Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 28514 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) |
4 | 3 | eleq2d 2895 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ 𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})) |
5 | fveq2 6663 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝐴‘𝑡) = (𝐴‘𝑇)) | |
6 | id 22 | . . . 4 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇) | |
7 | 5, 6 | eqeq12d 2834 | . . 3 ⊢ (𝑡 = 𝑇 → ((𝐴‘𝑡) = 𝑡 ↔ (𝐴‘𝑇) = 𝑇)) |
8 | 7 | elrab 3677 | . 2 ⊢ (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)) |
9 | 4, 8 | syl6bb 288 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 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: (None) |
Copyright terms: Public domain | W3C validator |