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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 7159 | . . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈)) | |
3 | 2 | anidms 569 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈)) |
4 | hmoval.9 | . . . . . 6 ⊢ 𝐴 = (𝑈adj𝑈) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴) |
6 | 5 | dmeqd 5768 | . . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴) |
7 | 5 | fveq1d 6666 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴‘𝑡)) |
8 | 7 | eqeq1d 2823 | . . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡)) |
9 | 6, 8 | rabeqbidv 3485 | . . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
10 | df-hmo 28522 | . . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}) | |
11 | ovex 7183 | . . . . . 6 ⊢ (𝑈adj𝑈) ∈ V | |
12 | 4, 11 | eqeltri 2909 | . . . . 5 ⊢ 𝐴 ∈ V |
13 | 12 | dmex 7610 | . . . 4 ⊢ dom 𝐴 ∈ V |
14 | 13 | rabex 5227 | . . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V |
15 | 9, 10, 14 | fvmpt 6762 | . 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
16 | 1, 15 | syl5eq 2868 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 NrmCVeccnv 28355 adjcaj 28519 HmOpchmo 28520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-hmo 28522 |
This theorem is referenced by: ishmo 28582 |
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