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Theorem lnopf 29953
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 29952 . 2 (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simplbi 501 1 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  wral 3062  wf 6385  cfv 6389  (class class class)co 7222  cc 10740  chba 29013   + cva 29014   · csm 29015  LinOpclo 29041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532  ax-hilex 29093
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-sbc 3704  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-br 5063  df-opab 5125  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-fv 6397  df-ov 7225  df-oprab 7226  df-mpo 7227  df-map 8519  df-lnop 29935
This theorem is referenced by:  bdopf  29956  elbdop2  29965  unopadj2  30032  lnop0  30060  lnopmul  30061  lnopfi  30063  homco2  30071  nmopun  30108  cnlnadjeui  30171  cnlnssadj  30174
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