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Theorem cnfnc 29713
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 29665 . . . 4 (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤)))
21simprbi 500 . . 3 (𝑇 ∈ ContFn → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤))
3 oveq2 7143 . . . . . . . 8 (𝑧 = 𝐴 → (𝑦 𝑧) = (𝑦 𝐴))
43fveq2d 6649 . . . . . . 7 (𝑧 = 𝐴 → (norm‘(𝑦 𝑧)) = (norm‘(𝑦 𝐴)))
54breq1d 5040 . . . . . 6 (𝑧 = 𝐴 → ((norm‘(𝑦 𝑧)) < 𝑥 ↔ (norm‘(𝑦 𝐴)) < 𝑥))
6 fveq2 6645 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑇𝑧) = (𝑇𝐴))
76oveq2d 7151 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑇𝑦) − (𝑇𝑧)) = ((𝑇𝑦) − (𝑇𝐴)))
87fveq2d 6649 . . . . . . 7 (𝑧 = 𝐴 → (abs‘((𝑇𝑦) − (𝑇𝑧))) = (abs‘((𝑇𝑦) − (𝑇𝐴))))
98breq1d 5040 . . . . . 6 (𝑧 = 𝐴 → ((abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤 ↔ (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤))
105, 9imbi12d 348 . . . . 5 (𝑧 = 𝐴 → (((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
1110rexralbidv 3260 . . . 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 5034 . . . . . 6 (𝑤 = 𝐵 → ((abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤 ↔ (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
1312imbi2d 344 . . . . 5 (𝑤 = 𝐵 → (((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1413rexralbidv 3260 . . . 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 3581 . . 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 1113 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 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107   class class class wbr 5030  wf 6320  cfv 6324  (class class class)co 7135  cc 10524   < clt 10664  cmin 10859  +crp 12377  abscabs 14585  chba 28702  normcno 28706   cmv 28708  ContFnccnfn 28736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-hilex 28782
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-cnfn 29630
This theorem is referenced by:  nmcfnexi  29834
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