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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhlmod | Structured version Visualization version GIF version | ||
| Description: The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a left module. (Contributed by NM, 23-May-2015.) |
| Ref | Expression |
|---|---|
| dvhlvec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| dvhlvec.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| dvhlvec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| Ref | Expression |
|---|---|
| dvhlmod | ⊢ (𝜑 → 𝑈 ∈ LMod) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvhlvec.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | dvhlvec.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 3 | dvhlvec.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 4 | 1, 2, 3 | dvhlvec 41091 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
| 5 | lveclmod 21122 | . 2 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
| 6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → 𝑈 ∈ LMod) |
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