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Theorem cnfnc 28917
Description: Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnfnc ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑇,𝑦

Proof of Theorem cnfnc
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnfn 28869 . . . 4 (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤)))
21simprbi 479 . . 3 (𝑇 ∈ ContFn → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤))
3 oveq2 6698 . . . . . . . 8 (𝑧 = 𝐴 → (𝑦 𝑧) = (𝑦 𝐴))
43fveq2d 6233 . . . . . . 7 (𝑧 = 𝐴 → (norm‘(𝑦 𝑧)) = (norm‘(𝑦 𝐴)))
54breq1d 4695 . . . . . 6 (𝑧 = 𝐴 → ((norm‘(𝑦 𝑧)) < 𝑥 ↔ (norm‘(𝑦 𝐴)) < 𝑥))
6 fveq2 6229 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑇𝑧) = (𝑇𝐴))
76oveq2d 6706 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑇𝑦) − (𝑇𝑧)) = ((𝑇𝑦) − (𝑇𝐴)))
87fveq2d 6233 . . . . . . 7 (𝑧 = 𝐴 → (abs‘((𝑇𝑦) − (𝑇𝑧))) = (abs‘((𝑇𝑦) − (𝑇𝐴))))
98breq1d 4695 . . . . . 6 (𝑧 = 𝐴 → ((abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤 ↔ (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤))
105, 9imbi12d 333 . . . . 5 (𝑧 = 𝐴 → (((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
1110rexralbidv 3087 . . . 4 (𝑧 = 𝐴 → (∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
12 breq2 4689 . . . . . 6 (𝑤 = 𝐵 → ((abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤 ↔ (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
1312imbi2d 329 . . . . 5 (𝑤 = 𝐵 → (((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1413rexralbidv 3087 . . . 4 (𝑤 = 𝐵 → (∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1511, 14rspc2v 3353 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → (∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
162, 15syl5com 31 . 2 (𝑇 ∈ ContFn → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
17163impib 1281 1 ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  cc 9972   < clt 10112  cmin 10304  +crp 11870  abscabs 14018  chil 27904  normcno 27908   cmv 27910  ContFnccnfn 27938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-cnfn 28834
This theorem is referenced by:  nmcfnexi  29038
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