P. 320 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pjoc1 31901	Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ 𝐻 ↔ ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ))
Theorem	pjomli 31902	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 31871. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ) → 𝐴 = 𝐵)
Theorem	pjoml 31903	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 31871. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ)) → 𝐴 = 𝐵)
Theorem	pjococi 31904	Proof of orthocomplement theorem using projections. Compare ococ 31873. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (⊥‘(⊥‘𝐻)) = 𝐻
Theorem	pjoc2i 31905	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    &   ⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (𝐴 ∈ (⊥‘𝐻) ↔ ((projℎ‘𝐻)‘𝐴) = 0ℎ)
Theorem	pjoc2 31906	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (⊥‘𝐻) ↔ ((projℎ‘𝐻)‘𝐴) = 0ℎ))
20.5.3  Hilbert lattice operations
Theorem	sh0le 31907	The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
Theorem	ch0le 31908	The zero subspace is the smallest member of Cℋ. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)
Theorem	shle0 31909	No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
Theorem	chle0 31910	No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
Theorem	chnlen0 31911	A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ))
Theorem	ch0pss 31912	The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ))
Theorem	orthin 31913	The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) = 0ℋ))
Theorem	ssjo 31914	The lattice join of a subset with its orthocomplement is the whole space. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ)
Theorem	shne0i 31915*	A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)
Theorem	shs0i 31916	Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝐴 +ℋ 0ℋ) = 𝐴
Theorem	shs00i 31917	Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 +ℋ 𝐵) = 0ℋ)
Theorem	ch0lei 31918	The closed subspace zero is the smallest member of Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ 0ℋ ⊆ 𝐴
Theorem	chle0i 31919	No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)
Theorem	chne0i 31920*	A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)
Theorem	chocini 31921	Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ (⊥‘𝐴)) = 0ℋ
Theorem	chj0i 31922	Join with lattice zero in Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 0ℋ) = 𝐴
Theorem	chm1i 31923	Meet with lattice one in Cℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ ℋ) = 𝐴
Theorem	chjcli 31924	Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ
Theorem	chsleji 31925	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	chseli 31926*	Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))
Theorem	chincli 31927	Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ
Theorem	chsscon3i 31928	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))
Theorem	chsscon1i 31929	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)
Theorem	chsscon2i 31930	Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))
Theorem	chcon2i 31931	Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))
Theorem	chcon1i 31932	Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)
Theorem	chcon3i 31933	Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))
Theorem	chunssji 31934	Union is smaller than Cℋ join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	chjcomi 31935	Commutative law for join in Cℋ. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)
Theorem	chub1i 31936	Cℋ join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵)
Theorem	chub2i 31937	Cℋ join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ 𝐴 ⊆ (𝐵 ∨ℋ 𝐴)
Theorem	chlubi 31938	Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∨ℋ 𝐵) ⊆ 𝐶)
Theorem	chlubii 31939	Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 31938). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∨ℋ 𝐵) ⊆ 𝐶)
Theorem	chlej1i 31940	Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶))
Theorem	chlej2i 31941	Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵))
Theorem	chlej12i 31942	Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐷))
Theorem	chlejb1i 31943	Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵)
Theorem	chdmm1i 31944	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))
Theorem	chdmm2i 31945	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 ∨ℋ (⊥‘𝐵))
Theorem	chdmm3i 31946	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ℋ 𝐵)
Theorem	chdmm4i 31947	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 ∨ℋ 𝐵)
Theorem	chdmj1i 31948	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵))
Theorem	chdmj2i 31949	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))
Theorem	chdmj3i 31950	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘(𝐴 ∨ℋ (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵)
Theorem	chdmj4i 31951	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵)
Theorem	chnlei 31952	Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝐵))
Theorem	chjassi 31953	Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶))
Theorem	chj00i 31954	Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 ∨ℋ 𝐵) = 0ℋ)
Theorem	chjoi 31955	The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ
Theorem	chj1i 31956	Join with Hilbert lattice one. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ ℋ) = ℋ
Theorem	chm0i 31957	Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ 0ℋ) = 0ℋ
Theorem	chm0 31958	Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ 0ℋ) = 0ℋ)
Theorem	shjshsi 31959	Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 +ℋ 𝐵)))
Theorem	shjshseli 31960	A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐴 +ℋ 𝐵) ∈ Cℋ ↔ (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵))
Theorem	chne0 31961*	A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ))
Theorem	chocin 31962	Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ)
Theorem	chssoc 31963	A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ (⊥‘𝐴) ↔ 𝐴 = 0ℋ))
Theorem	chj0 31964	Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴)
Theorem	chslej 31965	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 +ℋ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵))
Theorem	chincl 31966	Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ )
Theorem	chsscon3 31967	Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)))
Theorem	chsscon1 31968	Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴))
Theorem	chsscon2 31969	Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))
Theorem	chpsscon3 31970	Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘𝐴)))
Theorem	chpsscon1 31971	Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴))
Theorem	chpsscon2 31972	Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊊ (⊥‘𝐵) ↔ 𝐵 ⊊ (⊥‘𝐴)))
Theorem	chjcom 31973	Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴))
Theorem	chub1 31974	Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ⊆ (𝐴 ∨ℋ 𝐵))
Theorem	chub2 31975	Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ⊆ (𝐵 ∨ℋ 𝐴))
Theorem	chlub 31976	Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∨ℋ 𝐵) ⊆ 𝐶))
Theorem	chlej1 31977	Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶))
Theorem	chlej2 31978	Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∨ℋ 𝐴) ⊆ (𝐶 ∨ℋ 𝐵))
Theorem	chlejb1 31979	Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵))
Theorem	chlejb2 31980	Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∨ℋ 𝐴) = 𝐵))
Theorem	chnle 31981	Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝐵)))
Theorem	chjo 31982	The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ)
Theorem	chabs1 31983	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) = 𝐴)
Theorem	chabs2 31984	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ (𝐴 ∨ℋ 𝐵)) = 𝐴)
Theorem	chabs1i 31985	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) = 𝐴
Theorem	chabs2i 31986	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 ∩ (𝐴 ∨ℋ 𝐵)) = 𝐴
Theorem	chjidm 31987	Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 𝐴) = 𝐴)
Theorem	chjidmi 31988	Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ 𝐴) = 𝐴
Theorem	chj12i 31989	A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) = (𝐵 ∨ℋ (𝐴 ∨ℋ 𝐶))
Theorem	chj4i 31990	Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    &   ⊢ 𝐷 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ (𝐶 ∨ℋ 𝐷)) = ((𝐴 ∨ℋ 𝐶) ∨ℋ (𝐵 ∨ℋ 𝐷))
Theorem	chjjdiri 31991	Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ 𝐶 ∈ Cℋ    ⇒   ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = ((𝐴 ∨ℋ 𝐶) ∨ℋ (𝐵 ∨ℋ 𝐶))
Theorem	chdmm1 31992	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))
Theorem	chdmm2 31993	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 ∨ℋ (⊥‘𝐵)))
Theorem	chdmm3 31994	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ℋ 𝐵))
Theorem	chdmm4 31995	De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 ∨ℋ 𝐵))
Theorem	chdmj1 31996	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)))
Theorem	chdmj2 31997	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘((⊥‘𝐴) ∨ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵)))
Theorem	chdmj3 31998	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘(𝐴 ∨ℋ (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵))
Theorem	chdmj4 31999	De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵))
Theorem	chjass 32000	Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)))