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Theorem ishil 20790
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 6663 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2syl6eqr 2871 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5767 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6663 . . . 4 ( = 𝐻 → (ClSubSp‘) = (ClSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (ClSubSp‘𝐻)
75, 6syl6eqr 2871 . . 3 ( = 𝐻 → (ClSubSp‘) = 𝐶)
84, 7eqeq12d 2834 . 2 ( = 𝐻 → (dom (proj‘) = (ClSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 20776 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}
108, 9elrab2 3680 1 (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  dom cdm 5548  cfv 6348  PreHilcphl 20696  ClSubSpccss 20733  projcpj 20772  Hilchil 20773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-dm 5558  df-iota 6307  df-fv 6356  df-hil 20776
This theorem is referenced by:  ishil2  20791  hlhil  23973
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