P. 284 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	jpi 28301*	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 28283 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))
19.7.2  Godowski's equation
Theorem	golem1 28302	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 28303	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 28304	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 28305*	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 28306*	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 28307*	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 28308*	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 28309*	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))
19.8  Cover relation, atoms, exchange axiom, and modular symmetry
19.8.1  Covers relation; modular pairs
Definition	df-cv 28310*	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 28313 and cvbr2 28314 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 28311*	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 28325 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 28312*	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 28330 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 28313*	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 28314*	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 28315	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 28316	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 28317	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 28318	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 28319	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 28320	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 28321	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 28322	The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → ¬ 𝐴 ⋖ℋ 𝐴)
Theorem	cvntr 28323	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 28324	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 28325*	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 28326	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 28327*	Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 28325. (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 28328*	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 28329*	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 28330*	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 28331	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 28332	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 28333	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 28334*	Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 28330. (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 28335	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 28336*	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 28337*	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 28338	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 28339*	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 28340*	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 28341	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 28342	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)
Theorem	ssmd2 28343	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) → 𝐵 𝑀ℋ 𝐴)
Theorem	ssdmd1 28344	Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → 𝐴 𝑀ℋ* 𝐵)
Theorem	ssdmd2 28345	Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) 𝑀ℋ (⊥‘𝐴))
Theorem	dmdsl3 28346	Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐴 ∨ℋ 𝐵))) → ((𝐶 ∩ 𝐵) ∨ℋ 𝐴) = 𝐶)
Theorem	mdsl3 28347	Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶)
Theorem	mdslle1i 28348	Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵)) → (𝐶 ⊆ 𝐷 ↔ (𝐶 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵)))
Theorem	mdslle2i 28349	Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → (𝐶 ⊆ 𝐷 ↔ (𝐶 ∨ℋ 𝐴) ⊆ (𝐷 ∨ℋ 𝐴)))
Theorem	mdslj1i 28350	Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵))) → ((𝐶 ∨ℋ 𝐷) ∩ 𝐵) = ((𝐶 ∩ 𝐵) ∨ℋ (𝐷 ∩ 𝐵)))
Theorem	mdslj2i 28351	Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → ((𝐶 ∩ 𝐷) ∨ℋ 𝐴) = ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)))
Theorem	mdsl1i 28352*	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) ↔ 𝐴 𝑀ℋ 𝐵)
Theorem	mdsl2i 28353*	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
Theorem	mdsl2bi 28354*	If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))
Theorem	cvmdi 28355	The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∩ 𝐵) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵)
Theorem	mdslmd1lem1 28356	Lemma for mdslmd1i 28360. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵)))) → (((𝑅 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑅 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑅 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑅 ∧ 𝑅 ⊆ (𝐷 ∩ 𝐵)) → ((𝑅 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑅 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))))
Theorem	mdslmd1lem2 28357	Lemma for mdslmd1i 28360. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    &   ⊢ 𝑅 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵)))) → (((𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵) → (((𝑅 ∩ 𝐵) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → ((𝑅 ∨ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨ℋ (𝐶 ∩ 𝐷)))))
Theorem	mdslmd1lem3 28358*	Lemma for mdslmd1i 28360. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝑥 ∈ Cℋ ∧ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵))))) → (((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))))
