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Theorem ulmcl 24968
 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 24966 . . . 4 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
2 ulmval 24967 . . . 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 269 . 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 1133 . . 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 3282 . 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 208   ∧ w3a 1083   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  Vcvv 3494   class class class wbr 5065  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ↑m cmap 8405  ℂcc 10534   < clt 10674   − cmin 10869  ℤcz 11980  ℤ≥cuz 12242  ℝ+crp 12388  abscabs 14592  ⇝𝑢culm 24963 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8407  df-pm 8408  df-neg 10872  df-z 11981  df-uz 12243  df-ulm 24964 This theorem is referenced by:  ulmi  24973  ulmclm  24974  ulmres  24975  ulmshftlem  24976  ulmuni  24979  ulmcau  24982  ulmss  24984  ulmbdd  24985  ulmcn  24986  ulmdvlem1  24987  ulmdvlem3  24989  ulmdv  24990  mbfulm  24993  iblulm  24994  itgulm  24995  itgulm2  24996  pserulm  25009  lgamgulmlem6  25610  lgamgulm2  25612  knoppcnlem9  33840
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