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Theorem elcncf1ii 22455
Description: Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
Hypotheses
Ref Expression
elcncf1i.1 𝐹:𝐴𝐵
elcncf1i.2 ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)
elcncf1i.3 (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))
Assertion
Ref Expression
elcncf1ii ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))
Distinct variable groups:   𝑥,𝑤,𝑦,𝐴   𝑤,𝐵,𝑥,𝑦   𝑤,𝐹,𝑥,𝑦   𝑤,𝑍
Allowed substitution hints:   𝑍(𝑥,𝑦)

Proof of Theorem elcncf1ii
StepHypRef Expression
1 elcncf1i.1 . . . 4 𝐹:𝐴𝐵
21a1i 11 . . 3 (⊤ → 𝐹:𝐴𝐵)
3 elcncf1i.2 . . . 4 ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)
43a1i 11 . . 3 (⊤ → ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))
5 elcncf1i.3 . . . 4 (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))
65a1i 11 . . 3 (⊤ → (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
72, 4, 6elcncf1di 22454 . 2 (⊤ → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵)))
87trud 1484 1 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wtru 1476  wcel 1977  wss 3540   class class class wbr 4578  wf 5786  cfv 5790  (class class class)co 6527  cc 9791   < clt 9931  cmin 10118  +crp 11667  abscabs 13771  cnccncf 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724  df-cncf 22437
This theorem is referenced by:  logcnlem5  24137
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