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Mirrors > Home > MPE Home > Th. List > ishmo | Structured version Visualization version GIF version |
Description: The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmoval.8 | ⊢ 𝐻 = (HmOp‘𝑈) |
hmoval.9 | ⊢ 𝐴 = (𝑈adj𝑈) |
Ref | Expression |
---|---|
ishmo | ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmoval.8 | . . . 4 ⊢ 𝐻 = (HmOp‘𝑈) | |
2 | hmoval.9 | . . . 4 ⊢ 𝐴 = (𝑈adj𝑈) | |
3 | 1, 2 | hmoval 30619 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
4 | 3 | eleq2d 2815 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ 𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})) |
5 | fveq2 6897 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝐴‘𝑡) = (𝐴‘𝑇)) | |
6 | id 22 | . . . 4 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇) | |
7 | 5, 6 | eqeq12d 2744 | . . 3 ⊢ (𝑡 = 𝑇 → ((𝐴‘𝑡) = 𝑡 ↔ (𝐴‘𝑇) = 𝑇)) |
8 | 7 | elrab 3682 | . 2 ⊢ (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)) |
9 | 4, 8 | bitrdi 287 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 dom cdm 5678 ‘cfv 6548 (class class class)co 7420 NrmCVeccnv 30393 adjcaj 30557 HmOpchmo 30558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-hmo 30560 |
This theorem is referenced by: (None) |
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