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Theorem ishil 20426
 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 6434 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2syl6eqr 2880 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5559 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6434 . . . 4 ( = 𝐻 → (ClSubSp‘) = (ClSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (ClSubSp‘𝐻)
75, 6syl6eqr 2880 . . 3 ( = 𝐻 → (ClSubSp‘) = 𝐶)
84, 7eqeq12d 2841 . 2 ( = 𝐻 → (dom (proj‘) = (ClSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 20412 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}
108, 9elrab2 3590 1 (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  dom cdm 5343  ‘cfv 6124  PreHilcphl 20332  ClSubSpccss 20369  projcpj 20408  Hilchil 20409 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-dm 5353  df-iota 6087  df-fv 6132  df-hil 20412 This theorem is referenced by:  ishil2  20427  hlhil  23612
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