HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  elcnop Structured version   Visualization version   GIF version

Theorem elcnop 31680
Description: Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elcnop (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝑇

Proof of Theorem elcnop
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6896 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑤) = (𝑇𝑤))
2 fveq1 6896 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
31, 2oveq12d 7438 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑤) − (𝑡𝑥)) = ((𝑇𝑤) − (𝑇𝑥)))
43fveq2d 6901 . . . . . . 7 (𝑡 = 𝑇 → (norm‘((𝑡𝑤) − (𝑡𝑥))) = (norm‘((𝑇𝑤) − (𝑇𝑥))))
54breq1d 5158 . . . . . 6 (𝑡 = 𝑇 → ((norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦 ↔ (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦))
65imbi2d 340 . . . . 5 (𝑡 = 𝑇 → (((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
76rexralbidv 3217 . . . 4 (𝑡 = 𝑇 → (∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
872ralbidv 3215 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
9 df-cnop 31663 . . 3 ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}
108, 9elrab2 3685 . 2 (𝑇 ∈ ContOp ↔ (𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
11 ax-hilex 30822 . . . 4 ℋ ∈ V
1211, 11elmap 8890 . . 3 (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
1312anbi1i 623 . 2 ((𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
1410, 13bitri 275 1 (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  wrex 3067   class class class wbr 5148  wf 6544  cfv 6548  (class class class)co 7420  m cmap 8845   < clt 11279  +crp 13007  chba 30742  normcno 30746   cmv 30748  ContOpccop 30769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-hilex 30822
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-cnop 31663
This theorem is referenced by:  cnopc  31736  0cnop  31802  idcnop  31804  lnopconi  31857
  Copyright terms: Public domain W3C validator