P. 307 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)

Type	Label	Description
Statement
19.8.2  Atoms
Definition	df-at 30601	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 30602 and elat2 30603 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
Theorem	ela 30602	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 30603*	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 30604	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 30605	An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)
Theorem	atssch 30606	Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ HAtoms ⊆ Cℋ
Theorem	atelch 30607	An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms → 𝐴 ∈ Cℋ )
Theorem	atne0 30608	An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ HAtoms → 𝐴 ≠ 0ℋ)
Theorem	atss 30609	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 30610	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 30611	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 30612	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 30613	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 30614	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 30615	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 30616*	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 30617*	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 30618	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 30619	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 30620	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 30621*	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 30622*	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 30623*	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 30624*	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 30625*	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 30626*	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 30627*	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 30628*	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 30629*	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 30630*	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 30631	Lemma for cvexchi 30632. (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 30632	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))
Theorem	chrelat2 30633*	A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)))
Theorem	chrelat3 30634*	A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)))
Theorem	chrelat3i 30635*	A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
Theorem	chrelat4i 30636*	A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
Theorem	cvexch 30637	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, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝐴 ∩ 𝐵) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵)))
Theorem	cvp 30638	The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → ((𝐴 ∩ 𝐵) = 0ℋ ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵)))
Theorem	atnssm0 30639	The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (¬ 𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 0ℋ))
Theorem	atnemeq0 30640	The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴 ≠ 𝐵 ↔ (𝐴 ∩ 𝐵) = 0ℋ))
Theorem	atssma 30641	The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ∈ HAtoms))
Theorem	atcv0eq 30642	Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (0ℋ ⋖ℋ (𝐴 ∨ℋ 𝐵) ↔ 𝐴 = 𝐵))
Theorem	atcv1 30643	Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐴 ⋖ℋ (𝐵 ∨ℋ 𝐶)) → (𝐴 = 0ℋ ↔ 𝐵 = 𝐶))
Theorem	atexch 30644	The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 30640 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐵 ⊆ (𝐴 ∨ℋ 𝐶) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → 𝐶 ⊆ (𝐴 ∨ℋ 𝐵)))
Theorem	atomli 30645	An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐵 ∈ HAtoms → ((𝐴 ∨ℋ 𝐵) ∩ (⊥‘𝐴)) ∈ (HAtoms ∪ {0ℋ}))
Theorem	atoml2i 30646	An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝐵 ∈ HAtoms ∧ ¬ 𝐵 ⊆ 𝐴) → ((𝐴 ∨ℋ 𝐵) ∩ (⊥‘𝐴)) ∈ HAtoms)
Theorem	atordi 30647	An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝐵 ∈ HAtoms ∧ 𝐴 𝐶ℋ 𝐵) → (𝐵 ⊆ 𝐴 ∨ 𝐵 ⊆ (⊥‘𝐴)))
Theorem	atcvatlem 30648	Lemma for atcvati 30649. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) → (¬ 𝐵 ⊆ 𝐴 → 𝐴 ∈ HAtoms))
Theorem	atcvati 30649	A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) → 𝐴 ∈ HAtoms))
Theorem	atcvat2i 30650	A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶 ∧ 𝐴 ⋖ℋ (𝐵 ∨ℋ 𝐶)) → 𝐴 ∈ HAtoms))
Theorem	atord 30651	An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ∧ 𝐴 𝐶ℋ 𝐵) → (𝐵 ⊆ 𝐴 ∨ 𝐵 ⊆ (⊥‘𝐴)))
Theorem	atcvat2 30652	A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶 ∧ 𝐴 ⋖ℋ (𝐵 ∨ℋ 𝐶)) → 𝐴 ∈ HAtoms))
19.8.5  Irreducibility
Theorem	chirredlem1 30653*	Lemma for chirredi 30657. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (((𝑝 ∈ HAtoms ∧ (𝑞 ∈ Cℋ ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ ((𝑟 ∈ HAtoms ∧ 𝑟 ⊆ 𝐴) ∧ 𝑟 ⊆ (𝑝 ∨ℋ 𝑞))) → (𝑝 ∩ (⊥‘𝑟)) = 0ℋ)
Theorem	chirredlem2 30654*	Lemma for chirredi 30657. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((((𝑝 ∈ HAtoms ∧ 𝑝 ⊆ 𝐴) ∧ (𝑞 ∈ Cℋ ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ ((𝑟 ∈ HAtoms ∧ 𝑟 ⊆ 𝐴) ∧ 𝑟 ⊆ (𝑝 ∨ℋ 𝑞))) → ((⊥‘𝑟) ∩ (𝑝 ∨ℋ 𝑞)) = 𝑞)
Theorem	chirredlem3 30655*	Lemma for chirredi 30657. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ (𝑥 ∈ Cℋ → 𝐴 𝐶ℋ 𝑥)    ⇒   ⊢ ((((𝑝 ∈ HAtoms ∧ 𝑝 ⊆ 𝐴) ∧ (𝑞 ∈ HAtoms ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ (𝑟 ∈ HAtoms ∧ 𝑟 ⊆ (𝑝 ∨ℋ 𝑞))) → (𝑟 ⊆ 𝐴 → 𝑟 = 𝑝))
Theorem	chirredlem4 30656*	Lemma for chirredi 30657. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ (𝑥 ∈ Cℋ → 𝐴 𝐶ℋ 𝑥)    ⇒   ⊢ ((((𝑝 ∈ HAtoms ∧ 𝑝 ⊆ 𝐴) ∧ (𝑞 ∈ HAtoms ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ (𝑟 ∈ HAtoms ∧ 𝑟 ⊆ (𝑝 ∨ℋ 𝑞))) → (𝑟 = 𝑝 ∨ 𝑟 = 𝑞))
Theorem	chirredi 30657*	The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ (𝑥 ∈ Cℋ → 𝐴 𝐶ℋ 𝑥)    ⇒   ⊢ (𝐴 = 0ℋ ∨ 𝐴 = ℋ)
Theorem	chirred 30658*	The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) → (𝐴 = 0ℋ ∨ 𝐴 = ℋ))
19.8.6  Atoms (cont.)
Theorem	atcvat3i 30659	A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) ∈ HAtoms))
Theorem	atcvat4i 30660*	A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐵 ⊆ (𝐴 ∨ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨ℋ 𝑥))))
Theorem	atdmd 30661	Two Hilbert lattice elements have the dual modular pair property if the first is an atom. Theorem 7.6(c) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → 𝐴 𝑀ℋ* 𝐵)
Theorem	atmd 30662	Two Hilbert lattice elements have the modular pair property if the first is an atom. Theorem 7.6(b) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → 𝐴 𝑀ℋ 𝐵)
Theorem	atmd2 30663	Two Hilbert lattice elements have the dual modular pair property if the second is an atom. Part of Exercise 6 of [Kalmbach] p. 103. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → 𝐴 𝑀ℋ 𝐵)
Theorem	atabsi 30664	Absorption of an incomparable atom. Similar to Exercise 7.1 of [MaedaMaeda] p. 34. (Contributed by NM, 15-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ (𝐴 ∨ℋ 𝐵) → ((𝐴 ∨ℋ 𝐶) ∩ 𝐵) = (𝐴 ∩ 𝐵)))
Theorem	atabs2i 30665	Absorption of an incomparable atom. (Contributed by NM, 18-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ (𝐴 ∨ℋ 𝐵) → ((𝐴 ∨ℋ 𝐶) ∩ (𝐴 ∨ℋ 𝐵)) = 𝐴))
19.8.7  Modular symmetry
Theorem	mdsymlem1 30666*	Lemma for mdsymi 30674. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ (((𝑝 ∈ Cℋ ∧ (𝐵 ∩ 𝐶) ⊆ 𝐴) ∧ (𝐵 𝑀ℋ* 𝐴 ∧ 𝑝 ⊆ (𝐴 ∨ℋ 𝐵))) → 𝑝 ⊆ 𝐴)
Theorem	mdsymlem2 30667*	Lemma for mdsymi 30674. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ (((𝑝 ∈ HAtoms ∧ (𝐵 ∩ 𝐶) ⊆ 𝐴) ∧ (𝐵 𝑀ℋ* 𝐴 ∧ 𝑝 ⊆ (𝐴 ∨ℋ 𝐵))) → (𝐵 ≠ 0ℋ → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))))
Theorem	mdsymlem3 30668*	Lemma for mdsymi 30674. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ ((((𝑝 ∈ HAtoms ∧ ¬ (𝐵 ∩ 𝐶) ⊆ 𝐴) ∧ 𝑝 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝐴 ≠ 0ℋ) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)))
Theorem	mdsymlem4 30669*	Lemma for mdsymi 30674. This is the forward direction of Lemma 4(i) of [Maeda] p. 168. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ (𝑝 ∈ HAtoms → ((𝐵 𝑀ℋ* 𝐴 ∧ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) ∧ 𝑝 ⊆ (𝐴 ∨ℋ 𝐵))) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))))
Theorem	mdsymlem5 30670*	Lemma for mdsymi 30674. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (¬ 𝑞 = 𝑝 → ((𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)) → (((𝑐 ∈ Cℋ ∧ 𝐴 ⊆ 𝑐) ∧ 𝑝 ∈ HAtoms) → (𝑝 ⊆ 𝑐 → 𝑝 ⊆ ((𝑐 ∩ 𝐵) ∨ℋ 𝐴))))))
