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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchub2i 27501 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 27502 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 27503 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 27502). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶𝐵𝐶) → (𝐴 𝐵) ⊆ 𝐶)
 
Theoremchlej1i 27504 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 27505 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 27506 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 27507 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵)
 
Theoremchdmm1i 27508 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 27509 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 27510 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 27511 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 27512 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 27513 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 27514 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 27515 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 27516 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 27517 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 27518 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 27519 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 (⊥‘𝐴)) = ℋ
 
Theoremchj1i 27520 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ℋ) = ℋ
 
Theoremchm0i 27521 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ 0) = 0
 
Theoremchm0 27522 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ∩ 0) = 0)
 
Theoremshjshsi 27523 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 27524 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 27525* 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 27526 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 27527 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 27528 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 0) = 𝐴)
 
Theoremchslej 27529 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 27530 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵) ∈ C )
 
Theoremchsscon3 27531 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)))
 
Theoremchsscon1 27532 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴))
 
Theoremchsscon2 27533 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))
 
Theoremchpsscon3 27534 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘𝐴)))
 
Theoremchpsscon1 27535 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴))
 
Theoremchpsscon2 27536 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊊ (⊥‘𝐵) ↔ 𝐵 ⊊ (⊥‘𝐴)))
 
Theoremchjcom 27537 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
 
Theoremchub1 27538 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 27539 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 27540 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 27541 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 27542 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 27543 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵))
 
Theoremchlejb2 27544 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐵 𝐴) = 𝐵))
 
Theoremchnle 27545 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐵𝐴𝐴 ⊊ (𝐴 𝐵)))
 
Theoremchjo 27546 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
(𝐴C → (𝐴 (⊥‘𝐴)) = ℋ)
 
Theoremchabs1 27547 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 (𝐴𝐵)) = 𝐴)
 
Theoremchabs2 27548 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ∩ (𝐴 𝐵)) = 𝐴)
 
Theoremchabs1i 27549 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐴𝐵)) = 𝐴
 
Theoremchabs2i 27550 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ∩ (𝐴 𝐵)) = 𝐴
 
Theoremchjidm 27551 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 𝐴) = 𝐴)
 
Theoremchjidmi 27552 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴C       (𝐴 𝐴) = 𝐴
 
Theoremchj12i 27553 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴 (𝐵 𝐶)) = (𝐵 (𝐴 𝐶))
 
Theoremchj4i 27554 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝐵) ∨ (𝐶 𝐷)) = ((𝐴 𝐶) ∨ (𝐵 𝐷))
 
Theoremchjjdiri 27555 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = ((𝐴 𝐶) ∨ (𝐵 𝐶))
 
Theoremchdmm1 27556 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴𝐵)) = ((⊥‘𝐴) ∨ (⊥‘𝐵)))
 
Theoremchdmm2 27557 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∩ 𝐵)) = (𝐴 (⊥‘𝐵)))
 
Theoremchdmm3 27558 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴 ∩ (⊥‘𝐵))) = ((⊥‘𝐴) ∨ 𝐵))
 
Theoremchdmm4 27559 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 𝐵))
 
Theoremchdmj1 27560 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)))
 
Theoremchdmj2 27561 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∨ 𝐵)) = (𝐴 ∩ (⊥‘𝐵)))
 
Theoremchdmj3 27562 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘(𝐴 (⊥‘𝐵))) = ((⊥‘𝐴) ∩ 𝐵))
 
Theoremchdmj4 27563 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (⊥‘((⊥‘𝐴) ∨ (⊥‘𝐵))) = (𝐴𝐵))
 
Theoremchjass 27564 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵) ∨ 𝐶) = (𝐴 (𝐵 𝐶)))
 
Theoremchj12 27565 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 (𝐵 𝐶)) = (𝐵 (𝐴 𝐶)))
 
Theoremchj4 27566 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
(((𝐴C𝐵C ) ∧ (𝐶C𝐷C )) → ((𝐴 𝐵) ∨ (𝐶 𝐷)) = ((𝐴 𝐶) ∨ (𝐵 𝐷)))
 
