P. 317 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hatomic 31601*	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 31602*	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 31603*	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 31604*	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 31605*	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 31606*	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 31607*	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 31608*	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 31609	Lemma for cvexchi 31610. (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 31610	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 31611*	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 31612*	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 31613*	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 31614*	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 31615	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 31616	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 31617	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 31618	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 31619	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 31620	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 31621	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 31622	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 31618 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 31623	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 31624	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 31625	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 31626	Lemma for atcvati 31627. (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 31627	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 31628	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 31629	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 31630	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))
20.8.5  Irreducibility
Theorem	chirredlem1 31631*	Lemma for chirredi 31635. (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 31632*	Lemma for chirredi 31635. (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 31633*	Lemma for chirredi 31635. (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 31634*	Lemma for chirredi 31635. (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 31635*	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 31636*	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))
20.8.6  Atoms (cont.)
Theorem	atcvat3i 31637	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 31638*	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 31639	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 31640	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 31641	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 31642	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 31643	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))
20.8.7  Modular symmetry
Theorem	mdsymlem1 31644*	Lemma for mdsymi 31652. (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 31645*	Lemma for mdsymi 31652. (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 31646*	Lemma for mdsymi 31652. (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 31647*	Lemma for mdsymi 31652. 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 31648*	Lemma for mdsymi 31652. (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 31649*	Lemma for mdsymi 31652. 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 31650*	Lemma for mdsymi 31652. 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 31651*	Lemma for mdsymi 31652. 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 31652	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 31653	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 31654	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 31655	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 31656	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 31657	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 31658	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 31659	Lemma for sumdmdi 31661. 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 31660*	Lemma for sumdmdi 31661. (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 31661	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 31662*	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 31663*	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 31664*	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 31665*	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 31666	Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝑀ℋ (⊥‘𝐴)
Theorem	mdoc2i 31667	Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) 𝑀ℋ 𝐴
Theorem	dmdoc1i 31668	Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝑀ℋ* (⊥‘𝐴)
Theorem	dmdoc2i 31669	Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) 𝑀ℋ* 𝐴
Theorem	mdcompli 31670	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 31671	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 31672*	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 31673*	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 31674*	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 31675*	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 31676*	Lemma for cdj3i 31682. 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 31677*	Lemma for cdj3i 31682. 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 31678*	Lemma for cdj3i 31682. 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))))
Theorem	cdj3lem3 31679*	Lemma for cdj3i 31682. Value of the second-component function 𝑇. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷)
Theorem	cdj3lem3a 31680*	Lemma for cdj3i 31682. Closure of the second-component function 𝑇. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘𝐶) ∈ 𝐵)
Theorem	cdj3lem3b 31681*	Lemma for cdj3i 31682. The second-component function 𝑇 is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 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‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
Theorem	cdj3i 31682*	Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    &   ⊢ (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))    &   ⊢ (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))    ⇒   ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓))
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
21.1  Mathboxes for user contributions
21.1.1  Mathbox guidelines
Theorem	mathbox 31683	(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.) A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm. Mathboxes are provided to help keep your work synchronized with changes in set.mm while allowing you to work independently without affecting other contributors. Even though in a sense your mathbox belongs to you, it is still part of the shared body of knowledge contained in set.mm, and occasionally other people may make maintenance edits to your mathbox for things like keeping it synchronized with the rest of set.mm, reducing proof lengths, moving your theorems to the main part of set.mm when needed, and fixing typos or other errors. If you want to preserve it the way you left it, you can keep a local copy or keep track of the GitHub commit number. Guidelines: 1. See conventions 29643 for our general style guidelines. For contributing via GitHub, see https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md 29643. The Metamath program command "verify markup *" will check that you have followed many of of the conventions we use. 2. If at all possible, please use only nullary class constants for new definitions, for example as in df-div 11869. 3. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command will take care of indentation conventions and line wrapping. 4. All mathbox content will be on public display and should hopefully reflect the overall quality of the website. 5. Mathboxes must be independent from one another (checked by "verify markup *"). If you need a theorem from another mathbox, typically it is moved to the main part of set.mm. New users should consult with more experienced users before doing this. 6. If a contributor is no longer active, we will continue the usual maintenance edits. As time goes on, often theorems will be moved to main or removed in favor of similar replacements. But we are also willing to maintain mathboxes in place, as work by others from years ago may form the foundation of future work; you could even argue that all of mathematics is like that. 7. For theorems of importance (for example, a Metamath 100 theorem or a dependency of one), we prefer to eventually move them out of mathboxes (although a mathbox is perfectly appropriate as proofs are being developed and refined). (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ 𝜑
21.2  Mathbox for Stefan Allan
Theorem	sa-abvi 31684	A theorem about the universal class. Inference associated with bj-abv 35775 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)
⊢ 𝜑    ⇒   ⊢ V = {𝑥 ∣ 𝜑}
Theorem	xfree 31685	A partial converse to 19.9t 2198. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	xfree2 31686	A partial converse to 19.9t 2198. (Contributed by Stefan Allan, 21-Dec-2008.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	addltmulALT 31687	A proof readability experiment for addltmul 12445. (Contributed by Stefan Allan, 30-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))
21.3  Mathbox for Thierry Arnoux
21.3.1  Propositional Calculus - misc additions
Theorem	bian1d 31688	Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	or3di 31689	Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))
Theorem	or3dir 31690	Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∨ 𝜏) ↔ ((𝜑 ∨ 𝜏) ∧ (𝜓 ∨ 𝜏) ∧ (𝜒 ∨ 𝜏)))
Theorem	3o1cs 31691	Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	3o2cs 31692	Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	3o3cs 31693	Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	13an22anass 31694	Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
21.3.2  Predicate Calculus
21.3.2.1  Predicate Calculus - misc additions
Theorem	sbc2iedf 31695*	Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
Theorem	rspc2daf 31696*	Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
21.3.2.2  Restricted quantification - misc additions
Theorem	ralcom4f 31697*	Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexcom4f 31698*	Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
Theorem	19.9d2rf 31699	A deduction version of one direction of 19.9 2199 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.9d2r 31700*	A deduction version of one direction of 19.9 2199 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)