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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembracnlnval 29101* 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 (𝑇𝑥) = (𝑥 ·ih 𝑦))))

Theoremcnvbramul 29102 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) · (bra‘𝑇)))

Theoremkbass1 29103 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ ) ∣ 𝐶⟩ = 𝐴⟩(⟨𝐵𝐶⟩) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since 𝐵𝐶 is a complex number, it is the first argument in the inner product · 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) · 𝐴))

Theoremkbass2 29104 Dirac bra-ket associative law (⟨𝐴𝐵⟩)⟨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)))

Theoremkbass3 29105 Dirac bra-ket associative law 𝐴𝐵 𝐶𝐷⟩ = (⟨𝐴𝐵 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))

Theoremkbass4 29106 Dirac bra-ket associative law 𝐴𝐵 𝐶𝐷⟩ = 𝐴 ∣ ( ∣ 𝐵 𝐶𝐷⟩). (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) · 𝐵)))

Theoremkbass5 29107 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))

Theoremkbass6 29108 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)))))

19.6.15  Positive operators (cont.)

Theoremleopg 29109* Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇𝐴𝑈𝐵) → (𝑇op 𝑈 ↔ ((𝑈op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))))

Theoremleop 29110* 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) → (𝑇op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))

Theoremleop2 29111* 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) → (𝑇op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝑥) ≤ ((𝑈𝑥) ·ih 𝑥)))

Theoremleop3 29112 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) → (𝑇op 𝑈 ↔ 0hopop (𝑈op 𝑇)))

Theoremleoppos 29113* 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 → ( 0hopop 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇𝑥) ·ih 𝑥)))

Theoremleoprf2 29114 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 𝑇op 𝑇)

Theoremleoprf 29115 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇op 𝑇)

Theoremleopsq 29116 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 0hopop (𝑇𝑇))

Theorem0leop 29117 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.)
0hopop 0hop

Theoremidleop 29118 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.)
0hopop Iop

Theoremleopadd 29119 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) ∧ ( 0hopop 𝑇 ∧ 0hopop 𝑈)) → 0hopop (𝑇 +op 𝑈))

Theoremleopmuli 29120 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 ∧ 0hopop 𝑇)) → 0hopop (𝐴 ·op 𝑇))

Theoremleopmul 29121 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) → ( 0hopop 𝑇 ↔ 0hopop (𝐴 ·op 𝑇)))

Theoremleopmul2i 29122 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴𝑇op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))

Theoremleoptri 29123 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) → ((𝑇op 𝑈𝑈op 𝑇) ↔ 𝑇 = 𝑈))

Theoremleoptr 29124 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) ∧ (𝑆op 𝑇𝑇op 𝑈)) → 𝑆op 𝑈)

Theoremleopnmid 29125 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 ∧ (normop𝑇) ∈ ℝ) → 𝑇op ((normop𝑇) ·op Iop ))

Theoremnmopleid 29126 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 ∧ (normop𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop𝑇)) ·op 𝑇) ≤op Iop )

Theoremopsqrlem1 29127* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (normop𝑇) ∈ ℝ    &    0hopop 𝑇    &   𝑅 = ((1 / (normop𝑇)) ·op 𝑇)    &   (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hopop 𝑢 ∧ (𝑢𝑢) = 𝑅))       (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hopop 𝑣 ∧ (𝑣𝑣) = 𝑇))

Theoremopsqrlem2 29128* Lemma for opsqri . 𝐹𝑁 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 (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       (𝐹‘1) = 0hop

Theoremopsqrlem3 29129* 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 (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))

Theoremopsqrlem4 29130* 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 (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       𝐹:ℕ⟶HrmOp

Theoremopsqrlem5 29131* 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 (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹𝑁) +op ((1 / 2) ·op (𝑇op ((𝐹𝑁) ∘ (𝐹𝑁))))))

Theoremopsqrlem6 29132* 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 (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))    &   𝑇op Iop       (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )

19.6.16  Projectors as operators

Theorempjhmopi 29133 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ HrmOp

Theorempjlnopi 29134 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ LinOp

Theorempjnmopi 29135 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.)
𝐻C       (𝐻 ≠ 0 → (normop‘(proj𝐻)) = 1)

Theorempjbdlni 29136 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ BndLinOp

Theorempjhmop 29137 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) ∈ HrmOp)

Theoremhmopidmchi 29138 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    &   (𝑇𝑇) = 𝑇       ran 𝑇C

