 Home Metamath Proof ExplorerTheorem List (p. 370 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdihglblem3N 36901* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}    &   𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑇)) = 𝑥𝑇 (𝐼𝑥))

Theoremdihglblem3aN 36902* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}    &   𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑆)) = 𝑥𝑇 (𝐼𝑥))

Theoremdihglblem4 36903* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}    &   𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅)) → (𝐼‘(𝐺𝑆)) ⊆ 𝑥𝑆 (𝐼𝑥))

Theoremdihglblem5 36904* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐵𝑇 ≠ ∅)) → 𝑥𝑇 (𝐼𝑥) ∈ 𝑆)

Theoremdihmeetlem2N 36905 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐺 = (𝑇 (𝑃) = 𝑞)    &    0 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

TheoremdihglbcpreN 36906* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane 𝑊. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐹 = (𝑔𝑇 (𝑔𝑃) = 𝑞)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ ¬ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))

TheoremdihglbcN 36907* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = (le‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ ¬ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))

TheoremdihmeetcN 36908 Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ ¬ (𝑋 𝑌) 𝑊) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

TheoremdihmeetbN 36909 Isomorphism H of a lattice meet when one element is under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵 ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

TheoremdihmeetbclemN 36910 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) → (𝐼‘(𝑋 𝑌)) = (((𝐼𝑋) ∩ (𝐼𝑌)) ∩ (𝐼𝑊)))

Theoremdihmeetlem3N 36911 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌) 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑌 𝑊)) = 𝑌)) → 𝑄𝑅)

Theoremdihmeetlem4preN 36912* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) = { 0 })

Theoremdihmeetlem4N 36913 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) = { 0 })

Theoremdihmeetlem5 36914 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴𝑄 𝑋)) → (𝑋 (𝑌 𝑄)) = ((𝑋 𝑌) 𝑄))

Theoremdihmeetlem6 36915 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → ¬ (𝑋 (𝑌 𝑄)) 𝑊)

Theoremdihmeetlem7N 36916 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑌)) → (((𝑋 𝑌) 𝑝) 𝑌) = (𝑋 𝑌))

Theoremdihjatc1 36917 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change order of (𝑋 𝑌) 𝑄 here and down? (Contributed by NM, 6-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 𝑋 ∧ (𝑋 𝑌) 𝑊)) → (𝐼‘((𝑋 𝑌) 𝑄)) = ((𝐼𝑄) (𝐼‘(𝑋 𝑌))))

Theoremdihjatc2N 36918 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 𝑋 ∧ (𝑋 𝑌) 𝑊)) → (𝐼‘(𝑄 (𝑋 𝑌))) = ((𝐼𝑄) (𝐼‘(𝑋 𝑌))))

Theoremdihjatc3 36919 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 𝑋 ∧ (𝑋 𝑌) 𝑊)) → (𝐼‘((𝑋 𝑌) 𝑄)) = ((𝐼‘(𝑋 𝑌)) (𝐼𝑄)))

Theoremdihmeetlem8N 36920 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change order of (𝑋 𝑌) 𝑝 here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑝 𝑋 ∧ (𝑋 𝑌) 𝑊)) → (𝐼‘((𝑋 𝑌) 𝑝)) = ((𝐼𝑝) (𝐼‘(𝑋 𝑌))))

Theoremdihmeetlem9N 36921 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝐼𝑝) (𝐼‘(𝑋 𝑌))) ∩ (𝐼𝑌)) = ((𝐼‘(𝑋 𝑌)) ((𝐼𝑝) ∩ (𝐼𝑌))))

Theoremdihmeetlem10N 36922 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ 𝑝 𝑋)) → (𝐼‘((𝑋 𝑌) 𝑝)) = ((𝐼𝑋) ∩ (𝐼‘(𝑌 𝑝))))

Theoremdihmeetlem11N 36923 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ 𝑝 𝑋)) → ((𝐼‘((𝑋 𝑌) 𝑝)) ∩ (𝐼𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

Theoremdihmeetlem12N 36924 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ 𝑝 𝑋 ∧ (𝑋 𝑌) 𝑊)) → ((𝐼‘(𝑋 𝑌)) ((𝐼𝑝) ∩ (𝐼𝑌))) = ((𝐼𝑋) ∩ (𝐼𝑌)))

Theoremdihmeetlem13N 36925* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) = { 0 })

Theoremdihmeetlem14N 36926 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌𝐵𝑝𝐵) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ 𝑟 𝑌 ∧ (𝑌 𝑝) 𝑊)) → ((𝐼‘(𝑌 𝑝)) ((𝐼𝑟) ∩ (𝐼𝑝))) = ((𝐼𝑌) ∩ (𝐼𝑝)))

Theoremdihmeetlem15N 36927 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    0 = (0g𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌𝐵 ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ 𝑟 𝑌 ∧ (𝑌 𝑝) 𝑊)) → ((𝐼𝑟) ∩ (𝐼𝑝)) = { 0 })

