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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 6918 | . . 3 ⊢ (eigvec‘𝑇) ∈ V | |
| 2 | 1 | mptex 7244 | . 2 ⊢ (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) ∈ V | 
| 3 | ax-hilex 31019 | . 2 ⊢ ℋ ∈ V | |
| 4 | fveq2 6905 | . . 3 ⊢ (𝑡 = 𝑇 → (eigvec‘𝑡) = (eigvec‘𝑇)) | |
| 5 | fveq1 6904 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥)) | |
| 6 | 5 | oveq1d 7447 | . . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·ih 𝑥) = ((𝑇‘𝑥) ·ih 𝑥)) | 
| 7 | 6 | oveq1d 7447 | . . 3 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)) = (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))) | 
| 8 | 4, 7 | mpteq12dv 5232 | . 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 31874 | . 2 ⊢ eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | |
| 10 | 2, 3, 3, 8, 9 | fvmptmap 8922 | 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 1539 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 / cdiv 11921 2c2 12322 ↑cexp 14103 ℋchba 30939 ·ih csp 30942 normℎcno 30943 eigveccei 30979 eigvalcel 30980 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-hilex 31019 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-eigval 31874 | 
| This theorem is referenced by: eigvalval 31980 | 
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