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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 6659 | . . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈)) | |
3 | 2 | anidms 677 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈)) |
4 | hmoval.9 | . . . . . 6 ⊢ 𝐴 = (𝑈adj𝑈) | |
5 | 3, 4 | syl6eqr 2674 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴) |
6 | 5 | dmeqd 5326 | . . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴) |
7 | 5 | fveq1d 6193 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴‘𝑡)) |
8 | 7 | eqeq1d 2624 | . . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡)) |
9 | 6, 8 | rabeqbidv 3195 | . . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
10 | df-hmo 27606 | . . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}) | |
11 | ovex 6678 | . . . . . 6 ⊢ (𝑈adj𝑈) ∈ V | |
12 | 4, 11 | eqeltri 2697 | . . . . 5 ⊢ 𝐴 ∈ V |
13 | 12 | dmex 7099 | . . . 4 ⊢ dom 𝐴 ∈ V |
14 | 13 | rabex 4813 | . . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V |
15 | 9, 10, 14 | fvmpt 6282 | . 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
16 | 1, 15 | syl5eq 2668 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 NrmCVeccnv 27439 adjcaj 27603 HmOpchmo 27604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-hmo 27606 |
This theorem is referenced by: ishmo 27666 |
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