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Theorem eigvalval 29664
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 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))

Proof of Theorem eigvalval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eigvalfval 29601 . . 3 (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
21fveq1d 6665 . 2 (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴))
3 fveq2 6663 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4oveq12d 7163 . . . 4 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑥) = ((𝑇𝐴) ·ih 𝐴))
6 fveq2 6663 . . . . 5 (𝑥 = 𝐴 → (norm𝑥) = (norm𝐴))
76oveq1d 7160 . . . 4 (𝑥 = 𝐴 → ((norm𝑥)↑2) = ((norm𝐴)↑2))
85, 7oveq12d 7163 . . 3 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
9 eqid 2818 . . 3 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
10 ovex 7178 . . 3 (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)) ∈ V
118, 9, 10fvmpt 6761 . 2 (𝐴 ∈ (eigvec‘𝑇) → ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
122, 11sylan9eq 2873 1 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145   / cdiv 11285  2c2 11680  cexp 13417  chba 28623   ·ih csp 28626  normcno 28627  eigveccei 28663  eigvalcel 28664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-eigval 29558
This theorem is referenced by:  eigvalcl  29665  eigvec1  29666
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