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Theorem ishil 21645
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.)
Hypotheses
Ref Expression
ishil.k 𝐾 = (proj‘𝐻)
ishil.c 𝐶 = (ClSubSp‘𝐻)
Assertion
Ref Expression
ishil (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Proof of Theorem ishil
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2eqtr4di 2786 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5902 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6891 . . . 4 ( = 𝐻 → (ClSubSp‘) = (ClSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (ClSubSp‘𝐻)
75, 6eqtr4di 2786 . . 3 ( = 𝐻 → (ClSubSp‘) = 𝐶)
84, 7eqeq12d 2744 . 2 ( = 𝐻 → (dom (proj‘) = (ClSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 21631 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}
108, 9elrab2 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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