Theorem hmoval 27665

Description: The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
hmoval	|- ( U e. NrmCVec -> H = { t e. dom A | ( A ` t ) = t } )

Distinct variable groups:   t, A   t, U

Allowed substitution hint:   H( t)

Proof of Theorem hmoval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmoval.8	. 2 |- H = ( HmOp ` U )
2		oveq12 6659	. . . . . . 7 |- ( ( u = U /\ u = U ) -> ( u adj u ) = ( U adj U ) )
3	2	anidms 677	. . . . . 6 |- ( u = U -> ( u adj u ) = ( U adj U ) )
4		hmoval.9	. . . . . 6 |- A = ( U adj U )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( u = U -> ( u adj u ) = A )
6	5	dmeqd 5326	. . . 4 |- ( u = U -> dom ( u adj u ) = dom A )
7	5	fveq1d 6193	. . . . 5 |- ( u = U -> ( ( u adj u ) ` t ) = ( A ` t ) )
8	7	eqeq1d 2624	. . . 4 |- ( u = U -> ( ( ( u adj u ) ` t ) = t <-> ( A ` t ) = t ) )
9	6, 8	rabeqbidv 3195	. . 3 |- ( u = U -> { t e. dom ( u adj u ) | ( ( u adj u ) ` t ) = t } = { t e. dom A | ( A ` t ) = t } )
10		df-hmo 27606	. . 3 |- HmOp = ( u e. NrmCVec |-> { t e. dom ( u adj u ) | ( ( u adj u ) ` t ) = t } )
11		ovex 6678	. . . . . 6 |- ( U adj U ) e. _V
12	4, 11	eqeltri 2697	. . . . 5 |- A e. _V
13	12	dmex 7099	. . . 4 |- dom A e. _V
14	13	rabex 4813	. . 3 |- { t e. dom A | ( A ` t ) = t } e. _V
15	9, 10, 14	fvmpt 6282	. 2 |- ( U e. NrmCVec -> ( HmOp ` U ) = { t e. dom A | ( A ` t ) = t } )
16	1, 15	syl5eq 2668	1 |- ( U e. NrmCVec -> H = { t e. dom A | ( A ` t ) = t } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   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:  ishmo  27666