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Theorem ulmcl 26424
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmcl (𝐹(⇝𝑢𝑆)𝐺𝐺:𝑆⟶ℂ)

Proof of Theorem ulmcl
Dummy variables 𝑗 𝑘 𝑛 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ulmscl 26422 . . . 4 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
2 ulmval 26423 . . . 4 (𝑆 ∈ V → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
31, 2syl 17 . . 3 (𝐹(⇝𝑢𝑆)𝐺 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
43ibi 267 . 2 (𝐹(⇝𝑢𝑆)𝐺 → ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
5 simp2 1138 . . 3 ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → 𝐺:𝑆⟶ℂ)
65rexlimivw 3151 . 2 (∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → 𝐺:𝑆⟶ℂ)
74, 6syl 17 1 (𝐹(⇝𝑢𝑆)𝐺𝐺:𝑆⟶ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087  wcel 2108  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153   < clt 11295  cmin 11492  cz 12613  cuz 12878  +crp 13034  abscabs 15273  𝑢culm 26419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-pm 8869  df-neg 11495  df-z 12614  df-uz 12879  df-ulm 26420
This theorem is referenced by:  ulmi  26429  ulmclm  26430  ulmres  26431  ulmshftlem  26432  ulmuni  26435  ulmcau  26438  ulmss  26440  ulmbdd  26441  ulmcn  26442  ulmdvlem1  26443  ulmdvlem3  26445  ulmdv  26446  mbfulm  26449  iblulm  26450  itgulm  26451  itgulm2  26452  pserulm  26465  lgamgulmlem6  27077  lgamgulm2  27079  knoppcnlem9  36502
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