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Theorem cnopc 29377
Description: Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnopc ((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑇,𝑦

Proof of Theorem cnopc
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnop 29321 . . . 4 (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤)))
21simprbi 497 . . 3 (𝑇 ∈ ContOp → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤))
3 oveq2 7031 . . . . . . . 8 (𝑧 = 𝐴 → (𝑦 𝑧) = (𝑦 𝐴))
43fveq2d 6549 . . . . . . 7 (𝑧 = 𝐴 → (norm‘(𝑦 𝑧)) = (norm‘(𝑦 𝐴)))
54breq1d 4978 . . . . . 6 (𝑧 = 𝐴 → ((norm‘(𝑦 𝑧)) < 𝑥 ↔ (norm‘(𝑦 𝐴)) < 𝑥))
6 fveq2 6545 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑇𝑧) = (𝑇𝐴))
76oveq2d 7039 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑇𝑦) − (𝑇𝑧)) = ((𝑇𝑦) − (𝑇𝐴)))
87fveq2d 6549 . . . . . . 7 (𝑧 = 𝐴 → (norm‘((𝑇𝑦) − (𝑇𝑧))) = (norm‘((𝑇𝑦) − (𝑇𝐴))))
98breq1d 4978 . . . . . 6 (𝑧 = 𝐴 → ((norm‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤 ↔ (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤))
105, 9imbi12d 346 . . . . 5 (𝑧 = 𝐴 → (((norm‘(𝑦 𝑧)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
1110rexralbidv 3266 . . . 4 (𝑧 = 𝐴 → (∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
12 breq2 4972 . . . . . 6 (𝑤 = 𝐵 → ((norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤 ↔ (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
1312imbi2d 342 . . . . 5 (𝑤 = 𝐵 → (((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1413rexralbidv 3266 . . . 4 (𝑤 = 𝐵 → (∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1511, 14rspc2v 3574 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → (∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
162, 15syl5com 31 . 2 (𝑇 ∈ ContOp → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
17163impib 1109 1 ((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1525  wcel 2083  wral 3107  wrex 3108   class class class wbr 4968  wf 6228  cfv 6232  (class class class)co 7023   < clt 10528  +crp 12243  chba 28383  normcno 28387   cmv 28389  ContOpccop 28410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-hilex 28463
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-map 8265  df-cnop 29304
This theorem is referenced by:  nmcopexi  29491
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