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Theorem climcn1lem 14377
 Description: The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
Hypotheses
Ref Expression
climcn1lem.1 𝑍 = (ℤ𝑀)
climcn1lem.2 (𝜑𝐹𝐴)
climcn1lem.4 (𝜑𝐺𝑊)
climcn1lem.5 (𝜑𝑀 ∈ ℤ)
climcn1lem.6 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
climcn1lem.7 𝐻:ℂ⟶ℂ
climcn1lem.8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((𝐻𝑧) − (𝐻𝐴))) < 𝑥))
climcn1lem.9 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐻‘(𝐹𝑘)))
Assertion
Ref Expression
climcn1lem (𝜑𝐺 ⇝ (𝐻𝐴))
Distinct variable groups:   𝑥,𝑘,𝑦,𝑧,𝐴   𝑘,𝐹,𝑦,𝑧   𝑘,𝐺,𝑥   𝜑,𝑘,𝑥,𝑦,𝑧   𝑘,𝑍,𝑦   𝑘,𝐻,𝑥,𝑦,𝑧   𝑘,𝑀
Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑦,𝑧)   𝑀(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧,𝑘)   𝑍(𝑥,𝑧)

Proof of Theorem climcn1lem
StepHypRef Expression
1 climcn1lem.1 . 2 𝑍 = (ℤ𝑀)
2 climcn1lem.5 . 2 (𝜑𝑀 ∈ ℤ)
3 climcn1lem.2 . . 3 (𝜑𝐹𝐴)
4 climcl 14274 . . 3 (𝐹𝐴𝐴 ∈ ℂ)
53, 4syl 17 . 2 (𝜑𝐴 ∈ ℂ)
6 climcn1lem.7 . . . 4 𝐻:ℂ⟶ℂ
76ffvelrni 6398 . . 3 (𝑧 ∈ ℂ → (𝐻𝑧) ∈ ℂ)
87adantl 481 . 2 ((𝜑𝑧 ∈ ℂ) → (𝐻𝑧) ∈ ℂ)
9 climcn1lem.4 . 2 (𝜑𝐺𝑊)
10 climcn1lem.8 . . 3 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((𝐻𝑧) − (𝐻𝐴))) < 𝑥))
115, 10sylan 487 . 2 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℂ ((abs‘(𝑧𝐴)) < 𝑦 → (abs‘((𝐻𝑧) − (𝐻𝐴))) < 𝑥))
12 climcn1lem.6 . 2 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
13 climcn1lem.9 . 2 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐻‘(𝐹𝑘)))
141, 2, 5, 8, 3, 9, 11, 12, 13climcn1 14366 1 (𝜑𝐺 ⇝ (𝐻𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   class class class wbr 4685  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℂcc 9972   < clt 10112   − cmin 10304  ℤcz 11415  ℤ≥cuz 11725  ℝ+crp 11870  abscabs 14018   ⇝ cli 14259 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-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-neg 10307  df-z 11416  df-uz 11726  df-clim 14263 This theorem is referenced by:  climabs  14378  climcj  14379  climre  14380  climim  14381  sinccvglem  31692
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