ∩ ((((𝐴 +ℋ 𝐹) ∩ (𝐵 +ℋ 𝐺)) ∩ (((𝐴 +ℋ 𝑅) ∩ (𝐵 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆)))) +ℋ (((𝐶 +ℋ 𝐹) ∩ (𝐷 +ℋ 𝐺)) ∩ (((𝐶 +ℋ 𝑅) ∩ (𝐷 +ℋ 𝑆)) +ℋ ((𝐹 +ℋ 𝑅) ∩ (𝐺 +ℋ 𝑆))))))))) | ||
Theorem | 5oai 29441 | Orthoarguesian law 5OA. This 8-variable inference is called 5OA because it can be converted to a 5-variable equation (see Quantum Logic Explorer). (Contributed by NM, 5-May-2000.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝐷 ∈ Cℋ & ⊢ 𝐹 ∈ Cℋ & ⊢ 𝐺 ∈ Cℋ & ⊢ 𝑅 ∈ Cℋ & ⊢ 𝑆 ∈ Cℋ & ⊢ 𝐴 ⊆ (⊥‘𝐵) & ⊢ 𝐶 ⊆ (⊥‘𝐷) & ⊢ 𝐹 ⊆ (⊥‘𝐺) & ⊢ 𝑅 ⊆ (⊥‘𝑆) ⇒ ⊢ (((𝐴 ∨ℋ 𝐵) ∩ (𝐶 ∨ℋ 𝐷)) ∩ ((𝐹 ∨ℋ 𝐺) ∩ (𝑅 ∨ℋ 𝑆))) ⊆ (𝐵 ∨ℋ (𝐴 ∩ (𝐶 ∨ℋ ((((𝐴 ∨ℋ 𝐶) ∩ (𝐵 ∨ℋ 𝐷)) ∩ (((𝐴 ∨ℋ 𝑅) ∩ (𝐵 ∨ℋ 𝑆)) ∨ℋ ((𝐶 ∨ℋ 𝑅) ∩ (𝐷 ∨ℋ 𝑆)))) ∩ ((((𝐴 ∨ℋ 𝐹) ∩ (𝐵 ∨ℋ 𝐺)) ∩ (((𝐴 ∨ℋ 𝑅) ∩ (𝐵 ∨ℋ 𝑆)) ∨ℋ ((𝐹 ∨ℋ 𝑅) ∩ (𝐺 ∨ℋ 𝑆)))) ∨ℋ (((𝐶 ∨ℋ 𝐹) ∩ (𝐷 ∨ℋ 𝐺)) ∩ (((𝐶 ∨ℋ 𝑅) ∩ (𝐷 ∨ℋ 𝑆)) ∨ℋ ((𝐹 ∨ℋ 𝑅) ∩ (𝐺 ∨ℋ 𝑆))))))))) | ||
Theorem | 3oalem1 29442* | Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝑅 ∈ Cℋ & ⊢ 𝑆 ∈ Cℋ ⇒ ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))) | ||
Theorem | 3oalem2 29443* | Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝑅 ∈ Cℋ & ⊢ 𝑆 ∈ Cℋ ⇒ ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → 𝑣 ∈ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))))) | ||
Theorem | 3oalem3 29444 | Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝑅 ∈ Cℋ & ⊢ 𝑆 ∈ Cℋ ⇒ ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) ⊆ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) | ||
Theorem | 3oalem4 29445 | Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ⇒ ⊢ 𝑅 ⊆ (⊥‘𝐵) | ||
Theorem | 3oalem5 29446 | Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) & ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) ⇒ ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) = ((𝐵 ∨ℋ 𝑅) ∩ (𝐶 ∨ℋ 𝑆)) | ||
Theorem | 3oalem6 29447 | Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) & ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) ⇒ ⊢ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))))) | ||
Theorem | 3oai 29448 | 3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ Cℋ & ⊢ 𝐵 ∈ Cℋ & ⊢ 𝐶 ∈ Cℋ & ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) & ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) ⇒ ⊢ ((𝐵 ∨ℋ 𝑅) ∩ (𝐶 ∨ℋ 𝑆)) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))))) | ||
Theorem | pjorthi 29449 | Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐻 ∈ Cℋ → (((projℎ‘𝐻)‘𝐴) ·ih ((projℎ‘(⊥‘𝐻))‘𝐵)) = 0) | ||
Theorem | pjch1 29450 | Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → ((projℎ‘ ℋ)‘𝐴) = 𝐴) | ||
Theorem | pjo 29451 | The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘(⊥‘𝐻))‘𝐴) = (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴))) | ||
Theorem | pjcompi 29452 | Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = 𝐴) | ||
Theorem | pjidmi 29453 | A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ ⇒ ⊢ ((projℎ‘𝐻)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴) | ||
Theorem | pjadjii 29454 | A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (((projℎ‘𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((projℎ‘𝐻)‘𝐵)) | ||
Theorem | pjaddii 29455 | Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) | ||
Theorem | pjinormii 29456 | The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) | ||
