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Theorem ishil 21669
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 6896 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2eqtr4di 2783 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5908 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6896 . . . 4 ( = 𝐻 → (ClSubSp‘) = (ClSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (ClSubSp‘𝐻)
75, 6eqtr4di 2783 . . 3 ( = 𝐻 → (ClSubSp‘) = 𝐶)
84, 7eqeq12d 2741 . 2 ( = 𝐻 → (dom (proj‘) = (ClSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 21655 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}
108, 9elrab2 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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