Theorem eigvalval 30322

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 30259	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))))
2	1	fveq1d 6776	. 2 ⊢ (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))‘𝐴))
3		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
4		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5	3, 4	oveq12d 7293	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih 𝑥) = ((𝑇‘𝐴) ·ih 𝐴))
6		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (normℎ‘𝑥) = (normℎ‘𝐴))
7	6	oveq1d 7290	. . . 4 ⊢ (𝑥 = 𝐴 → ((normℎ‘𝑥)↑2) = ((normℎ‘𝐴)↑2))
8	5, 7	oveq12d 7293	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)))
9		eqid 2738	. . 3 ⊢ (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))
10		ovex 7308	. . 3 ⊢ (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ∈ V
11	8, 9, 10	fvmpt 6875	. 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 2798	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   / cdiv 11632  2c2 12028  ↑cexp 13782   ℋchba 29281   ·_ih csp 29284  norm_ℎcno 29285  eigveccei 29321  eigvalcel 29322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-hilex 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-eigval 30216

This theorem is referenced by:  eigvalcl  30323  eigvec1  30324