Theorem | pjmulii 29457 | Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((projℎ‘𝐻)‘(𝐶 ·ℎ 𝐴)) = (𝐶 ·ℎ ((projℎ‘𝐻)‘𝐴)) | ||
Theorem | pjsubii 29458 | Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((projℎ‘𝐻)‘(𝐴 −ℎ 𝐵)) = (((projℎ‘𝐻)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐵)) | ||
Theorem | pjsslem 29459 | Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐺 ∈ Cℋ ⇒ ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) −ℎ ((projℎ‘(⊥‘𝐺))‘𝐴)) = (((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) | ||
Theorem | pjss2i 29460 | Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐺 ∈ Cℋ ⇒ ⊢ (𝐻 ⊆ 𝐺 → ((projℎ‘𝐻)‘((projℎ‘𝐺)‘𝐴)) = ((projℎ‘𝐻)‘𝐴)) | ||
Theorem | pjssmii 29461 | Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐺 ∈ Cℋ ⇒ ⊢ (𝐻 ⊆ 𝐺 → (((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)) | ||
Theorem | pjssge0ii 29462 | Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐺 ∈ Cℋ ⇒ ⊢ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴)) | ||
Theorem | pjdifnormii 29463 | Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐺 ∈ Cℋ ⇒ ⊢ (0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴) ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘((projℎ‘𝐺)‘𝐴))) | ||
Theorem | pjcji 29464 | The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐺 ∈ Cℋ ⇒ ⊢ (𝐻 ⊆ (⊥‘𝐺) → ((projℎ‘(𝐻 ∨ℋ 𝐺))‘𝐴) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐺)‘𝐴))) | ||
Theorem | pjadji 29465 | A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((projℎ‘𝐻)‘𝐵))) | ||
Theorem | pjaddi 29466 | Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵))) | ||
Theorem | pjinormi 29467 | The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2)) | ||
Theorem | pjsubi 29468 | Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((projℎ‘𝐻)‘(𝐴 −ℎ 𝐵)) = (((projℎ‘𝐻)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐵))) | ||
Theorem | pjmuli 29469 | Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((projℎ‘𝐻)‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ ((projℎ‘𝐻)‘𝐵))) | ||
Theorem | pjige0i 29470 | The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → 0 ≤ (((projℎ‘𝐻)‘𝐴) ·ih 𝐴)) | ||
Theorem | pjige0 29471 | The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → 0 ≤ (((projℎ‘𝐻)‘𝐴) ·ih 𝐴)) | ||
Theorem | pjcjt2 29472 | The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Cℋ ∧ 𝐺 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((projℎ‘(𝐻 ∨ℋ 𝐺))‘𝐴) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐺)‘𝐴)))) | ||
Theorem | pj0i 29473 | The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐻)‘0ℎ) = 0ℎ | ||
Theorem | pjch 29474 | Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ 𝐻 ↔ ((projℎ‘𝐻)‘𝐴) = 𝐴)) | ||
Theorem | pjid 29475 | The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → ((projℎ‘𝐻)‘𝐴) = 𝐴) | ||
Theorem | pjvec 29476* | The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → 𝐻 = {𝑥 ∈ ℋ ∣ ((projℎ‘𝐻)‘𝑥) = 𝑥}) | ||
Theorem | pjocvec 29477* | The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ((projℎ‘𝐻)‘𝑥) = 0ℎ}) | ||
Theorem | pjocini 29478 | Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ (⊥‘(𝐺 ∩ 𝐻)) → ((projℎ‘𝐺)‘𝐴) ∈ (⊥‘(𝐺 ∩ 𝐻))) | ||
Theorem | pjini 29479 | Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ (𝐺 ∩ 𝐻) → ((projℎ‘𝐺)‘𝐴) ∈ (𝐺 ∩ 𝐻)) | ||
