P. 324 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-hst 32301*	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 32302*	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 32303*	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 32304	[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 32305	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 32306	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 32307	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 32308	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 32309	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 32310	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 32311	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 32312	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 32313	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 32314	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 32315	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 32316	The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 1 ↔ (𝑆‘𝐴) = (𝑆‘ ℋ)))
Theorem	hst0h 32317	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 32318	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 32319	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 32320	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 32321	A Hilbert-space-valued state orthogonal to the state of the lattice one 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 32322	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 32323	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 32324	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 32325	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 32326	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 32327	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))
Theorem	stle0i 32328	If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ≤ 0 ↔ (𝑆‘𝐴) = 0))
Theorem	stlei 32329	Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → (𝐴 ⊆ 𝐵 → (𝑆‘𝐴) ≤ (𝑆‘𝐵)))
Theorem	stlesi 32330	Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → (𝐴 ⊆ 𝐵 → ((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1)))
Theorem	stji1i 32331	Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → (𝑆‘((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵))) = ((𝑆‘(⊥‘𝐴)) + (𝑆‘(𝐴 ∩ 𝐵))))
Theorem	stm1i 32332	State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((𝑆‘(𝐴 ∩ 𝐵)) = 1 → (𝑆‘𝐴) = 1))
Theorem	stm1ri 32333	State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((𝑆‘(𝐴 ∩ 𝐵)) = 1 → (𝑆‘𝐵) = 1))
Theorem	stm1addi 32334	Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((𝑆‘(𝐴 ∩ 𝐵)) = 1 → ((𝑆‘𝐴) + (𝑆‘𝐵)) = 2))
Theorem	staddi 32335	If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → (((𝑆‘𝐴) + (𝑆‘𝐵)) = 2 → (𝑆‘𝐴) = 1))
Theorem	stm1add3i 32336	Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((𝑆‘((𝐴 ∩ 𝐵) ∩ 𝐶)) = 1 → (((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3))
Theorem	stadd3i 32337	If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝑆 ∈ States → ((((𝑆‘𝐴) + (𝑆‘𝐵)) + (𝑆‘𝐶)) = 3 → (𝑆‘𝐴) = 1))
Theorem	st0 32338	The state of the zero subspace. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝑆 ∈ States → (𝑆‘0ℋ) = 0)
Theorem	strlem1 32339*	Lemma for strong state theorem: if closed subspace 𝐴 is not contained in 𝐵, there is a unit vector 𝑢 in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1)
Theorem	strlem2 32340*	Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    ⇒   ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))
Theorem	strlem3a 32341*	Lemma for strong state theorem: the function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    ⇒   ⊢ ((𝑢 ∈ ℋ ∧ (normℎ‘𝑢) = 1) → 𝑆 ∈ States)
Theorem	strlem3 32342*	Lemma for strong state theorem: the function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → 𝑆 ∈ States)
Theorem	strlem4 32343*	Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → (𝑆‘𝐴) = 1)
Theorem	strlem5 32344*	Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → (𝑆‘𝐵) < 1)
Theorem	strlem6 32345*	Lemma for strong state theorem. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → ¬ ((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1))
Theorem	stri 32346*	Strong state theorem. The states on a Hilbert lattice define an ordering. Remark in [Mayet] p. 370. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (∀𝑓 ∈ States ((𝑓‘𝐴) = 1 → (𝑓‘𝐵) = 1) → 𝐴 ⊆ 𝐵)
Theorem	strb 32347*	Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (∀𝑓 ∈ States ((𝑓‘𝐴) = 1 → (𝑓‘𝐵) = 1) ↔ 𝐴 ⊆ 𝐵)
Theorem	hstrlem2 32348*	Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))    ⇒   ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((projℎ‘𝐶)‘𝑢))
