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Theorem ishil 19881
 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 = (CSubSp‘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 6103 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2syl6eqr 2662 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5248 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6103 . . . 4 ( = 𝐻 → (CSubSp‘) = (CSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (CSubSp‘𝐻)
75, 6syl6eqr 2662 . . 3 ( = 𝐻 → (CSubSp‘) = 𝐶)
84, 7eqeq12d 2625 . 2 ( = 𝐻 → (dom (proj‘) = (CSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 19867 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (CSubSp‘)}
108, 9elrab2 3333 1 (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  dom cdm 5038  ‘cfv 5804  PreHilcphl 19788  CSubSpccss 19824  projcpj 19863  Hilchs 19864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-hil 19867 This theorem is referenced by:  ishil2  19882  hlhil  23022
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