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Theorem ishil 19267
 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 5877 . . . . 5
2 ishil.k . . . . 5
31, 2syl6eqr 2481 . . . 4
43dmeqd 5052 . . 3
5 fveq2 5877 . . . 4
6 ishil.c . . . 4
75, 6syl6eqr 2481 . . 3
84, 7eqeq12d 2444 . 2
9 df-hil 19253 . 2
108, 9elrab2 3231 1
 Colors of variables: wff setvar class Syntax hints:   wb 187   wa 370   wceq 1437   wcel 1868   cdm 4849  cfv 5597  cphl 19177  ccss 19210  cpj 19249  chs 19250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-dm 4859  df-iota 5561  df-fv 5605  df-hil 19253 This theorem is referenced by:  ishil2  19268  hlhil  22383
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