Theorem lnfnf 29083

Description: A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
lnfnf	⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Proof of Theorem lnfnf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellnfn 29082	. 2 ⊢ (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧))))
2	1	simplbi 485	1 ⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  ℂcc 10136   + caddc 10141   · cmul 10143   ℋchil 28116   +_ℎ cva 28117   ·_ℎ csm 28118  LinFnclf 28151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-hilex 28196

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-lnfn 29047

This theorem is referenced by:  nmfn0  29186  lnfnfi  29240  rnbra  29306