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Mirrors > Home > HSE Home > Th. List > hmopf | Structured version Visualization version GIF version |
Description: A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmopf | ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elhmop 30235 | . 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℋchba 29281 ·ih csp 29284 HrmOpcho 29312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-hilex 29361 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-hmop 30206 |
This theorem is referenced by: hmopex 30237 hmopre 30285 hmopadj 30301 hmdmadj 30302 hmoplin 30304 eighmre 30325 eighmorth 30326 hmops 30382 hmopm 30383 hmopd 30384 hmopco 30385 leop2 30486 leoppos 30488 leoprf 30490 leopsq 30491 leopadd 30494 leopmuli 30495 leopmul 30496 leopmul2i 30497 leopnmid 30500 nmopleid 30501 opsqrlem1 30502 opsqrlem6 30507 elpjrn 30552 |
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