Theorem	hstrlem3a 32349*	Lemma for strong set of CH states theorem: the function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))    ⇒   ⊢ ((𝑢 ∈ ℋ ∧ (normℎ‘𝑢) = 1) → 𝑆 ∈ CHStates)
Theorem	hstrlem3 32350*	Lemma for strong set of CH states theorem: the function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a state. This lemma restates the hypotheses in a more convenient form to work with. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → 𝑆 ∈ CHStates)
Theorem	hstrlem4 32351*	Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → (normℎ‘(𝑆‘𝐴)) = 1)
Theorem	hstrlem5 32352*	Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → (normℎ‘(𝑆‘𝐵)) < 1)
Theorem	hstrlem6 32353*	Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))    &   ⊢ (𝜑 ↔ (𝑢 ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘𝑢) = 1))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → ¬ ((normℎ‘(𝑆‘𝐴)) = 1 → (normℎ‘(𝑆‘𝐵)) = 1))
Theorem	hstri 32354*	Hilbert space admits a strong set of Hilbert-space-valued states (CH-states). Theorem in [Mayet3] p. 10. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (∀𝑓 ∈ CHStates ((normℎ‘(𝑓‘𝐴)) = 1 → (normℎ‘(𝑓‘𝐵)) = 1) → 𝐴 ⊆ 𝐵)
Theorem	hstrbi 32355*	Strong CH-state theorem (bidirectional version). Theorem in [Mayet3] p. 10 and its converse. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (∀𝑓 ∈ CHStates ((normℎ‘(𝑓‘𝐴)) = 1 → (normℎ‘(𝑓‘𝐵)) = 1) ↔ 𝐴 ⊆ 𝐵)
Theorem	largei 32356*	A Hilbert lattice admits a largei set of states. Remark in [Mayet] p. 370. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (¬ 𝐴 = 0ℋ ↔ ∃𝑓 ∈ States (𝑓‘𝐴) = 1)
Theorem	jplem1 32357	Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝑢 ∈ ℋ ∧ (normℎ‘𝑢) = 1) → (𝑢 ∈ 𝐴 ↔ ((normℎ‘((projℎ‘𝐴)‘𝑢))↑2) = 1))
Theorem	jplem2 32358*	Lemma for Jauch-Piron theorem. (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    &   ⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝑢 ∈ ℋ ∧ (normℎ‘𝑢) = 1) → (𝑢 ∈ 𝐴 ↔ (𝑆‘𝐴) = 1))
Theorem	jpi 32359*	The function 𝑆, that maps a closed subspace to the square of the norm of its projection onto a unit vector, is a Jauch-Piron state. Remark in [Mayet] p. 370. (See strlem3a 32341 for the proof that 𝑆 is a state.) (Contributed by NM, 8-Apr-2001.) (New usage is discouraged.)
⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝑢 ∈ ℋ ∧ (normℎ‘𝑢) = 1) → (((𝑆‘𝐴) = 1 ∧ (𝑆‘𝐵) = 1) ↔ (𝑆‘(𝐴 ∩ 𝐵)) = 1))
20.7.2  Godowski's equation
Theorem	golem1 32360	Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐹 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵))    &   ⊢ 𝐺 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶))    &   ⊢ 𝐻 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴))    &   ⊢ 𝐷 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴))    &   ⊢ 𝑅 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐵))    &   ⊢ 𝑆 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐶))    ⇒   ⊢ (𝑓 ∈ States → (((𝑓‘𝐹) + (𝑓‘𝐺)) + (𝑓‘𝐻)) = (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆)))
Theorem	golem2 32361	Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐹 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵))    &   ⊢ 𝐺 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶))    &   ⊢ 𝐻 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴))    &   ⊢ 𝐷 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴))    &   ⊢ 𝑅 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐵))    &   ⊢ 𝑆 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐶))    ⇒   ⊢ (𝑓 ∈ States → ((𝑓‘((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 → (𝑓‘𝐷) = 1))
Theorem	goeqi 32362	Godowski's equation, shown here as a variant equivalent to Equation SF of [Godowski] p. 730. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐹 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵))    &   ⊢ 𝐺 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶))    &   ⊢ 𝐻 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴))    &   ⊢ 𝐷 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴))    ⇒   ⊢ ((𝐹 ∩ 𝐺) ∩ 𝐻) ⊆ 𝐷
Theorem	stcltr1i 32363*	Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))
Theorem	stcltr2i 32364*	Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))    &   ⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ))
Theorem	stcltrlem1 32365*	Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → ((𝑆‘𝐵) = 1 → (𝑆‘((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵))) = 1))
Theorem	stcltrlem2 32366*	Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))    &   ⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝜑 → 𝐵 ⊆ ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵)))