Theoremhmopidmpji 29139 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 29138 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (𝑇𝑇) = 𝑇       𝑇 = (proj‘ran 𝑇)

Theoremhmopidmch 29140 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 ∧ (𝑇𝑇) = 𝑇) → ran 𝑇C )

Theoremhmopidmpj 29141 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 ∧ (𝑇𝑇) = 𝑇) → 𝑇 = (proj‘ran 𝑇))

Theorempjsdii 29142 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((proj𝐻) ∘ (𝑆 +op 𝑇)) = (((proj𝐻) ∘ 𝑆) +op ((proj𝐻) ∘ 𝑇))

Theorempjddii 29143 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((proj𝐻) ∘ (𝑆op 𝑇)) = (((proj𝐻) ∘ 𝑆) −op ((proj𝐻) ∘ 𝑇))

Theorempjsdi2i 29144 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅𝑆) +op (𝑅𝑇)) → (((proj𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((proj𝐻) ∘ 𝑅) ∘ 𝑆) +op (((proj𝐻) ∘ 𝑅) ∘ 𝑇)))

Theorempjcoi 29145 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) = ((proj𝐺)‘((proj𝐻)‘𝐴)))

Theorempjcocli 29146 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) ∈ 𝐺)

Theorempjcohcli 29147 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) ∈ ℋ)

Theorempjadjcoi 29148 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.)
𝐺C    &   𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((proj𝐺) ∘ (proj𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih (((proj𝐻) ∘ (proj𝐺))‘𝐵)))

Theorempjcofni 29149 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((proj𝐺) ∘ (proj𝐻)) Fn ℋ

Theorempjss1coi 29150 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) ∘ (proj𝐺)) = (proj𝐺))

Theorempjss2coi 29151 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐺) ∘ (proj𝐻)) = (proj𝐺))

Theorempjssmi 29152 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.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (𝐻𝐺 → (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)))

Theorempjssge0i 29153 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴)))

Theorempjdifnormi 29154 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴) ↔ (norm‘((proj𝐻)‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴))))

Theorempjnormssi 29155* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ∀𝑥 ∈ ℋ (norm‘((proj𝐺)‘𝑥)) ≤ (norm‘((proj𝐻)‘𝑥)))

Theorempjorthcoi 29156 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.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → ((proj𝐺) ∘ (proj𝐻)) = 0hop )

Theorempjscji 29157 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → (proj‘(𝐺 𝐻)) = ((proj𝐺) +op (proj𝐻)))

Theorempjssumi 29158 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → (proj‘(𝐺 + 𝐻)) = ((proj𝐺) +op (proj𝐻)))

Theorempjssposi 29159* 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.)
𝐺C    &   𝐻C       (∀𝑥 ∈ ℋ 0 ≤ ((((proj𝐻) −op (proj𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺𝐻)

Theorempjordi 29160* 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 𝐺𝐻; see pjssposi 29159). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (∀𝑥 ∈ ℋ 0 ≤ ((((proj𝐻) −op (proj𝐺))‘𝑥) ·ih 𝑥) ↔ ((proj𝐺) “ ℋ) ⊆ ((proj𝐻) “ ℋ))

Theorempjssdif2i 29161 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 29159). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) −op (proj𝐺)) = (proj‘(𝐻 ∩ (⊥‘𝐺))))

Theorempjssdif1i 29162 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 29159). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) −op (proj𝐺)) ∈ ran proj)

Theorempjimai 29163 The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective" http://www.arxiv.org/pdf/quant-ph/0701113 p. 20. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
𝐴S    &   𝐵C       ((proj𝐵) “ 𝐴) = ((𝐴 + (⊥‘𝐵)) ∩ 𝐵)

Theorempjidmcoi 29164 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) ∘ (proj𝐻)) = (proj𝐻)

Theorempjoccoi 29165 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) ∘ (proj‘(⊥‘𝐻))) = 0hop

Theorempjtoi 29166 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) +op (proj‘(⊥‘𝐻))) = (proj‘ ℋ)

Theorempjoci 29167 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj‘ ℋ) −op (proj𝐻)) = (proj‘(⊥‘𝐻))

Theorempjidmco 29168 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → ((proj𝐻) ∘ (proj𝐻)) = (proj𝐻))

Theoremdfpjop 29169 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 28929. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝑇 ∈ ran proj ↔ (𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇))

