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Theorem ishil 20838
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 6646 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2syl6eqr 2873 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5750 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6646 . . . 4 ( = 𝐻 → (ClSubSp‘) = (ClSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (ClSubSp‘𝐻)
75, 6syl6eqr 2873 . . 3 ( = 𝐻 → (ClSubSp‘) = 𝐶)
84, 7eqeq12d 2836 . 2 ( = 𝐻 → (dom (proj‘) = (ClSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 20824 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}
108, 9elrab2 3663 1 (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  wcel 2114  dom cdm 5531  cfv 6331  PreHilcphl 20744  ClSubSpccss 20781  projcpj 20820  Hilchil 20821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-dm 5541  df-iota 6290  df-fv 6339  df-hil 20824
This theorem is referenced by:  ishil2  20839  hlhil  24026
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