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Mirrors > Home > HSE Home > Th. List > eigvalval | Structured version Visualization version GIF version |
Description: The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eigvalval | ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eigvalfval 29680 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | |
2 | 1 | fveq1d 6647 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))‘𝐴)) |
3 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
4 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | 3, 4 | oveq12d 7153 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih 𝑥) = ((𝑇‘𝐴) ·ih 𝐴)) |
6 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 𝐴 → (normℎ‘𝑥) = (normℎ‘𝐴)) | |
7 | 6 | oveq1d 7150 | . . . 4 ⊢ (𝑥 = 𝐴 → ((normℎ‘𝑥)↑2) = ((normℎ‘𝐴)↑2)) |
8 | 5, 7 | oveq12d 7153 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) |
9 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) | |
10 | ovex 7168 | . . 3 ⊢ (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ∈ V | |
11 | 8, 9, 10 | fvmpt 6745 | . 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 2853 | 1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 / cdiv 11286 2c2 11680 ↑cexp 13425 ℋchba 28702 ·ih csp 28705 normℎcno 28706 eigveccei 28742 eigvalcel 28743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-eigval 29637 |
This theorem is referenced by: eigvalcl 29744 eigvec1 29745 |
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