![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > elhmop | Structured version Visualization version GIF version |
Description: Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elhmop | ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6901 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦)) | |
2 | 1 | oveq2d 7442 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑥 ·ih (𝑡‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦))) |
3 | fveq1 6901 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥)) | |
4 | 3 | oveq1d 7441 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦)) |
5 | 2, 4 | eqeq12d 2744 | . . . 4 ⊢ (𝑡 = 𝑇 → ((𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
6 | 5 | 2ralbidv 3216 | . . 3 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
7 | df-hmop 31674 | . . 3 ⊢ HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)} | |
8 | 6, 7 | elrab2 3687 | . 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
9 | ax-hilex 30829 | . . . 4 ⊢ ℋ ∈ V | |
10 | 9, 9 | elmap 8896 | . . 3 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ) |
11 | 10 | anbi1i 622 | . 2 ⊢ ((𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
12 | 8, 11 | bitri 274 | 1 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8851 ℋchba 30749 ·ih csp 30752 HrmOpcho 30780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-hilex 30829 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-hmop 31674 |
This theorem is referenced by: hmopf 31704 hmop 31752 hmopadj2 31771 idhmop 31812 0hmop 31813 lnophmi 31848 hmops 31850 hmopm 31851 hmopco 31853 pjhmopi 31976 |
Copyright terms: Public domain | W3C validator |