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Theorem ishil 16937
 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
ishil.c
Assertion
Ref Expression
ishil

Proof of Theorem ishil
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . 5
2 ishil.k . . . . 5
31, 2syl6eqr 2485 . . . 4
43dmeqd 5064 . . 3
5 fveq2 5720 . . . 4
6 ishil.c . . . 4
75, 6syl6eqr 2485 . . 3
84, 7eqeq12d 2449 . 2
9 df-hil 16923 . 2
108, 9elrab2 3086 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725   cdm 4870  cfv 5446  cphl 16847  ccss 16880  cpj 16919  chs 16920 This theorem is referenced by:  ishil2  16938  hlhil  19336 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-dm 4880  df-iota 5410  df-fv 5454  df-hil 16923
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