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Mirrors > Home > MPE Home > Th. List > hmoval | Structured version Visualization version GIF version |
Description: The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmoval.8 | ⊢ 𝐻 = (HmOp‘𝑈) |
hmoval.9 | ⊢ 𝐴 = (𝑈adj𝑈) |
Ref | Expression |
---|---|
hmoval | ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmoval.8 | . 2 ⊢ 𝐻 = (HmOp‘𝑈) | |
2 | oveq12 7154 | . . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈)) | |
3 | 2 | anidms 567 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈)) |
4 | hmoval.9 | . . . . . 6 ⊢ 𝐴 = (𝑈adj𝑈) | |
5 | 3, 4 | syl6eqr 2871 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴) |
6 | 5 | dmeqd 5767 | . . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴) |
7 | 5 | fveq1d 6665 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴‘𝑡)) |
8 | 7 | eqeq1d 2820 | . . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡)) |
9 | 6, 8 | rabeqbidv 3483 | . . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
10 | df-hmo 28455 | . . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}) | |
11 | ovex 7178 | . . . . . 6 ⊢ (𝑈adj𝑈) ∈ V | |
12 | 4, 11 | eqeltri 2906 | . . . . 5 ⊢ 𝐴 ∈ V |
13 | 12 | dmex 7605 | . . . 4 ⊢ dom 𝐴 ∈ V |
14 | 13 | rabex 5226 | . . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V |
15 | 9, 10, 14 | fvmpt 6761 | . 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
16 | 1, 15 | syl5eq 2865 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 NrmCVeccnv 28288 adjcaj 28452 HmOpchmo 28453 |
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-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
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-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 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-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-hmo 28455 |
This theorem is referenced by: ishmo 28515 |
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