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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchsleji 27701 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 27702* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremchincli 27703 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ∈ C

Theoremchsscon3i 27704 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴))

Theoremchsscon1i 27705 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴)

Theoremchsscon2i 27706 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴))

Theoremchcon2i 27707 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = (⊥‘𝐵) ↔ 𝐵 = (⊥‘𝐴))

Theoremchcon1i 27708 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((⊥‘𝐴) = 𝐵 ↔ (⊥‘𝐵) = 𝐴)

Theoremchcon3i 27709 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = 𝐵 ↔ (⊥‘𝐵) = (⊥‘𝐴))

Theoremchunssji 27710 Union is smaller than C join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵) ⊆ (𝐴 𝐵)

Theoremchjcomi 27711 Commutative law for join in C. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (𝐵 𝐴)

Theoremchub1i 27712 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 27713 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 27714 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 27715 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 27714). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴𝐶𝐵𝐶) → (𝐴 𝐵) ⊆ 𝐶)

Theoremchlej1i 27716 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 27717 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 27718 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 27719 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵)

Theoremchdmm1i 27720 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 27721 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 27722 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 27723 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 27724 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 27725 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 27726 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 27727 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 27728 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 27729 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 27730 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 27731 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
𝐴C       (𝐴 (⊥‘𝐴)) = ℋ

Theoremchj1i 27732 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ℋ) = ℋ

Theoremchm0i 27733 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
𝐴C       (𝐴 ∩ 0) = 0

Theoremchm0 27734 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
(𝐴C → (𝐴 ∩ 0) = 0)

Theoremshjshsi 27735 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 27736 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 27737* 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 27738 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 27739 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 27740 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 0) = 𝐴)

Theoremchslej 27741 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 27742 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵) ∈ C )

Theoremchsscon3 27743 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Theoremchsscon1 27744 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊆ 𝐵 ↔ (⊥‘𝐵) ⊆ 𝐴))

Theoremchsscon2 27745 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))

Theoremchpsscon3 27746 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘𝐴)))

Theoremchpsscon1 27747 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴))

Theoremchpsscon2 27748 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ⊊ (⊥‘𝐵) ↔ 𝐵 ⊊ (⊥‘𝐴)))

Theoremchjcom 27749 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))

Theoremchub1 27750 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 27751 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 27752 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 27753 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 27754 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 27755 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐴 𝐵) = 𝐵))

Theoremchlejb2 27756 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ (𝐵 𝐴) = 𝐵))

Theoremchnle 27757 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 27758 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 27759 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 27760 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 27761 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 27762 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 27763 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 𝐴) = 𝐴)

Theoremchjidmi 27764 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
𝐴C       (𝐴 𝐴) = 𝐴

Theoremchj12i 27765 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 27766 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 27767 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 27768 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 27769 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 27770 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 27771 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 27772 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 27773 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 27774 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 27775 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 27776 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 27777 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → (𝐴 (𝐵 𝐶)) = (𝐵 (𝐴 𝐶)))

Theoremchj4 27778 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 27779 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 27780 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 27781 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 27782 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 27783 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐶C ) → ((𝐴𝐵) ∨ (𝐴𝐶)) ⊆ (𝐴 ∩ (𝐵 𝐶)))

20.5.4  Span (cont.) and one-dimensional subspaces

Theoremspansn0 27784 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 27785 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 27786* 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 27787 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 27788 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 27789 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 27790 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 27791 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
( ‘∅) = 0

Theoremh1deoi 27792 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 27793* 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 27794 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 27795 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 27796 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 27797 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 27798* Lemma for h1de2ci 27799. (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 27799* 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 27800 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}))

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