Theoremdihmeetlem16N 36928 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌𝐵 ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ 𝑟 𝑌 ∧ (𝑌 𝑝) 𝑊)) → (𝐼‘(𝑌 𝑝)) = ((𝐼𝑌) ∩ (𝐼𝑝)))

Theoremdihmeetlem17N 36929 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    0 = (0.‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑝 𝑋)) → (𝑌 𝑝) = 0 )

Theoremdihmeetlem18N 36930 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    0 = (0g𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝑌𝐵) ∧ ((𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ (𝑝 𝑋𝑟 𝑌 ∧ (𝑋 𝑌) 𝑊))) → ((𝐼𝑌) ∩ (𝐼𝑝)) = { 0 })

Theoremdihmeetlem19N 36931 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝑌𝐵) ∧ ((𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ (𝑝 𝑋𝑟 𝑌 ∧ (𝑋 𝑌) 𝑊))) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

Theoremdihmeetlem20N 36932 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑌𝐵 ∧ ¬ 𝑌 𝑊) ∧ (𝑋 𝑌) 𝑊)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

TheoremdihmeetALTN 36933 Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

Theoremdih1dimatlem0 36934* Lemma for dih1dimat 36936. (Contributed by NM, 11-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝐹 = (Scalar‘𝑈)    &   𝐽 = (invr𝐹)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &    0 = (0g𝑈)    &   𝐺 = (𝑇 (𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝑖 = (𝑝𝐺) ∧ 𝑝𝐸) ↔ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))))

Theoremdih1dimatlem 36935* Lemma for dih1dimat 36936. (Contributed by NM, 10-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝐹 = (Scalar‘𝑈)    &   𝐽 = (invr𝐹)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &    0 = (0g𝑈)    &   𝐺 = (𝑇 (𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝐴) → 𝐷 ∈ ran 𝐼)

Theoremdih1dimat 36936 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → 𝑃 ∈ ran 𝐼)

Theoremdihlsprn 36937 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)

TheoremdihlspsnssN 36938 A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇𝑆𝑇 ∈ ran 𝐼))

Theoremdihlspsnat 36939 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉𝑋0 ) → (𝐼‘(𝑁‘{𝑋})) ∈ 𝐴)

Theoremdihatlat 36940 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐿 = (LSAtoms‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → (𝐼𝑄) ∈ 𝐿)

Theoremdihat 36941 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐼𝑃) ∈ 𝐴)

TheoremdihpN 36942* The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆 (the zero translation ltrnid 35739 and a nonzero member of the endomorphism ring). In particular, 𝑆 can be replaced with the ring unit ( I ↾ 𝑇). (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑆𝐸𝑆𝑂))       (𝜑 → (𝐼𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))

Theoremdihlatat 36943 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐿 = (LSAtoms‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐿) → (𝐼𝑄) ∈ 𝐴)

Theoremdihatexv 36944* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐵)       (𝜑 → (𝑄𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼𝑄) = (𝑁‘{𝑥})))

Theoremdihatexv2 36945* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑄𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (𝐼‘(𝑁‘{𝑥}))))

Theoremdihglblem6 36946* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑃 = (LSubSp‘𝑈)    &   𝐷 = (LSAtoms‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅)) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))

Theoremdihglb 36947* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅)) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))

Theoremdihglb2 36948* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝑉) → (𝐼‘(𝐺‘{𝑥𝐵𝑆 ⊆ (𝐼𝑥)})) = {𝑦 ∈ ran 𝐼𝑆𝑦})

Theoremdihmeet 36949 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

Theoremdihintcl 36950 The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆 ⊆ ran 𝐼𝑆 ≠ ∅)) → 𝑆 ∈ ran 𝐼)

Theoremdihmeetcl 36951 Closure of closed subspace meet for DVecH vector space. (Contributed by NM, 5-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ ran 𝐼𝑌 ∈ ran 𝐼)) → (𝑋𝑌) ∈ ran 𝐼)

Theoremdihmeet2 36952 Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
= (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝐼‘(𝑋𝑌)) = ((𝐼𝑋) (𝐼𝑌)))

Syntaxcoch 36953 Extend class notation with subspace orthocomplement for DVecH vector space.
class ocH

Definitiondf-doch 36954* Define subspace orthocomplement for DVecH vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 14-Mar-2014.)
ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))

Theoremdochffval 36955* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))

Theoremdochfval 36956* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))

Theoremdochval 36957* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))

Theoremdochval2 36958* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Apr-2014.)
= (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))))

Theoremdochcl 36959 Closure of subspace orthocomplement for DVecH vector space. (Contributed by NM, 9-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → ( 𝑋) ∈ ran 𝐼)

Theoremdochlss 36960 A subspace orthocomplement is a subspace of the DVecH vector space. (Contributed by NM, 22-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (LSubSp‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → ( 𝑋) ∈ 𝑆)

Theoremdochssv 36961 A subspace orthocomplement belongs to the DVecH vector space. (Contributed by NM, 22-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → ( 𝑋) ⊆ 𝑉)

TheoremdochfN 36962 Domain and codomain of the subspace orthocomplement for the DVecH vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 :𝒫 𝑉⟶ran 𝐼)

Theoremdochvalr 36963 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
= (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁𝑋) = (𝐼‘( ‘(𝐼𝑋))))

Theoremdoch0 36964 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ‘{ 0 }) = 𝑉)

Theoremdoch1 36965 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( 𝑉) = { 0 })

Theoremdochoc0 36966 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ( ‘( ‘{ 0 })) = { 0 })

Theoremdochoc1 36967 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ( ‘( 𝑉)) = 𝑉)

Theoremdochvalr2 36968 Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝑁‘(𝐼𝑋)) = (𝐼‘( 𝑋)))

Theoremdochvalr3 36969 Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
= (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)       (𝜑 → ( ‘(𝐼𝑋)) = (𝐼‘(𝑁𝑋)))

Theoremdoch2val2 36970* Double orthocomplement for DVecH vector space. (Contributed by NM, 26-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘( 𝑋)) = {𝑧 ∈ ran 𝐼𝑋𝑧})

Theoremdochss 36971 Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))

Theoremdochocss 36972 Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → 𝑋 ⊆ ( ‘( 𝑋)))

Theoremdochoc 36973 Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ‘( 𝑋)) = 𝑋)

Theoremdochsscl 36974 If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝑌 ↔ ( ‘( 𝑋)) ⊆ 𝑌))

Theoremdochoccl 36975 A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋 ∈ ran 𝐼 ↔ ( ‘( 𝑋)) = 𝑋))

Theoremdochord 36976 Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝑌 ↔ ( 𝑌) ⊆ ( 𝑋)))

Theoremdochord2N 36977 Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (( 𝑋) ⊆ 𝑌 ↔ ( 𝑌) ⊆ 𝑋))

Theoremdochord3 36978 Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋 ⊆ ( 𝑌) ↔ 𝑌 ⊆ ( 𝑋)))

Theoremdoch11 36979 Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (( 𝑋) = ( 𝑌) ↔ 𝑋 = 𝑌))

TheoremdochsordN 36980 Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝑌 ↔ ( 𝑌) ⊊ ( 𝑋)))

Theoremdochn0nv 36981 An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (( 𝑋) ≠ { 0 } ↔ ( ‘( 𝑋)) ≠ 𝑉))

Theoremdihoml4c 36982 Version of dihoml4 36983 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

Theoremdihoml4 36983 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 35557 analog.) (Contributed by NM, 15-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑 → ( ‘( 𝑌)) = 𝑌)    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋)))

Theoremdochspss 36984 The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁𝑋) ⊆ ( ‘( 𝑋)))

Theoremdochocsp 36985 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘(𝑁𝑋)) = ( 𝑋))

TheoremdochspocN 36986 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁‘( 𝑋)) = ( ‘(𝑁𝑋)))

Theoremdochocsn 36987 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘( ‘{𝑋})) = (𝑁‘{𝑋}))

Theoremdochsncom 36988 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 ∈ ( ‘{𝑌}) ↔ 𝑌 ∈ ( ‘{𝑋})))

Theoremdochsat 36989 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝑆)       (𝜑 → (( ‘( 𝑄)) ∈ 𝐴𝑄𝐴))

Theoremdochshpncl 36990 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘( 𝑋)) ≠ 𝑋 ↔ ( ‘( 𝑋)) = 𝑉))

Theoremdochlkr 36991 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ∈ 𝑌 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ (𝐿𝐺) ∈ 𝑌)))

Theoremdochkrshp 36992 The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ ( ‘( ‘(𝐿𝐺))) ∈ 𝑌))

Theoremdochkrshp2 36993 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ (𝐿𝐺) ∈ 𝑌)))

Theoremdochkrshp3 36994 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ (𝐿𝐺) ≠ 𝑉)))

Theoremdochkrshp4 36995 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ∨ (𝐿𝐺) = 𝑉)))

Theoremdochdmj1 36996 De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉𝑌𝑉) → ( ‘(𝑋𝑌)) = (( 𝑋) ∩ ( 𝑌)))

Theoremdochnoncon 36997 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    0 = (0g𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑆) → (𝑋 ∩ ( 𝑋)) = { 0 })

Theoremdochnel2 36998 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    0 = (0g𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)    &   (𝜑𝑋 ∈ (𝑇 ∖ { 0 }))       (𝜑 → ¬ 𝑋 ∈ ( 𝑇))

Theoremdochnel 36999 A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ¬ 𝑋 ∈ ( ‘{𝑋}))

Syntaxcdjh 37000 Extend class notation with subspace join for DVecH vector space.
class joinH

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >