P. 323 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)

Type	Label	Description
Statement
20.6.14  Dirac bra-ket notation (cont.)
Theorem	branmfn 32201	The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴))
Theorem	brabn 32202	The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) ∈ ℝ)
Theorem	rnbra 32203	The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ ran bra = (LinFn ∩ ContFn)
Theorem	bra11 32204	The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ bra: ℋ–1-1-onto→(LinFn ∩ ContFn)
Theorem	bracnln 32205	A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn))
Theorem	cnvbraval 32206*	Value of the converse of the bra function. Based on the Riesz Lemma riesz4 32160, this very important theorem not only justifies the Dirac bra-ket notation, but allows to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from ℋ to ℂ). (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) = (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)))
Theorem	cnvbracl 32207	Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) ∈ ℋ)
Theorem	cnvbrabra 32208	The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (◡bra‘(bra‘𝐴)) = 𝐴)
Theorem	bracnvbra 32209	The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (bra‘(◡bra‘𝑇)) = 𝑇)
Theorem	bracnlnval 32210*	The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → 𝑇 = (bra‘(℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))))
Theorem	cnvbramul 32211	Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (LinFn ∩ ContFn)) → (◡bra‘(𝐴 ·fn 𝑇)) = ((∗‘𝐴) ·ℎ (◡bra‘𝑇)))
Theorem	kbass1 32212	Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩ = ∣ 𝐴⟩(⟨𝐵 ∣ 𝐶⟩), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since ⟨𝐵 ∣ 𝐶⟩ is a complex number, it is the first argument in the inner product ·ℎ that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) ·ℎ 𝐴))
Theorem	kbass2 32213	Dirac bra-ket associative law (⟨𝐴 ∣ 𝐵⟩)⟨𝐶 ∣ = ⟨𝐴 ∣ ( ∣ 𝐵⟩⟨𝐶 ∣ ), i.e., the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶)) = ((bra‘𝐴) ∘ (𝐵 ketbra 𝐶)))
Theorem	kbass3 32214	Dirac bra-ket associative law ⟨𝐴 ∣ 𝐵⟩⟨𝐶 ∣ 𝐷⟩ = (⟨𝐴 ∣ 𝐵⟩⟨𝐶 ∣ ) ∣ 𝐷⟩. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶))‘𝐷))
Theorem	kbass4 32215	Dirac bra-ket associative law ⟨𝐴 ∣ 𝐵⟩⟨𝐶 ∣ 𝐷⟩ = ⟨𝐴 ∣ ( ∣ 𝐵⟩⟨𝐶 ∣ 𝐷⟩). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((bra‘𝐴)‘(((bra‘𝐶)‘𝐷) ·ℎ 𝐵)))
Theorem	kbass5 32216	Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ )( ∣ 𝐶⟩⟨𝐷 ∣ ) = (( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩)⟨𝐷 ∣. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷))
Theorem	kbass6 32217	Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ )( ∣ 𝐶⟩⟨𝐷 ∣ ) = ∣ 𝐴⟩(⟨𝐵 ∣ ( ∣ 𝐶⟩⟨𝐷 ∣ )). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷)))))
20.6.15  Positive operators (cont.)
Theorem	leopg 32218*	Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))))
Theorem	leop 32219*	Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))
Theorem	leop2 32220*	Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)))
Theorem	leop3 32221	Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇)))
Theorem	leoppos 32222*	Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥)))
Theorem	leoprf2 32223	The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → 𝑇 ≤op 𝑇)
Theorem	leoprf 32224	The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇)
Theorem	leopsq 32225	The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 0hop ≤op (𝑇 ∘ 𝑇))
Theorem	0leop 32226	The zero operator is a positive operator. (The literature calls it "positive", even though in some sense it is really "nonnegative".) Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ 0hop ≤op 0hop
Theorem	idleop 32227	The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ 0hop ≤op Iop
Theorem	leopadd 32228	The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈))
Theorem	leopmuli 32229	The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇))
Theorem	leopmul 32230	The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 0 < 𝐴) → ( 0hop ≤op 𝑇 ↔ 0hop ≤op (𝐴 ·op 𝑇)))
Theorem	leopmul2i 32231	Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))
Theorem	leoptri 32232	The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑇 ≤op 𝑈 ∧ 𝑈 ≤op 𝑇) ↔ 𝑇 = 𝑈))
Theorem	leoptr 32233	The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈)
Theorem	leopnmid 32234	A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ (normop‘𝑇) ∈ ℝ) → 𝑇 ≤op ((normop‘𝑇) ·op Iop ))
Theorem	nmopleid 32235	A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ (normop‘𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop‘𝑇)) ·op 𝑇) ≤op Iop )
Theorem	opsqrlem1 32236*	Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ (normop‘𝑇) ∈ ℝ    &   ⊢ 0hop ≤op 𝑇    &   ⊢ 𝑅 = ((1 / (normop‘𝑇)) ·op 𝑇)    &   ⊢ (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hop ≤op 𝑢 ∧ (𝑢 ∘ 𝑢) = 𝑅))    ⇒   ⊢ (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hop ≤op 𝑣 ∧ (𝑣 ∘ 𝑣) = 𝑇))
Theorem	opsqrlem2 32237*	Lemma for opsqri . 𝐹‘𝑁 is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ (𝐹‘1) = 0hop
Theorem	opsqrlem3 32238*	Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))
Theorem	opsqrlem4 32239*	Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ 𝐹:ℕ⟶HrmOp
Theorem	opsqrlem5 32240*	Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁))))))
Theorem	opsqrlem6 32241*	Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    &   ⊢ 𝑇 ≤op Iop    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )
20.6.16  Projectors as operators
Theorem	pjhmopi 32242	A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ HrmOp
Theorem	pjlnopi 32243	A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ LinOp
Theorem	pjnmopi 32244	The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐻 ≠ 0ℋ → (normop‘(projℎ‘𝐻)) = 1)
Theorem	pjbdlni 32245	A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ BndLinOp
Theorem	pjhmop 32246	A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp)
Theorem	hmopidmchi 32247	An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ (𝑇 ∘ 𝑇) = 𝑇    ⇒   ⊢ ran 𝑇 ∈ Cℋ
Theorem	hmopidmpji 32248	An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that 𝐻 is a closed subspace, which is not trivial as hmopidmchi 32247 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ (𝑇 ∘ 𝑇) = 𝑇    ⇒   ⊢ 𝑇 = (projℎ‘ran 𝑇)
Theorem	hmopidmch 32249	An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ (𝑇 ∘ 𝑇) = 𝑇) → ran 𝑇 ∈ Cℋ )
Theorem	hmopidmpj 32250	An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ (𝑇 ∘ 𝑇) = 𝑇) → 𝑇 = (projℎ‘ran 𝑇))
Theorem	pjsdii 32251	Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((projℎ‘𝐻) ∘ (𝑆 +op 𝑇)) = (((projℎ‘𝐻) ∘ 𝑆) +op ((projℎ‘𝐻) ∘ 𝑇))
Theorem	pjddii 32252	Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((projℎ‘𝐻) ∘ (𝑆 −op 𝑇)) = (((projℎ‘𝐻) ∘ 𝑆) −op ((projℎ‘𝐻) ∘ 𝑇))
Theorem	pjsdi2i 32253	Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → (((projℎ‘𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) +op (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇)))
Theorem	pjcoi 32254	Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴)))
Theorem	pjcocli 32255	Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ∈ 𝐺)
Theorem	pjcohcli 32256	Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ∈ ℋ)
Theorem	pjadjcoi 32257	Adjoint of composition of projections. Special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝐵)))
Theorem	pjcofni 32258	Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) Fn ℋ
Theorem	pjss1coi 32259	Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) = (projℎ‘𝐺))
Theorem	pjss2coi 32260	Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺))
Theorem	pjssmi 32261	Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (𝐻 ⊆ 𝐺 → (((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)))
Theorem	pjssge0i 32262	Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴)))
Theorem	pjdifnormi 32263	Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴) ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘((projℎ‘𝐺)‘𝐴))))
Theorem	pjnormssi 32264*	Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ 𝐻 ↔ ∀𝑥 ∈ ℋ (normℎ‘((projℎ‘𝐺)‘𝑥)) ≤ (normℎ‘((projℎ‘𝐻)‘𝑥)))
Theorem	pjorthcoi 32265	Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop )
Theorem	pjscji 32266	The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ (⊥‘𝐻) → (projℎ‘(𝐺 ∨ℋ 𝐻)) = ((projℎ‘𝐺) +op (projℎ‘𝐻)))
Theorem	pjssumi 32267	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    ⇒   ⊢ (𝐺 ⊆ (⊥‘𝐻) → (projℎ‘(𝐺 +ℋ 𝐻)) = ((projℎ‘𝐺) +op (projℎ‘𝐻)))
Theorem	pjssposi 32268*	Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺 ⊆ 𝐻)
Theorem	pjordi 32269*	The definition of projector ordering in [Halmos] p. 42 is equivalent to the definition of projector ordering in [Beran] p. 110. (We will usually express projector ordering with the even simpler equivalent 𝐺 ⊆ 𝐻; see pjssposi 32268). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ ((projℎ‘𝐺) “ ℋ) ⊆ ((projℎ‘𝐻) “ ℋ))
