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Mirrors > Home > HSE Home > Th. List > eigvalfval | Structured version Visualization version GIF version |
Description: The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eigvalfval | ⊢ (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6676 | . . 3 ⊢ (eigvec‘𝑇) ∈ V | |
2 | 1 | mptex 6977 | . 2 ⊢ (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) ∈ V |
3 | ax-hilex 28703 | . 2 ⊢ ℋ ∈ V | |
4 | fveq2 6663 | . . 3 ⊢ (𝑡 = 𝑇 → (eigvec‘𝑡) = (eigvec‘𝑇)) | |
5 | fveq1 6662 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥)) | |
6 | 5 | oveq1d 7160 | . . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·ih 𝑥) = ((𝑇‘𝑥) ·ih 𝑥)) |
7 | 6 | oveq1d 7160 | . . 3 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)) = (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) |
8 | 4, 7 | mpteq12dv 5142 | . 2 ⊢ (𝑡 = 𝑇 → (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) |
9 | df-eigval 29558 | . 2 ⊢ eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | |
10 | 2, 3, 3, 8, 9 | fvmptmap 8434 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 / cdiv 11285 2c2 11680 ↑cexp 13417 ℋchba 28623 ·ih csp 28626 normℎcno 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: eigvalval 29664 |
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