Theorem lnopf 29642

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.

Step	Hyp	Ref	Expression
1		ellnop 29641	. 2 ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
2	1	simplbi 501	1 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℂcc 10524   ℋchba 28702   +_ℎ cva 28703   ·_ℎ csm 28704  LinOpclo 28730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-hilex 28782

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-lnop 29624

This theorem is referenced by:  bdopf  29645  elbdop2  29654  unopadj2  29721  lnop0  29749  lnopmul  29750  lnopfi  29752  homco2  29760  nmopun  29797  cnlnadjeui  29860  cnlnssadj  29863