Theoremledii 27567 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐵) ∨ (𝐴𝐶)) ⊆ (𝐴 ∩ (𝐵 𝐶))
 
Theoremlediri 27568 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶) ∨ (𝐵𝐶)) ⊆ ((𝐴 𝐵) ∩ 𝐶)
 
Theoremlejdii 27569 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       (𝐴 (𝐵𝐶)) ⊆ ((𝐴 𝐵) ∩ (𝐴 𝐶))
 
Theoremlejdiri 27570 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐵) ∨ 𝐶) ⊆ ((𝐴 𝐶) ∩ (𝐵 𝐶))
 
Theoremledi 27571 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴𝐵) ∨ (𝐴𝐶)) ⊆ (𝐴 ∩ (𝐵 𝐶)))
 
19.5.4  Span (cont.) and one-dimensional subspaces
 
Theoremspansn0 27572 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
(span‘{0}) = 0
 
Theoremspan0 27573 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(span‘∅) = 0
 
Theoremelspani 27574* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴 ⊆ ℋ → (𝐵 ∈ (span‘𝐴) ↔ ∀𝑥S (𝐴𝑥𝐵𝑥)))
 
Theoremspanuni 27575 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (span‘(𝐴𝐵)) = ((span‘𝐴) + (span‘𝐵))
 
Theoremspanun 27576 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴𝐵)) = ((span‘𝐴) + (span‘𝐵)))
 
Theoremsshhococi 27577 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (𝐴 𝐵) = ((⊥‘(⊥‘𝐴)) ∨ (⊥‘(⊥‘𝐵)))
 
Theoremhne0 27578 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
( ℋ ≠ 0 ↔ ∃𝑥 ∈ ℋ 𝑥 ≠ 0)
 
Theoremchsup0 27579 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
( ‘∅) = 0
 
Theoremh1deoi 27580 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)
𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))
 
Theoremh1dei 27581* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ ℋ ((𝐵 ·ih 𝑥) = 0 → (𝐴 ·ih 𝑥) = 0)))
 
Theoremh1did 27582 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴})))
 
Theoremh1dn0 27583 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (⊥‘(⊥‘{𝐴})) ≠ 0)
 
Theoremh1de2i 27584 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·ih 𝐵) · 𝐴) = ((𝐴 ·ih 𝐵) · 𝐵))
 
Theoremh1de2bi 27585 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐵 ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) · 𝐵)))
 
Theoremh1de2ctlem 27586* Lemma for h1de2ci 27587. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 · 𝐵))
 
Theoremh1de2ci 27587* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐵 ∈ ℋ       (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 · 𝐵))
 
Theoremspansni 27588 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴 ∈ ℋ       (span‘{𝐴}) = (⊥‘(⊥‘{𝐴}))
 
Theoremelspansni 27589* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴 ∈ ℋ       (𝐵 ∈ (span‘{𝐴}) ↔ ∃𝑥 ∈ ℂ 𝐵 = (𝑥 · 𝐴))
 
Theoremspansn 27590 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (span‘{𝐴}) = (⊥‘(⊥‘{𝐴})))
 
Theoremspansnch 27591 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (span‘{𝐴}) ∈ C )
 
Theoremspansnsh 27592 The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (span‘{𝐴}) ∈ S )
 
Theoremspansnchi 27593 The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
𝐴 ∈ ℋ       (span‘{𝐴}) ∈ C
 
Theoremspansnid 27594 A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴}))
 
Theoremspansnmul 27595 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 · 𝐴) ∈ (span‘{𝐴}))
 
Theoremelspansncl 27596 A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ (span‘{𝐴})) → 𝐵 ∈ ℋ)
 
Theoremelspansn 27597* Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐵 ∈ (span‘{𝐴}) ↔ ∃𝑥 ∈ ℂ 𝐵 = (𝑥 · 𝐴)))
 
Theoremelspansn2 27598 Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) · 𝐵)))
 
Theoremspansncol 27599 The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 · 𝐴)}) = (span‘{𝐴}))
 
Theoremspansneleqi 27600 Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((span‘{𝐴}) = (span‘{𝐵}) → 𝐴 ∈ (span‘{𝐵})))
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