Theorem eigvalval 29391

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) = (((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 29328	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))))
2	1	fveq1d 6448	. 2 ⊢ (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))‘𝐴))
3		fveq2 6446	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
4		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	oveq12d 6940	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih 𝑥) = ((𝑇‘𝐴) ·ih 𝐴))
6		fveq2 6446	. . . . 5 ⊢ (𝑥 = 𝐴 → (normℎ‘𝑥) = (normℎ‘𝐴))
7	6	oveq1d 6937	. . . 4 ⊢ (𝑥 = 𝐴 → ((normℎ‘𝑥)↑2) = ((normℎ‘𝐴)↑2))
8	5, 7	oveq12d 6940	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)))
9		eqid 2778	. . 3 ⊢ (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))
10		ovex 6954	. . 3 ⊢ (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ∈ V
11	8, 9, 10	fvmpt 6542	. 2 ⊢ (𝐴 ∈ (eigvec‘𝑇) → ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)))
12	2, 11	sylan9eq 2834	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ↦ cmpt 4965  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   / cdiv 11032  2c2 11430  ↑cexp 13178   ℋchba 28348   ·_ih csp 28351  norm_ℎcno 28352  eigveccei 28388  eigvalcel 28389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-hilex 28428

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-eigval 29285

This theorem is referenced by:  eigvalcl  29392  eigvec1  29393