Theorem ishmo 27666

Description: The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hmoval.8	|- H = ( HmOp ` U )
hmoval.9	|- A = ( U adj U )

Assertion

Ref	Expression
ishmo	|- ( U e. NrmCVec -> ( T e. H <-> ( T e. dom A /\ ( A ` T ) = T ) ) )

Proof of Theorem ishmo

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmoval.8	. . . 4 |- H = ( HmOp ` U )
2		hmoval.9	. . . 4 |- A = ( U adj U )
3	1, 2	hmoval 27665	. . 3 |- ( U e. NrmCVec -> H = { t e. dom A | ( A ` t ) = t } )
4	3	eleq2d 2687	. 2 |- ( U e. NrmCVec -> ( T e. H <-> T e. { t e. dom A | ( A ` t ) = t } ) )
5		fveq2 6191	. . . 4 |- ( t = T -> ( A ` t ) = ( A ` T ) )
6		id 22	. . . 4 |- ( t = T -> t = T )
7	5, 6	eqeq12d 2637	. . 3 |- ( t = T -> ( ( A ` t ) = t <-> ( A ` T ) = T ) )
8	7	elrab 3363	. 2 |- ( T e. { t e. dom A | ( A ` t ) = t } <-> ( T e. dom A /\ ( A ` T ) = T ) )
9	4, 8	syl6bb 276	1 |- ( U e. NrmCVec -> ( T e. H <-> ( T e. dom A /\ ( A ` T ) = T ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   dom cdm 5114   ` cfv 5888  (class class class)co 6650   NrmCVeccnv 27439   adjcaj 27603   HmOpchmo 27604

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-hmo 27606

This theorem is referenced by: (None)