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Mirrors > Home > MPE Home > Th. List > ishil | Structured version Visualization version GIF version |
Description: The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
ishil.k | ⊢ 𝐾 = (proj‘𝐻) |
ishil.c | ⊢ 𝐶 = (ClSubSp‘𝐻) |
Ref | Expression |
---|---|
ishil | ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻)) | |
2 | ishil.k | . . . . 5 ⊢ 𝐾 = (proj‘𝐻) | |
3 | 1, 2 | eqtr4di 2783 | . . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾) |
4 | 3 | dmeqd 5908 | . . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾) |
5 | fveq2 6896 | . . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻)) | |
6 | ishil.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻) | |
7 | 5, 6 | eqtr4di 2783 | . . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶) |
8 | 4, 7 | eqeq12d 2741 | . 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶)) |
9 | df-hil 21655 | . 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)} | |
10 | 8, 9 | elrab2 3682 | 1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 dom cdm 5678 ‘cfv 6549 PreHilcphl 21573 ClSubSpccss 21610 projcpj 21651 Hilchil 21652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-dm 5688 df-iota 6501 df-fv 6557 df-hil 21655 |
This theorem is referenced by: ishil2 21670 hlhil 25415 |
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