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Theorem elcnop 29938
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 6716 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑤) = (𝑇𝑤))
2 fveq1 6716 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
31, 2oveq12d 7231 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑤) − (𝑡𝑥)) = ((𝑇𝑤) − (𝑇𝑥)))
43fveq2d 6721 . . . . . . 7 (𝑡 = 𝑇 → (norm‘((𝑡𝑤) − (𝑡𝑥))) = (norm‘((𝑇𝑤) − (𝑇𝑥))))
54breq1d 5063 . . . . . 6 (𝑡 = 𝑇 → ((norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦 ↔ (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦))
65imbi2d 344 . . . . 5 (𝑡 = 𝑇 → (((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
76rexralbidv 3220 . . . 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 3120 . . 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 29921 . . 3 ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}
108, 9elrab2 3605 . 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 29080 . . . 4 ℋ ∈ V
1211, 11elmap 8552 . . 3 (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
1312anbi1i 627 . 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 278 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 209  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062   class class class wbr 5053  wf 6376  cfv 6380  (class class class)co 7213  m cmap 8508   < clt 10867  +crp 12586  chba 29000  normcno 29004   cmv 29006  ContOpccop 29027
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 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-hilex 29080
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 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-cnop 29921
This theorem is referenced by:  cnopc  29994  0cnop  30060  idcnop  30062  lnopconi  30115
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