Theorem eigvalval 28819

Description: The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
eigvalval	|- ( ( T : ~H --> ~H /\ A e. ( eigvec ` T ) ) -> ( ( eigval ` T ) ` A ) = ( ( ( T ` A ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) )

Proof of Theorem eigvalval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eigvalfval 28756	. . 3 |- ( T : ~H --> ~H -> ( eigval ` T ) = ( x e. ( eigvec ` T ) |-> ( ( ( T ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) ) )
2	1	fveq1d 6193	. 2 |- ( T : ~H --> ~H -> ( ( eigval ` T ) ` A ) = ( ( x e. ( eigvec ` T ) |-> ( ( ( T ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) ) ` A ) )
3		fveq2 6191	. . . . 5 |- ( x = A -> ( T ` x ) = ( T ` A ) )
4		id 22	. . . . 5 |- ( x = A -> x = A )
5	3, 4	oveq12d 6668	. . . 4 |- ( x = A -> ( ( T ` x ) .ih x ) = ( ( T ` A ) .ih A ) )
6		fveq2 6191	. . . . 5 |- ( x = A -> ( normh ` x ) = ( normh ` A ) )
7	6	oveq1d 6665	. . . 4 |- ( x = A -> ( ( normh ` x ) ^ 2 ) = ( ( normh ` A ) ^ 2 ) )
8	5, 7	oveq12d 6668	. . 3 |- ( x = A -> ( ( ( T ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) = ( ( ( T ` A ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) )
9		eqid 2622	. . 3 |- ( x e. ( eigvec ` T ) |-> ( ( ( T ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) ) = ( x e. ( eigvec ` T ) |-> ( ( ( T ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) )
10		ovex 6678	. . 3 |- ( ( ( T ` A ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) e. _V
11	8, 9, 10	fvmpt 6282	. 2 |- ( A e. ( eigvec ` T ) -> ( ( x e. ( eigvec ` T ) |-> ( ( ( T ` x ) .ih x ) / ( ( normh ` x ) ^ 2 ) ) ) ` A ) = ( ( ( T ` A ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) )
12	2, 11	sylan9eq 2676	1 |- ( ( T : ~H --> ~H /\ A e. ( eigvec ` T ) ) -> ( ( eigval ` T ) ` A ) = ( ( ( T ` A ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   / cdiv 10684   2c2 11070   ^cexp 12860   ~Hchil 27776   .ih csp 27779   normhcno 27780   eigveccei 27816   eigvalcel 27817

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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-eigval 28713

This theorem is referenced by:  eigvalcl  28820  eigvec1  28821