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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshlub 28401 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S𝐶C ) → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶))

Theoremshlubi 28402 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶C       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶)

19.5.2  Projectors (cont.)

Theorempjhtheu2 28403* Uniqueness of 𝑦 for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥𝐻 𝐴 = (𝑥 + 𝑦))

Theorempjcli 28404 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ 𝐻)

Theorempjhcli 28405 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → ((proj𝐻)‘𝐴) ∈ ℋ)

Theorempjpjpre 28406 Decomposition of a vector into projections. This formulation of axpjpj 28407 avoids pjhth 28380. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝜑𝐻C )    &   (𝜑𝐴 ∈ (𝐻 + (⊥‘𝐻)))       (𝜑𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))

Theoremaxpjpj 28407 Decomposition of a vector into projections. See comment in axpjcl 28387. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → 𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴)))

Theorempjclii 28408 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ 𝐻

Theorempjhclii 28409 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) ∈ ℋ

Theorempjpj0i 28410 Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴))

Theorempjpji 28411 Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴 = (((proj𝐻)‘𝐴) + ((proj‘(⊥‘𝐻))‘𝐴))

Theorempjpjhth 28412* Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ∃𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦))

Theorempjpjhthi 28413* Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐻C       𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)

Theorempjop 28414 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴)))

Theorempjpo 28415 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴)))

Theorempjopi 28416 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj‘(⊥‘𝐻))‘𝐴) = (𝐴 ((proj𝐻)‘𝐴))

Theorempjpoi 28417 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘𝐴) = (𝐴 ((proj‘(⊥‘𝐻))‘𝐴))

Theorempjoc1i 28418 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0)

Theorempjchi 28419 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴𝐻 ↔ ((proj𝐻)‘𝐴) = 𝐴)

Theorempjoccl 28420 The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴 ((proj𝐻)‘𝐴)) ∈ (⊥‘𝐻))

Theorempjoc1 28421 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ ((proj‘(⊥‘𝐻))‘𝐴) = 0))

Theorempjomli 28422 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 28391. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
𝐴C    &   𝐵S       ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)

Theorempjoml 28423 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 28391. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵S ) ∧ (𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0)) → 𝐴 = 𝐵)

Theorempjococi 28424 Proof of orthocomplement theorem using projections. Compare ococ 28393. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
𝐻C       (⊥‘(⊥‘𝐻)) = 𝐻

Theorempjoc2i 28425 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.)
𝐻C    &   𝐴 ∈ ℋ       (𝐴 ∈ (⊥‘𝐻) ↔ ((proj𝐻)‘𝐴) = 0)

Theorempjoc2 28426 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.)
((𝐻C𝐴 ∈ ℋ) → (𝐴 ∈ (⊥‘𝐻) ↔ ((proj𝐻)‘𝐴) = 0))

19.5.3  Hilbert lattice operations

Theoremsh0le 28427 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴S → 0𝐴)

Theoremch0le 28428 The zero subspace is the smallest member of C. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → 0𝐴)

Theoremshle0 28429 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
(𝐴S → (𝐴 ⊆ 0𝐴 = 0))

Theoremchle0 28430 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
(𝐴C → (𝐴 ⊆ 0𝐴 = 0))

Theoremchnlen0 28431 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
(𝐵C → (¬ 𝐴𝐵 → ¬ 𝐴 = 0))

Theoremch0pss 28432 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴C → (0𝐴𝐴 ≠ 0))

Theoremorthin 28433 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴𝐵) = 0))

Theoremssjo 28434 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 (⊥‘𝐴)) = ℋ)

Theoremshne0i 28435* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴S       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)

Theoremshs0i 28436 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
𝐴S       (𝐴 + 0) = 𝐴

Theoremshs00i 28437 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐴 = 0𝐵 = 0) ↔ (𝐴 + 𝐵) = 0)

Theoremch0lei 28438 The closed subspace zero is the smallest member of C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       0𝐴

Theoremchle0i 28439 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
𝐴C       (𝐴 ⊆ 0𝐴 = 0)

Theoremchne0i 28440* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
𝐴C       (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)

Theoremchocini 28441 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.)
𝐴C       (𝐴 ∩ (⊥‘𝐴)) = 0

Theoremchj0i 28442 Join with lattice zero in C. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 0) = 𝐴

Theoremchm1i 28443 Meet with lattice one in C. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ ℋ) = 𝐴