Theorempjhmopidm 29170 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 proj = {𝑡 ∈ HrmOp ∣ (𝑡𝑡) = 𝑡}

Theoremelpjidm 29171 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 proj → (𝑇𝑇) = 𝑇)

Theoremelpjhmop 29172 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 proj𝑇 ∈ HrmOp)

Theorem0leopj 29173 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
(𝑇 ∈ ran proj → 0hopop 𝑇)

Theorempjadj2 29174 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj → (adj𝑇) = 𝑇)

Theorempjadj3 29175 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremelpjch 29176 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 proj → (ran 𝑇C𝑇 = (proj‘ran 𝑇)))

Theoremelpjrn 29177* 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 proj → ran 𝑇 = {𝑥 ∈ ℋ ∣ (𝑇𝑥) = 𝑥})

Theorempjinvari 29178 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    &   𝐻C    &   𝑇 = (proj𝐻)       ((𝑆𝑇): ℋ⟶𝐻 ↔ (𝑆𝑇) = (𝑇 ∘ (𝑆𝑇)))

Theorempjin1i 29179 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (proj‘(𝐺𝐻)) = ((proj𝐺) ∘ (proj‘(𝐺𝐻)))

Theorempjin2i 29180 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) = ((proj𝐺) ∘ (proj𝐻)) ∧ (proj𝐻) = ((proj𝐻) ∘ (proj𝐺))) ↔ (proj𝐺) = (proj𝐻))

Theorempjin3i 29181 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((proj𝐹) = ((proj𝐹) ∘ (proj𝐺)) ∧ (proj𝐹) = ((proj𝐹) ∘ (proj𝐻))) ↔ (proj𝐹) = ((proj𝐹) ∘ (proj‘(𝐺𝐻))))

Theorempjclem1 29182 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 → ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))

Theorempjclem2 29183 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 → ((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)))

Theorempjclem3 29184 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) → ((proj𝐺) ∘ (proj‘(⊥‘𝐻))) = ((proj‘(⊥‘𝐻)) ∘ (proj𝐺)))

Theorempjclem4a 29185 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (𝐺𝐻) → (((proj𝐺) ∘ (proj𝐻))‘𝐴) = 𝐴)

Theorempjclem4 29186 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) → ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))

Theorempjci 29187 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 ↔ ((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)))

Theorempjcmul1i 29188 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.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) ↔ ((proj𝐺) ∘ (proj𝐻)) ∈ ran proj)

Theorempjcmul2i 29189 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.)
𝐺C    &   𝐻C       (((proj𝐺) ∘ (proj𝐻)) = ((proj𝐻) ∘ (proj𝐺)) ↔ ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))

Theorempjcohocli 29190 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)
𝐻C    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → (((proj𝐻) ∘ 𝑇)‘𝐴) ∈ 𝐻)

Theorempjadj2coi 29191 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.)
𝐹C    &   𝐺C    &   𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹))‘𝐵)))

Theorempj2cocli 29192 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (𝐴 ∈ ℋ → ((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻))‘𝐴) ∈ 𝐹)

Theorempj3lem1 29193 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (𝐴 ∈ ((𝐹𝐺) ∩ 𝐻) → ((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻))‘𝐴) = 𝐴)

Theorempj3si 29194 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹)) ∧ ran (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) ⊆ 𝐺) → (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (proj‘((𝐹𝐺) ∩ 𝐻)))

Theorempj3i 29195 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹)) ∧ (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐺) ∘ (proj𝐹)) ∘ (proj𝐻))) → (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (proj‘((𝐹𝐺) ∩ 𝐻)))

Theorempj3cor1i 29196 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐺)) ∘ (proj𝐹)) ∧ (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐺) ∘ (proj𝐹)) ∘ (proj𝐻))) → (((proj𝐹) ∘ (proj𝐺)) ∘ (proj𝐻)) = (((proj𝐻) ∘ (proj𝐹)) ∘ (proj𝐺)))

Theorempjs14i 29197 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (norm‘(((proj𝐻) ∘ (proj𝐺))‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴)))

19.7  States on a Hilbert lattice and Godowski's equation

19.7.1  States on a Hilbert lattice

Definitiondf-st 29198* Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
States = {𝑓 ∈ ((0[,]1) ↑𝑚 C ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))}

Definitiondf-hst 29199* Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
CHStates = {𝑓 ∈ ( ℋ ↑𝑚 C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}

Theoremisst 29200* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))

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