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Theorem ishil 19981
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 6148 . . . . 5 ( = 𝐻 → (proj‘) = (proj‘𝐻))
2 ishil.k . . . . 5 𝐾 = (proj‘𝐻)
31, 2syl6eqr 2673 . . . 4 ( = 𝐻 → (proj‘) = 𝐾)
43dmeqd 5286 . . 3 ( = 𝐻 → dom (proj‘) = dom 𝐾)
5 fveq2 6148 . . . 4 ( = 𝐻 → (CSubSp‘) = (CSubSp‘𝐻))
6 ishil.c . . . 4 𝐶 = (CSubSp‘𝐻)
75, 6syl6eqr 2673 . . 3 ( = 𝐻 → (CSubSp‘) = 𝐶)
84, 7eqeq12d 2636 . 2 ( = 𝐻 → (dom (proj‘) = (CSubSp‘) ↔ dom 𝐾 = 𝐶))
9 df-hil 19967 . 2 Hil = { ∈ PreHil ∣ dom (proj‘) = (CSubSp‘)}
108, 9elrab2 3348 1 (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  dom cdm 5074  cfv 5847  PreHilcphl 19888  CSubSpccss 19924  projcpj 19963  Hilchs 19964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855  df-hil 19967
This theorem is referenced by:  ishil2  19982  hlhil  23122
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