P. 318 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31701-31800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	qlax4i 31701	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 31702	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 31703	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 31704	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 31705	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 31706	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 31707	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
20.5.8  Orthogonal subspaces
Theorem	chscllem1 31708*	Lemma for chscl 31712. (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 31709*	Lemma for chscl 31712. (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 31710*	Lemma for chscl 31712. (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 31711*	Lemma for chscl 31712. (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 31712	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 31713	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 31464, although "the hard part" of this proof, chscl 31712, 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 31714	Corollary of osumi 31713. (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 31715	Corollary of osumi 31713, 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 31716	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 31717	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 31718	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 31719	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 31720	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 31721	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 31722	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 31723	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 31724	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}))))
20.5.9  Orthoarguesian laws 5OA and 3OA
Theorem	5oalem1 31725	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 31726	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 31727	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 31728	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 31729	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 31730	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 31731	Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) TODO: replace uses of ee4anv 2355 with 4exdistrv 1958 as in 3oalem3 31735. (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 31732	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 31733*	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 31734*	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 31735	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 31736	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 31737	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 31738	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 31739	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)))))
20.5.10  Projectors (cont.)
Theorem	pjorthi 31740	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 31741	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 31742	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 31743	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 31744	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 31745	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 31746	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 31747	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 31748	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 31749	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 31750	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 31751	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 31752	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 31753	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 31754	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 31755	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 31756	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 31757	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 31758	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 31759	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 31760	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 31761	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 31762	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 31763	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 31764	The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((projℎ‘𝐻)‘0ℎ) = 0ℎ
Theorem	pjch 31765	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 31766	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 31767*	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 31768*	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 31769	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 31770	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 31771*	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 31772	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 31773	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 31774	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 31775	The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻): ℋ⟶ ℋ
Theorem	pjvi 31776	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 31777	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 31778	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 31779	The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻): ℋ⟶ ℋ)
Theorem	pjfn 31780	Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) Fn ℋ)
Theorem	pjsumi 31781	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 31782	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 31783	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 31784	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 31785	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 31786	Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ projℎ Fn Cℋ
Theorem	pjmf1 31787	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 31788	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 31789	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 31790	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 31791	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))))
Theorem	pjnormi 31792	The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘𝐴)
Theorem	pjpythi 31793	Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((normℎ‘𝐴)↑2) = (((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) + ((normℎ‘((projℎ‘(⊥‘𝐻))‘𝐴))↑2))
Theorem	pjneli 31794	If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (¬ 𝐴 ∈ 𝐻 ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴))
Theorem	pjnorm 31795	The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘𝐴))
Theorem	pjpyth 31796	Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((normℎ‘𝐴)↑2) = (((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) + ((normℎ‘((projℎ‘(⊥‘𝐻))‘𝐴))↑2)))
Theorem	pjnel 31797	If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (¬ 𝐴 ∈ 𝐻 ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴)))
Theorem	pjnorm2 31798	A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 31765 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ 𝐻 ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) = (normℎ‘𝐴)))
20.5.11  Mayet's equation E_3
Theorem	mayete3i 31799	Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    &   ⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐴 ⊆ (⊥‘𝐶)    &   ⊢ 𝐴 ⊆ (⊥‘𝐹)    &   ⊢ 𝐶 ⊆ (⊥‘𝐹)    &   ⊢ 𝐴 ⊆ (⊥‘𝐵)    &   ⊢ 𝐶 ⊆ (⊥‘𝐷)    &   ⊢ 𝐹 ⊆ (⊥‘𝐺)    &   ⊢ 𝑋 = ((𝐴 ∨ℋ 𝐶) ∨ℋ 𝐹)    &   ⊢ 𝑌 = (((𝐴 ∨ℋ 𝐵) ∩ (𝐶 ∨ℋ 𝐷)) ∩ (𝐹 ∨ℋ 𝐺))    &   ⊢ 𝑍 = ((𝐵 ∨ℋ 𝐷) ∨ℋ 𝐺)    ⇒   ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑍
Theorem	mayetes3i 31800	Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    &   ⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    &   ⊢ 𝐴 ⊆ (⊥‘𝐶)    &   ⊢ 𝐴 ⊆ (⊥‘𝐹)    &   ⊢ 𝐶 ⊆ (⊥‘𝐹)    &   ⊢ 𝐴 ⊆ (⊥‘𝐵)    &   ⊢ 𝐶 ⊆ (⊥‘𝐷)    &   ⊢ 𝐹 ⊆ (⊥‘𝐺)    &   ⊢ 𝑅 ⊆ (⊥‘𝑋)    &   ⊢ 𝑋 = ((𝐴 ∨ℋ 𝐶) ∨ℋ 𝐹)    &   ⊢ 𝑌 = (((𝐴 ∨ℋ 𝐵) ∩ (𝐶 ∨ℋ 𝐷)) ∩ (𝐹 ∨ℋ 𝐺))    &   ⊢ 𝑍 = ((𝐵 ∨ℋ 𝐷) ∨ℋ 𝐺)    ⇒   ⊢ ((𝑋 ∨ℋ 𝑅) ∩ 𝑌) ⊆ (𝑍 ∨ℋ 𝑅)