Theorem	pjssdif2i 32270	The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 32268). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺))))
Theorem	pjssdif1i 32271	A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 32268). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ)
Theorem	pjimai 32272	The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective", https://doi.org/10.48550/arXiv.quant-ph/0701113. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((projℎ‘𝐵) “ 𝐴) = ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵)
Theorem	pjidmcoi 32273	A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((projℎ‘𝐻) ∘ (projℎ‘𝐻)) = (projℎ‘𝐻)
Theorem	pjoccoi 32274	Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((projℎ‘𝐻) ∘ (projℎ‘(⊥‘𝐻))) = 0hop
Theorem	pjtoi 32275	Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))) = (projℎ‘ ℋ)
Theorem	pjoci 32276	Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((projℎ‘ ℋ) −op (projℎ‘𝐻)) = (projℎ‘(⊥‘𝐻))
Theorem	pjidmco 32277	A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻) ∘ (projℎ‘𝐻)) = (projℎ‘𝐻))
Theorem	dfpjop 32278	Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 32038. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ ↔ (𝑇 ∈ HrmOp ∧ (𝑇 ∘ 𝑇) = 𝑇))
Theorem	pjhmopidm 32279	Two ways to express the set of all projection operators. (Contributed by NM, 24-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ ran projℎ = {𝑡 ∈ HrmOp ∣ (𝑡 ∘ 𝑡) = 𝑡}
Theorem	elpjidm 32280	A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → (𝑇 ∘ 𝑇) = 𝑇)
Theorem	elpjhmop 32281	A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → 𝑇 ∈ HrmOp)
Theorem	0leopj 32282	A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → 0hop ≤op 𝑇)
Theorem	pjadj2 32283	A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → (adjℎ‘𝑇) = 𝑇)
Theorem	pjadj3 32284	A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (adjℎ‘(projℎ‘𝐻)) = (projℎ‘𝐻))
Theorem	elpjch 32285	Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → (ran 𝑇 ∈ Cℋ ∧ 𝑇 = (projℎ‘ran 𝑇)))
Theorem	elpjrn 32286*	Reconstruction of the subspace of a projection operator. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → ran 𝑇 = {𝑥 ∈ ℋ ∣ (𝑇‘𝑥) = 𝑥})
Theorem	pjinvari 32287	A closed subspace 𝐻 with projection 𝑇 is invariant under an operator 𝑆 iff 𝑆𝑇 = 𝑇𝑆𝑇. Theorem 27.1 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝑇 = (projℎ‘𝐻)    ⇒   ⊢ ((𝑆 ∘ 𝑇): ℋ⟶𝐻 ↔ (𝑆 ∘ 𝑇) = (𝑇 ∘ (𝑆 ∘ 𝑇)))
Theorem	pjin1i 32288	Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘(𝐺 ∩ 𝐻)) = ((projℎ‘𝐺) ∘ (projℎ‘(𝐺 ∩ 𝐻)))
Theorem	pjin2i 32289	Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((projℎ‘𝐺) = ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∧ (projℎ‘𝐻) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺))) ↔ (projℎ‘𝐺) = (projℎ‘𝐻))
Theorem	pjin3i 32290	Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((projℎ‘𝐹) = ((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∧ (projℎ‘𝐹) = ((projℎ‘𝐹) ∘ (projℎ‘𝐻))) ↔ (projℎ‘𝐹) = ((projℎ‘𝐹) ∘ (projℎ‘(𝐺 ∩ 𝐻))))
Theorem	pjclem1 32291	Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 𝐶ℋ 𝐻 → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘(𝐺 ∩ 𝐻)))
Theorem	pjclem2 32292	Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 𝐶ℋ 𝐻 → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)))
Theorem	pjclem3 32293	Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = ((projℎ‘(⊥‘𝐻)) ∘ (projℎ‘𝐺)))
Theorem	pjclem4a 32294	Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ (𝐺 ∩ 𝐻) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = 𝐴)
Theorem	pjclem4 32295	Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘(𝐺 ∩ 𝐻)))
Theorem	pjci 32296	Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐺 𝐶ℋ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)))
Theorem	pjcmul1i 32297	A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∈ ran projℎ)
Theorem	pjcmul2i 32298	The projection subspace of the difference between two projectors. Part 2 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘(𝐺 ∩ 𝐻)))
Theorem	pjcohocli 32299	Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐻) ∘ 𝑇)‘𝐴) ∈ 𝐻)
Theorem	pjadj2coi 32300	Adjoint of double composition of projections. Generalization of special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹))‘𝐵)))