Theorem ishil 20062

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 e. Hil <-> ( H e. PreHil /\ dom K = C ) )

Proof of Theorem ishil

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( h = H -> ( proj ` h ) = ( proj ` H ) )
2		ishil.k	. . . . 5 |- K = ( proj ` H )
3	1, 2	syl6eqr 2674	. . . 4 |- ( h = H -> ( proj ` h ) = K )
4	3	dmeqd 5326	. . 3 |- ( h = H -> dom ( proj ` h ) = dom K )
5		fveq2 6191	. . . 4 |- ( h = H -> ( CSubSp ` h ) = ( CSubSp ` H ) )
6		ishil.c	. . . 4 |- C = ( CSubSp ` H )
7	5, 6	syl6eqr 2674	. . 3 |- ( h = H -> ( CSubSp ` h ) = C )
8	4, 7	eqeq12d 2637	. 2 |- ( h = H -> ( dom ( proj ` h ) = ( CSubSp ` h ) <-> dom K = C ) )
9		df-hil 20048	. 2 |- Hil = { h e. PreHil | dom ( proj ` h ) = ( CSubSp ` h ) }
10	8, 9	elrab2 3366	1 |- ( H e. Hil <-> ( H e. PreHil /\ dom K = C ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   dom cdm 5114   ` cfv 5888   PreHilcphl 19969   CSubSpccss 20005   projcpj 20044   Hilchs 20045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-hil 20048

This theorem is referenced by:  ishil2  20063  hlhil  23214