Theorem eleigvec 28816

Description: Membership in 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
eleigvec	|- ( T : ~H --> ~H -> ( A e. ( eigvec ` T ) <-> ( A e. ~H /\ A =/= 0h /\ E. x e. CC ( T ` A ) = ( x .h A ) ) ) )

Distinct variable groups:   x, A   x, T

Proof of Theorem eleigvec

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eigvecval 28755	. . 3 |- ( T : ~H --> ~H -> ( eigvec ` T ) = { y e. ( ~H \ 0H ) | E. x e. CC ( T ` y ) = ( x .h y ) } )
2	1	eleq2d 2687	. 2 |- ( T : ~H --> ~H -> ( A e. ( eigvec ` T ) <-> A e. { y e. ( ~H \ 0H ) | E. x e. CC ( T ` y ) = ( x .h y ) } ) )
3		eldif 3584	. . . . 5 |- ( A e. ( ~H \ 0H ) <-> ( A e. ~H /\ -. A e. 0H ) )
4		elch0 28111	. . . . . . 7 |- ( A e. 0H <-> A = 0h )
5	4	necon3bbii 2841	. . . . . 6 |- ( -. A e. 0H <-> A =/= 0h )
6	5	anbi2i 730	. . . . 5 |- ( ( A e. ~H /\ -. A e. 0H ) <-> ( A e. ~H /\ A =/= 0h ) )
7	3, 6	bitri 264	. . . 4 |- ( A e. ( ~H \ 0H ) <-> ( A e. ~H /\ A =/= 0h ) )
8	7	anbi1i 731	. . 3 |- ( ( A e. ( ~H \ 0H ) /\ E. x e. CC ( T ` A ) = ( x .h A ) ) <-> ( ( A e. ~H /\ A =/= 0h ) /\ E. x e. CC ( T ` A ) = ( x .h A ) ) )
9		fveq2 6191	. . . . . 6 |- ( y = A -> ( T ` y ) = ( T ` A ) )
10		oveq2 6658	. . . . . 6 |- ( y = A -> ( x .h y ) = ( x .h A ) )
11	9, 10	eqeq12d 2637	. . . . 5 |- ( y = A -> ( ( T ` y ) = ( x .h y ) <-> ( T ` A ) = ( x .h A ) ) )
12	11	rexbidv 3052	. . . 4 |- ( y = A -> ( E. x e. CC ( T ` y ) = ( x .h y ) <-> E. x e. CC ( T ` A ) = ( x .h A ) ) )
13	12	elrab 3363	. . 3 |- ( A e. { y e. ( ~H \ 0H ) | E. x e. CC ( T ` y ) = ( x .h y ) } <-> ( A e. ( ~H \ 0H ) /\ E. x e. CC ( T ` A ) = ( x .h A ) ) )
14		df-3an 1039	. . 3 |- ( ( A e. ~H /\ A =/= 0h /\ E. x e. CC ( T ` A ) = ( x .h A ) ) <-> ( ( A e. ~H /\ A =/= 0h ) /\ E. x e. CC ( T ` A ) = ( x .h A ) ) )
15	8, 13, 14	3bitr4i 292	. 2 |- ( A e. { y e. ( ~H \ 0H ) | E. x e. CC ( T ` y ) = ( x .h y ) } <-> ( A e. ~H /\ A =/= 0h /\ E. x e. CC ( T ` A ) = ( x .h A ) ) )
16	2, 15	syl6bb 276	1 |- ( T : ~H --> ~H -> ( A e. ( eigvec ` T ) <-> ( A e. ~H /\ A =/= 0h /\ E. x e. CC ( T ` A ) = ( x .h A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   ~Hchil 27776   .h csm 27778   0hc0v 27781   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  ax-hv0cl 27860

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-ne 2795  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-ch0 28110  df-eigvec 28712

This theorem is referenced by:  eleigvec2  28817  eigvalcl  28820