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Theorem hmopf 29822
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 29821 . 2 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simplbi 501 1 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wral 3054  wf 6346  cfv 6350  (class class class)co 7183  chba 28867   ·ih csp 28870  HrmOpcho 28898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-hilex 28947
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-map 8452  df-hmop 29792
This theorem is referenced by:  hmopex  29823  hmopre  29871  hmopadj  29887  hmdmadj  29888  hmoplin  29890  eighmre  29911  eighmorth  29912  hmops  29968  hmopm  29969  hmopd  29970  hmopco  29971  leop2  30072  leoppos  30074  leoprf  30076  leopsq  30077  leopadd  30080  leopmuli  30081  leopmul  30082  leopmul2i  30083  leopnmid  30086  nmopleid  30087  opsqrlem1  30088  opsqrlem6  30093  elpjrn  30138
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