Theorem	mdslmd1lem4 28359*	Lemma for mdslmd1i 28360. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝑥 ∈ Cℋ ∧ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵))))) → (((𝑥 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵) → (((𝑥 ∩ 𝐵) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑥 ∩ 𝐵) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐷) → ((𝑥 ∨ℋ 𝐶) ∩ 𝐷) ⊆ (𝑥 ∨ℋ (𝐶 ∩ 𝐷)))))
Theorem	mdslmd1i 28360	Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵))) → (𝐶 𝑀ℋ 𝐷 ↔ (𝐶 ∩ 𝐵) 𝑀ℋ (𝐷 ∩ 𝐵)))
Theorem	mdslmd2i 28361	Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀ℋ 𝐷 ↔ (𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴)))
Theorem	mdsldmd1i 28362	Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵))) → (𝐶 𝑀ℋ* 𝐷 ↔ (𝐶 ∩ 𝐵) 𝑀ℋ* (𝐷 ∩ 𝐵)))
Theorem	mdslmd3i 28363	Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) 𝑀ℋ 𝐶) ∧ ((𝐴 ∩ 𝐶) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐴)) → 𝐷 𝑀ℋ (𝐵 ∩ 𝐶))
Theorem	mdslmd4i 28364	Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)) → 𝐶 𝑀ℋ 𝐷)
Theorem	csmdsymi 28365*	Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((∀𝑐 ∈ Cℋ (𝑐 𝑀ℋ 𝐵 → 𝐵 𝑀ℋ* 𝑐) ∧ 𝐴 𝑀ℋ 𝐵) → 𝐵 𝑀ℋ 𝐴)
Theorem	mdexchi 28366	An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐶 𝑀ℋ (𝐴 ∨ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨ℋ 𝐴) 𝑀ℋ 𝐵 ∧ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵)))
Theorem	cvmd 28367	The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ (𝐴 ∩ 𝐵) ⋖ℋ 𝐵) → 𝐴 𝑀ℋ 𝐵)
Theorem	cvdmd 28368	The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐵 ⋖ℋ (𝐴 ∨ℋ 𝐵)) → 𝐴 𝑀ℋ* 𝐵)
19.8.2  Atoms
Definition	df-at 28369	Define the set of atoms in a Hilbert lattice. An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. Definition of atom in [Kalmbach] p. 15. See ela 28370 and elat2 28371 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
Theorem	ela 28370	Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
Theorem	elat2 28371*	Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ (𝐴 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ)))))
Theorem	elatcv0 28372	A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴))
Theorem	atcv0 28373	An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)
Theorem	atssch 28374	Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ HAtoms ⊆ Cℋ
Theorem	atelch 28375	An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms → 𝐴 ∈ Cℋ )
Theorem	atne0 28376	An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms → 𝐴 ≠ 0ℋ)
Theorem	atss 28377	A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ)))
Theorem	atsseq 28378	Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	atcveq0 28379	A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 ⋖ℋ 𝐵 ↔ 𝐴 = 0ℋ))
Theorem	h1da 28380	A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (⊥‘(⊥‘{𝐴})) ∈ HAtoms)
Theorem	spansna 28381	The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (span‘{𝐴}) ∈ HAtoms)
Theorem	sh1dle 28382	A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴) → (⊥‘(⊥‘{𝐵})) ⊆ 𝐴)
Theorem	ch1dle 28383	A 1-dimensional subspace is less than or equal to any member of Cℋ containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ 𝐴) → (⊥‘(⊥‘{𝐵})) ⊆ 𝐴)
Theorem	atom1d 28384*	The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms ↔ ∃𝑥 ∈ ℋ (𝑥 ≠ 0ℎ ∧ 𝐴 = (span‘{𝑥})))
19.8.3  Superposition principle
Theorem	superpos 28385*	Superposition Principle. If 𝐴 and 𝐵 are distinct atoms, there exists a third atom, distinct from 𝐴 and 𝐵, that is the superposition of 𝐴 and 𝐵. Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ∧ 𝐴 ≠ 𝐵) → ∃𝑥 ∈ HAtoms (𝑥 ≠ 𝐴 ∧ 𝑥 ≠ 𝐵 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)))
19.8.4  Atoms, exchange and covering properties, atomicity
Theorem	chcv1 28386	The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵)))
Theorem	chcv2 28387	The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 ⊊ (𝐴 ∨ℋ 𝐵) ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵)))
Theorem	chjatom 28388	The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if 𝐴 or 𝐵 is finite-dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	shatomici 28389*	The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝐴 ≠ 0ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
Theorem	hatomici 28390*	The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ≠ 0ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
Theorem	hatomic 28391*	A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐴 ≠ 0ℋ) → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
Theorem	shatomistici 28392*	The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ 𝐴 = (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
Theorem	hatomistici 28393*	Cℋ is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 = ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
Theorem	chpssati 28394*	Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ 𝐴))
Theorem	chrelati 28395*	The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 ∈ HAtoms (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵))
Theorem	chrelat2i 28396*	A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
Theorem	cvati 28397*	If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⋖ℋ 𝐵 → ∃𝑥 ∈ HAtoms (𝐴 ∨ℋ 𝑥) = 𝐵)
Theorem	cvbr4i 28398*	An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∃𝑥 ∈ HAtoms (𝐴 ∨ℋ 𝑥) = 𝐵))
Theorem	cvexchlem 28399	Lemma for cvexchi 28400. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∩ 𝐵) ⋖ℋ 𝐵 → 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵))
Theorem	cvexchi 28400	The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∩ 𝐵) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵))