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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 6891 | . . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻)) | |
2 | ishil.k | . . . . 5 ⊢ 𝐾 = (proj‘𝐻) | |
3 | 1, 2 | eqtr4di 2786 | . . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾) |
4 | 3 | dmeqd 5902 | . . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾) |
5 | fveq2 6891 | . . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻)) | |
6 | ishil.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻) | |
7 | 5, 6 | eqtr4di 2786 | . . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶) |
8 | 4, 7 | eqeq12d 2744 | . 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶)) |
9 | df-hil 21631 | . 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)} | |
10 | 8, 9 | elrab2 3684 | 1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 dom cdm 5672 ‘cfv 6542 PreHilcphl 21549 ClSubSpccss 21586 projcpj 21627 Hilchil 21628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-dm 5682 df-iota 6494 df-fv 6550 df-hil 21631 |
This theorem is referenced by: ishil2 21646 hlhil 25364 |
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