Theorem	mdsymlem6 30671*	Lemma for mdsymi 30674. This is the converse direction of Lemma 4(i) of [Maeda] p. 168, and is based on the proof of Theorem 1(d) to (e) of [Maeda] p. 167. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ (∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 ∨ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))) → 𝐵 𝑀ℋ* 𝐴)
Theorem	mdsymlem7 30672*	Lemma for mdsymi 30674. Lemma 4(i) of [Maeda] p. 168. Note that Maeda's 1965 definition of dual modular pair has reversed arguments compared to the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, which is the one that we use. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (𝐵 𝑀ℋ* 𝐴 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 ∨ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)))))
Theorem	mdsymlem8 30673*	Lemma for mdsymi 30674. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝)    ⇒   ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (𝐵 𝑀ℋ* 𝐴 ↔ 𝐴 𝑀ℋ* 𝐵))
Theorem	mdsymi 30674	M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴)
Theorem	mdsym 30675	M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴))
Theorem	dmdsym 30676	Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ 𝐵 𝑀ℋ* 𝐴))
Theorem	atdmd2 30677	Two Hilbert lattice elements have the dual modular pair property if the second is an atom. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → 𝐴 𝑀ℋ* 𝐵)
Theorem	sumdmdii 30678	If the subspace sum of two Hilbert lattice elements is closed, then the elements are a dual modular pair. Remark in [MaedaMaeda] p. 139. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵) → 𝐴 𝑀ℋ* 𝐵)
Theorem	cmmdi 30679	Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵)
Theorem	cmdmdi 30680	Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝐶ℋ 𝐵 → 𝐴 𝑀ℋ* 𝐵)
Theorem	sumdmdlem 30681	Lemma for sumdmdi 30683. The span of vector 𝐶 not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ (𝐴 +ℋ 𝐵)) → ((𝐵 +ℋ (span‘{𝐶})) ∩ 𝐴) = (𝐵 ∩ 𝐴))
Theorem	sumdmdlem2 30682*	Lemma for sumdmdi 30683. (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (∀𝑥 ∈ HAtoms ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	sumdmdi 30683	The subspace sum of two Hilbert lattice elements is closed iff the elements are a dual modular pair. Theorem 2 of [Holland] p. 1519. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵) ↔ 𝐴 𝑀ℋ* 𝐵)
Theorem	dmdbr4ati 30684*	Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))
Theorem	dmdbr5ati 30685*	Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))
Theorem	dmdbr6ati 30686*	Dual modular pair property in terms of atoms. The modular law takes the form of the shearing identity. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥))
Theorem	dmdbr7ati 30687*	Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))
Theorem	mdoc1i 30688	Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝑀ℋ (⊥‘𝐴)
Theorem	mdoc2i 30689	Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) 𝑀ℋ 𝐴
Theorem	dmdoc1i 30690	Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝑀ℋ* (⊥‘𝐴)
Theorem	dmdoc2i 30691	Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) 𝑀ℋ* 𝐴
Theorem	mdcompli 30692	A condition equivalent to the modular pair property. Part of proof of Theorem 1.14 of [MaedaMaeda] p. 4. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))
Theorem	dmdcompli 30693	A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))
Theorem	mddmdin0i 30694*	If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦)    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 → 𝐴 𝑀ℋ 𝐵)
Theorem	cdjreui 30695*	A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐶 ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))
Theorem	cdj1i 30696*	Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (∃𝑤 ∈ ℝ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
Theorem	cdj3lem1 30697*	A property of "𝐴 and 𝐵 are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) = 0ℋ)
Theorem	cdj3lem2 30698*	Lemma for cdj3i 30704. Value of the first-component function 𝑆. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝐶 +ℎ 𝐷)) = 𝐶)
Theorem	cdj3lem2a 30699*	Lemma for cdj3i 30704. Closure of the first-component function 𝑆. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘𝐶) ∈ 𝐴)
Theorem	cdj3lem2b 30700*	Lemma for cdj3i 30704. The first-component function 𝑆 is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))