P. 306 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nmopleid 30501	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 30502*	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 30503*	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 30504*	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 30505*	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 30506*	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 30507*	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 )
19.6.16  Projectors as operators
Theorem	pjhmopi 30508	A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ HrmOp
Theorem	pjlnopi 30509	A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ LinOp
Theorem	pjnmopi 30510	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 30511	A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ BndLinOp
Theorem	pjhmop 30512	A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp)
Theorem	hmopidmchi 30513	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 30514	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 30513 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 30515	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 30516	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 30517	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 30518	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 30519	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 30520	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 30521	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 30522	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 30523	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 30524	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 30525	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 30526	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 30527	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 30528	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 30529	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 30530*	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 30531	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 30532	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 30533	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 30534*	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 30535*	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 30534). (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 30536	The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 30534). (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 30537	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 30534). (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 30538	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 30539	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 30540	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 30541	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 30542	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 30543	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 30544	Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 30304. (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 30545	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 30546	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 30547	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 30548	A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
⊢ (𝑇 ∈ ran projℎ → 0hop ≤op 𝑇)
Theorem	pjadj2 30549	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 30550	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 30551	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 30552*	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 30553	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 30554	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 30555	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 30556	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 30557	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 30558	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 30559	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 30560	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 30561	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 30562	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 30563	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 30564	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 30565	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 30566	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)))
Theorem	pj2cocli 30567	Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻))‘𝐴) ∈ 𝐹)
Theorem	pj3lem1 30568	Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ((𝐹 ∩ 𝐺) ∩ 𝐻) → ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻))‘𝐴) = 𝐴)
Theorem	pj3si 30569	Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹)) ∧ ran (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) ⊆ 𝐺) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (projℎ‘((𝐹 ∩ 𝐺) ∩ 𝐻)))
Theorem	pj3i 30570	Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹)) ∧ (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻))) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (projℎ‘((𝐹 ∩ 𝐺) ∩ 𝐻)))
Theorem	pj3cor1i 30571	Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)
⊢ 𝐹 ∈ Cℋ    &   ⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹)) ∧ (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻))) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐺)))
Theorem	pjs14i 30572	Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
⊢ 𝐺 ∈ Cℋ    &   ⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (normℎ‘(((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝐴)) ≤ (normℎ‘((projℎ‘𝐺)‘𝐴)))
19.7  States on a Hilbert lattice and Godowski's equation
19.7.1  States on a Hilbert lattice
Definition	df-st 30573*	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 Cℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))}
Definition	df-hst 30574*	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 Cℋ ) ∣ ((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))}
Theorem	isst 30575*	Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦)))))
Theorem	ishst 30576*	Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ (𝑆 ∈ CHStates ↔ (𝑆: Cℋ ⟶ ℋ ∧ (normℎ‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦))))))
Theorem	sticl 30577	[0, 1] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ (0[,]1)))
Theorem	stcl 30578	Real closure of the value of a state. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ))
Theorem	hstcl 30579	Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ ℋ)
Theorem	hst1a 30580	Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1)
Theorem	hstel2 30581	Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))
Theorem	hstorth 30582	Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0)
Theorem	hstosum 30583	Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))
Theorem	hstoc 30584	Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((𝑆‘𝐴) +ℎ (𝑆‘(⊥‘𝐴))) = (𝑆‘ ℋ))
Theorem	hstnmoc 30585	Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) = 1)
Theorem	stge0 30586	The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → 0 ≤ (𝑆‘𝐴)))
Theorem	stle1 30587	The value of a state is less than or equal to one. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ≤ 1))
Theorem	hstle1 30588	The norm of the value of a Hilbert-space-valued state is less than or equal to one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (normℎ‘(𝑆‘𝐴)) ≤ 1)
Theorem	hst1h 30589	The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 1 ↔ (𝑆‘𝐴) = (𝑆‘ ℋ)))
Theorem	hst0h 30590	The norm of a Hilbert-space-valued state equals zero iff the state value equals zero. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 0 ↔ (𝑆‘𝐴) = 0ℎ))
Theorem	hstpyth 30591	Pythagorean property of a Hilbert-space-valued state for orthogonal vectors 𝐴 and 𝐵. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ((normℎ‘(𝑆‘(𝐴 ∨ℋ 𝐵)))↑2) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘𝐵))↑2)))
Theorem	hstle 30592	Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵)) → (normℎ‘(𝑆‘𝐴)) ≤ (normℎ‘(𝑆‘𝐵)))
Theorem	hstles 30593	Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵)) → ((normℎ‘(𝑆‘𝐴)) = 1 → (normℎ‘(𝑆‘𝐵)) = 1))
Theorem	hstoh 30594	A Hilbert-space-valued state orthogonal to the state of the lattice unit is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ∧ ((𝑆‘𝐴) ·ih (𝑆‘ ℋ)) = 0) → (𝑆‘𝐴) = 0ℎ)
Theorem	hst0 30595	A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ (𝑆 ∈ CHStates → (𝑆‘0ℋ) = 0ℎ)
Theorem	sthil 30596	The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (𝑆‘ ℋ) = 1)
Theorem	stj 30597	The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘𝐵)) → (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) + (𝑆‘𝐵))))
Theorem	sto1i 30598	The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐴))) = 1)
Theorem	sto2i 30599	The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → (𝑆‘(⊥‘𝐴)) = (1 − (𝑆‘𝐴)))
Theorem	stge1i 30600	If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → (1 ≤ (𝑆‘𝐴) ↔ (𝑆‘𝐴) = 1))