Theorem eigvecval 28755

Description: The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
eigvecval	|- ( T : ~H --> ~H -> ( eigvec ` T ) = { x e. ( ~H \ 0H ) | E. y e. CC ( T ` x ) = ( y .h x ) } )

Distinct variable group:   x, y, T

Proof of Theorem eigvecval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 27856	. . . 4 |- ~H e. _V
2		difexg 4808	. . . 4 |- ( ~H e. _V -> ( ~H \ 0H ) e. _V )
3	1, 2	ax-mp 5	. . 3 |- ( ~H \ 0H ) e. _V
4	3	rabex 4813	. 2 |- { x e. ( ~H \ 0H ) | E. y e. CC ( T ` x ) = ( y .h x ) } e. _V
5		fveq1 6190	. . . . 5 |- ( t = T -> ( t ` x ) = ( T ` x ) )
6	5	eqeq1d 2624	. . . 4 |- ( t = T -> ( ( t ` x ) = ( y .h x ) <-> ( T ` x ) = ( y .h x ) ) )
7	6	rexbidv 3052	. . 3 |- ( t = T -> ( E. y e. CC ( t ` x ) = ( y .h x ) <-> E. y e. CC ( T ` x ) = ( y .h x ) ) )
8	7	rabbidv 3189	. 2 |- ( t = T -> { x e. ( ~H \ 0H ) | E. y e. CC ( t ` x ) = ( y .h x ) } = { x e. ( ~H \ 0H ) | E. y e. CC ( T ` x ) = ( y .h x ) } )
9		df-eigvec 28712	. 2 |- eigvec = ( t e. ( ~H ^m ~H ) |-> { x e. ( ~H \ 0H ) | E. y e. CC ( t ` x ) = ( y .h x ) } )
10	4, 1, 1, 8, 9	fvmptmap 7894	1 |- ( T : ~H --> ~H -> ( eigvec ` T ) = { x e. ( ~H \ 0H ) | E. y e. CC ( T ` x ) = ( y .h x ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   ~Hchil 27776   .h csm 27778   0Hc0h 27792   eigveccei 27816

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-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

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-pw 4160  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-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-eigvec 28712

This theorem is referenced by:  eleigvec  28816