Theorem | pjjsi 29480* | A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ (∀𝑥 ∈ (𝐺 ∨ℋ 𝐻)((projℎ‘(⊥‘𝐺))‘𝑥) ∈ 𝐻 → (𝐺 ∨ℋ 𝐻) = (𝐺 +ℋ 𝐻)) | ||
Theorem | pjfni 29481 | Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) Fn ℋ | ||
Theorem | pjrni 29482 | The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ran (projℎ‘𝐻) = 𝐻 | ||
Theorem | pjfoi 29483 | A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻): ℋ–onto→𝐻 | ||
Theorem | pjfi 29484 | The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻): ℋ⟶ ℋ | ||
Theorem | pjvi 29485 | The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = 𝐴) | ||
Theorem | pjhfo 29486 | A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻): ℋ–onto→𝐻) | ||
Theorem | pjrn 29487 | The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → ran (projℎ‘𝐻) = 𝐻) | ||
Theorem | pjhf 29488 | The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻): ℋ⟶ ℋ) | ||
Theorem | pjfn 29489 | Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) Fn ℋ) | ||
Theorem | pjsumi 29490 | The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘(𝐺 +ℋ 𝐻))‘𝐴) = (((projℎ‘𝐺)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐴)))) | ||
Theorem | pj11i 29491 | One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐺) = (projℎ‘𝐻) ↔ 𝐺 = 𝐻) | ||
Theorem | pjdsi 29492 | Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ (𝐺 ∨ℋ 𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻)) → 𝐴 = (((projℎ‘𝐺)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐴))) | ||
Theorem | pjds3i 29493 | Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐹 ∈ Cℋ & ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (((𝐴 ∈ ((𝐹 ∨ℋ 𝐺) ∨ℋ 𝐻) ∧ 𝐹 ⊆ (⊥‘𝐺)) ∧ (𝐹 ⊆ (⊥‘𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻))) → 𝐴 = ((((projℎ‘𝐹)‘𝐴) +ℎ ((projℎ‘𝐺)‘𝐴)) +ℎ ((projℎ‘𝐻)‘𝐴))) | ||
Theorem | pj11 29494 | One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ ((𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ) → ((projℎ‘𝐺) = (projℎ‘𝐻) ↔ 𝐺 = 𝐻)) | ||
Theorem | pjmfn 29495 | Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ projℎ Fn Cℋ | ||
Theorem | pjmf1 29496 | The projector function maps one-to-one into the set of Hilbert space operators. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ projℎ: Cℋ –1-1→( ℋ ↑m ℋ) | ||
Theorem | pjoi0 29497 | The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.) |
⊢ (((𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝐺 ⊆ (⊥‘𝐻)) → (((projℎ‘𝐺)‘𝐴) ·ih ((projℎ‘𝐻)‘𝐴)) = 0) | ||
Theorem | pjoi0i 29498 | The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (𝐺 ⊆ (⊥‘𝐻) → (((projℎ‘𝐺)‘𝐴) ·ih ((projℎ‘𝐻)‘𝐴)) = 0) | ||
Theorem | pjopythi 29499 | Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ ℋ ⇒ ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((normℎ‘(((projℎ‘𝐺)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐴)))↑2) = (((normℎ‘((projℎ‘𝐺)‘𝐴))↑2) + ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2))) | ||
Theorem | pjopyth 29500 | Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Cℋ ∧ 𝐺 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((normℎ‘(((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐺)‘𝐴)))↑2) = (((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) + ((normℎ‘((projℎ‘𝐺)‘𝐴))↑2)))) |
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