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Theorem hmopf 30236
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 30235 . 2 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simplbi 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  chba 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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