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Theorem hmopf 31903
Description: A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopf (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)

Proof of Theorem hmopf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elhmop 31902 . 2 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simplbi 497 1 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  chba 30948   ·ih csp 30951  HrmOpcho 30979
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-hmop 31873
This theorem is referenced by:  hmopex  31904  hmopre  31952  hmopadj  31968  hmdmadj  31969  hmoplin  31971  eighmre  31992  eighmorth  31993  hmops  32049  hmopm  32050  hmopd  32051  hmopco  32052  leop2  32153  leoppos  32155  leoprf  32157  leopsq  32158  leopadd  32161  leopmuli  32162  leopmul  32163  leopmul2i  32164  leopnmid  32167  nmopleid  32168  opsqrlem1  32169  opsqrlem6  32174  elpjrn  32219
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