Theorem	stcltrthi 32367*	Theorem for classically strong set of states. If there exists a "classically strong set of states" on lattice Cℋ (or actually any ortholattice, which would have an identical proof), then any two elements of the lattice commute, i.e., the lattice is distributive. (Proof due to Mladen Pavicic.) Theorem 3.3 of [MegPav2000] p. 2344. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ ∃𝑠 ∈ States ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑠‘𝑥) = 1 → (𝑠‘𝑦) = 1) → 𝑥 ⊆ 𝑦)    ⇒   ⊢ 𝐵 ⊆ ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵))
20.8  Cover relation, atoms, exchange axiom, and modular symmetry
20.8.1  Covers relation; modular pairs
Definition	df-cv 32368*	Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation 𝐴 ⋖ℋ 𝐵 is read "𝐵 covers 𝐴 " or "𝐴 is covered by 𝐵 " , and it means that 𝐵 is larger than 𝐴 and there is nothing in between. See cvbr 32371 and cvbr2 32372 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
⊢ ⋖ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈ Cℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))}
Definition	df-md 32369*	Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 32383 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
⊢ 𝑀ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))}
Definition	df-dmd 32370*	Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 32388 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝑀ℋ* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑦 ⊆ 𝑧 → ((𝑧 ∩ 𝑥) ∨ℋ 𝑦) = (𝑧 ∩ (𝑥 ∨ℋ 𝑦))))}
Theorem	cvbr 32371*	Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
Theorem	cvbr2 32372*	Binary relation expressing 𝐵 covers 𝐴. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵))))
Theorem	cvcon3 32373	Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (⊥‘𝐵) ⋖ℋ (⊥‘𝐴)))
Theorem	cvpss 32374	The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
Theorem	cvnbtwn 32375	The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
Theorem	cvnbtwn2 32376	The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)))
Theorem	cvnbtwn3 32377	The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)))
Theorem	cvnbtwn4 32378	The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))))
Theorem	cvnsym 32379	The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ 𝐵 ⋖ℋ 𝐴))
Theorem	cvnref 32380	The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴)
Theorem	cvntr 32381	The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⋖ℋ 𝐵 ∧ 𝐵 ⋖ℋ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))
Theorem	spansncv2 32382	Hilbert space has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ) → (¬ (span‘{𝐵}) ⊆ 𝐴 → 𝐴 ⋖ℋ (𝐴 ∨ℋ (span‘{𝐵}))))
Theorem	mdbr 32383*	Binary relation expressing ⟨𝐴, 𝐵⟩ is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
Theorem	mdi 32384	Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))
Theorem	mdbr2 32385*	Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 32383. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
Theorem	mdbr3 32386*	Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) = ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
Theorem	mdbr4 32387*	Binary relation expressing the modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∩ 𝐵) ∨ℋ 𝐴) ∩ 𝐵) ⊆ ((𝑥 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐵))))
Theorem	dmdbr 32388*	Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
Theorem	dmdmd 32389	The dual modular pair property expressed in terms of the modular pair property, that hold in Hilbert lattices. Remark 29.6 of [MaedaMaeda] p. 130. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ (⊥‘𝐴) 𝑀ℋ (⊥‘𝐵)))
Theorem	mddmd 32390	The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵)))
Theorem	dmdi 32391	Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))
Theorem	dmdbr2 32392*	Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 32388. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))))
Theorem	dmdi2 32393	Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))
Theorem	dmdbr3 32394*	Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))))
Theorem	dmdbr4 32395*	Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))
Theorem	dmdi4 32396	Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → ((𝐶 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝐶 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))
Theorem	dmdbr5 32397*	Binary relation expressing the dual modular pair property. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))))
Theorem	mddmd2 32398*	Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (∀𝑥 ∈ Cℋ 𝐴 𝑀ℋ 𝑥 ↔ ∀𝑥 ∈ Cℋ 𝐴 𝑀ℋ* 𝑥))
Theorem	mdsl0 32399	A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ (𝐶 ∈ Cℋ ∧ 𝐷 ∈ Cℋ )) → ((((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ 𝐴 𝑀ℋ 𝐵) → 𝐶 𝑀ℋ 𝐷))
Theorem	ssmd1 32400	Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → 𝐴 𝑀ℋ 𝐵)