Theoremchjcli 28444 Closure of C join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) ∈ C

Theoremchsleji 28445 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 + 𝐵) ⊆ (𝐴 𝐵)

Theoremchseli 28446* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremchincli 28447 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ∈ C

Theoremchsscon3i 28448 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))

Theoremchsscon1i 28449 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)

Theoremchsscon2i 28450 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))

Theoremchcon2i 28451 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))

Theoremchcon1i 28452 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)

Theoremchcon3i 28453 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))

Theoremchunssji 28454 Union is smaller than C join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ⊆ (𝐴 𝐵)

Theoremchjcomi 28455 Commutative law for join in C. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (𝐵 𝐴)

Theoremchub1i 28456 C join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 ⊆ (𝐴 𝐵)

Theoremchub2i 28457 C join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 ⊆ (𝐵 𝐴)

Theoremchlubi 28458 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶)

Theoremchlubii 28459 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 28458). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶𝐵𝐶) → (𝐴 𝐵) ⊆ 𝐶)

Theoremchlej1i 28460 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴𝐵 → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremchlej2i 28461 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴𝐵 → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremchlej12i 28462 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴𝐵𝐶𝐷) → (𝐴 𝐶) ⊆ (𝐵 𝐷))

Theoremchlejb1i 28463 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵)

Theoremchdmm1i 28464 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴𝐵)) = ((⊥‘𝐴) ∨ (⊥‘𝐵))

Theoremchdmm2i 28465 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 (⊥‘𝐵))

Theoremchdmm3i 28466 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ 𝐵)

Theoremchdmm4i 28467 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 𝐵)

Theoremchdmj1i 28468 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵))

Theoremchdmj2i 28469 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∨ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))

Theoremchdmj3i 28470 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘(𝐴 (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵)

Theoremchdmj4i 28471 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (⊥‘((⊥‘𝐴) ∨ (⊥‘𝐵))) = (𝐴𝐵)

Theoremchnlei 28472 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵𝐴𝐴 ⊊ (𝐴 𝐵))

Theoremchjassi 28473 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = (𝐴 (𝐵 𝐶))

Theoremchj00i 28474 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴 = 0𝐵 = 0) ↔ (𝐴 𝐵) = 0)

Theoremchjoi 28475 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 (⊥‘𝐴)) = ℋ

Theoremchj1i 28476 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ℋ) = ℋ

Theoremchm0i 28477 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ 0) = 0

Theoremchm0 28478 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ∩ 0) = 0)

Theoremshjshsi 28479 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)
𝐴S    &   𝐵S       (𝐴 𝐵) = (⊥‘(⊥‘(𝐴 + 𝐵)))

Theoremshjshseli 28480 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.)
𝐴S    &   𝐵S       ((𝐴 + 𝐵) ∈ C ↔ (𝐴 + 𝐵) = (𝐴 𝐵))

Theoremchne0 28481* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0))

Theoremchocin 28482 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.)
(𝐴C → (𝐴 ∩ (⊥‘𝐴)) = 0)

Theoremchssoc 28483 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ⊆ (⊥‘𝐴) ↔ 𝐴 = 0))

Theoremchj0 28484 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 0) = 𝐴)

Theoremchslej 28485 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 + 𝐵) ⊆ (𝐴 𝐵))

Theoremchincl 28486 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵) ∈ C )

Theoremchsscon3 28487 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Theoremchsscon1 28488 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴))

Theoremchsscon2 28489 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))

Theoremchpsscon3 28490 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘𝐴)))

Theoremchpsscon1 28491 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴))

Theoremchpsscon2 28492 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊊ (⊥‘𝐵) ↔ 𝐵 ⊊ (⊥‘𝐴)))

Theoremchjcom 28493 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))

Theoremchub1 28494 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → 𝐴 ⊆ (𝐴 𝐵))

Theoremchub2 28495 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → 𝐴 ⊆ (𝐵 𝐴))

Theoremchlub 28496 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 𝐵) ⊆ 𝐶))

Theoremchlej1 28497 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ 𝐴𝐵) → (𝐴 𝐶) ⊆ (𝐵 𝐶))

Theoremchlej2 28498 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ 𝐴𝐵) → (𝐶 𝐴) ⊆ (𝐶 𝐵))

Theoremchlejb1 28499 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵))

Theoremchlejb2 28500 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐵 𝐴) = 𝐵))

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