P. 326 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hatomici 32501*	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 32502*	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 32503*	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 32504*	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 32505*	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 32506*	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 32507*	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 32508*	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 32509*	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 32510	Lemma for cvexchi 32511. (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 32511	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 32512*	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 32513*	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 32514*	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 32515*	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 32516	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 32517	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 32518	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 32519	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 32520	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 32521	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 32522	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 32523	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 32519 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 32524	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 32525	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 32526	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 32527	Lemma for atcvati 32528. (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 32528	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 32529	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 32530	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 32531	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 32532*	Lemma for chirredi 32536. (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 32533*	Lemma for chirredi 32536. (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 32534*	Lemma for chirredi 32536. (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 32535*	Lemma for chirredi 32536. (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 32536*	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 32537*	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 32538	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 32539*	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 32540	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 32541	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 32542	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 32543	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 32544	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 32545*	Lemma for mdsymi 32553. (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 32546*	Lemma for mdsymi 32553. (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 32547*	Lemma for mdsymi 32553. (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 32548*	Lemma for mdsymi 32553. 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 32549*	Lemma for mdsymi 32553. (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 32550*	Lemma for mdsymi 32553. 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 32551*	Lemma for mdsymi 32553. 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 32552*	Lemma for mdsymi 32553. 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 32553	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 32554	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 32555	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 32556	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 32557	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 32558	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 32559	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 32560	Lemma for sumdmdi 32562. 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 32561*	Lemma for sumdmdi 32562. (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 32562	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 32563*	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 32564*	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 32565*	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 32566*	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 32567	Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝑀ℋ (⊥‘𝐴)
Theorem	mdoc2i 32568	Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) 𝑀ℋ 𝐴
Theorem	dmdoc1i 32569	Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 𝐴 𝑀ℋ* (⊥‘𝐴)
Theorem	dmdoc2i 32570	Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (⊥‘𝐴) 𝑀ℋ* 𝐴
Theorem	mdcompli 32571	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 32572	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 32573*	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 32574*	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 32575*	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 32576*	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 32577*	Lemma for cdj3i 32583. 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 32578*	Lemma for cdj3i 32583. 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 32579*	Lemma for cdj3i 32583. 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 32580*	Lemma for cdj3i 32583. 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 32581*	Lemma for cdj3i 32583. 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 32582*	Lemma for cdj3i 32583. 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 32583*	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 32584	(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 30541 for our general style guidelines. For contributing via GitHub, see https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md 30541. The Metamath program command "verify markup *" will check that you have followed many 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 11835. 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 32585	A theorem about the universal class. Inference associated with bj-abv 37339 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)
⊢ 𝜑    ⇒   ⊢ V = {𝑥 ∣ 𝜑}
Theorem	xfree 32586	A partial converse to 19.9t 2233. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	xfree2 32587	A partial converse to 19.9t 2233. (Contributed by Stefan Allan, 21-Dec-2008.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	addltmulALT 32588	A proof readability experiment for addltmul 12447. (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	ad11antr 32589	Deduction adding 11 conjuncts to antecedent. (Contributed by Thierry Arnoux, 27-Sep-2025.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((((((((((((𝜑 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ 𝜈) → 𝜓)
Theorem	simp-12l 32590	Simplification of a conjunction. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ 𝜈) → 𝜑)
Theorem	simp-12r 32591	Simplification of a conjunction. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ 𝜈) → 𝜓)
Theorem	an52ds 32592	Inference exchanging the last antecedent with the second. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜒) ∧ 𝜃) ∧ 𝜓) → 𝜂)
Theorem	an62ds 32593	Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((((((𝜑 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜁)
Theorem	an72ds 32594	Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜎)    ⇒   ⊢ (((((((𝜑 ∧ 𝜁) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜎)
Theorem	an82ds 32595	Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ ((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜌)    ⇒   ⊢ ((((((((𝜑 ∧ 𝜎) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜓) → 𝜌)
Theorem	syl22anbrc 32596	Syllogism inference. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜂 ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)))    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	bian1dOLD 32597	Obsolete version of bian1d 587 as of 29-Jun-2025. (Contributed by Thierry Arnoux, 26-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	orim12da 32598	Deduce a disjunction from another one. Variation on orim12d 975. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜏)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (𝜃 ∨ 𝜏))
Theorem	or3di 32599	Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))
Theorem	or3dir 32600	Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∨ 𝜏) ↔ ((𝜑 ∨ 𝜏) ∧ (𝜓 ∨ 𝜏) ∧ (𝜒 ∨ 𝜏)))