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Theorem elcnop 28556
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 6149 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑤) = (𝑇𝑤))
2 fveq1 6149 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
31, 2oveq12d 6623 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑤) − (𝑡𝑥)) = ((𝑇𝑤) − (𝑇𝑥)))
43fveq2d 6154 . . . . . . 7 (𝑡 = 𝑇 → (norm‘((𝑡𝑤) − (𝑡𝑥))) = (norm‘((𝑇𝑤) − (𝑇𝑥))))
54breq1d 4628 . . . . . 6 (𝑡 = 𝑇 → ((norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦 ↔ (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦))
65imbi2d 330 . . . . 5 (𝑡 = 𝑇 → (((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
76rexralbidv 3056 . . . 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 2988 . . 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 28539 . . 3 ContOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}
108, 9elrab2 3353 . 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 27696 . . . 4 ℋ ∈ V
1211, 11elmap 7831 . . 3 (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
1312anbi1i 730 . 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 264 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 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913   class class class wbr 4618  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803   < clt 10019  +crp 11776  chil 27616  normcno 27620   cmv 27622  ContOpccop 27643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-hilex 27696
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-cnop 28539
This theorem is referenced by:  cnopc  28612  0cnop  28678  idcnop  28680  lnopconi  28733
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