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Theorem lnopf 29294
Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnopf (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)

Proof of Theorem lnopf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnop 29293 . 2 (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simplbi 493 1 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wral 3090  wf 6133  cfv 6137  (class class class)co 6924  cc 10272  chba 28352   + cva 28353   · csm 28354  LinOpclo 28380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-hilex 28432
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-lnop 29276
This theorem is referenced by:  bdopf  29297  elbdop2  29306  unopadj2  29373  lnop0  29401  lnopmul  29402  lnopfi  29404  homco2  29412  nmopun  29449  cnlnadjeui  29512  cnlnssadj  29515
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