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Theorem ulmf 24356
 Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmf (𝐹(⇝𝑢𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
Distinct variable groups:   𝑛,𝐹   𝑛,𝐺   𝑆,𝑛

Proof of Theorem ulmf
Dummy variables 𝑗 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ulmscl 24353 . . . 4 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
2 ulmval 24354 . . . 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 256 . 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 simp1 1130 . . 3 ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
65reximi 3159 . 2 (∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → ∃𝑛 ∈ ℤ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
74, 6syl 17 1 (𝐹(⇝𝑢𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1071   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   class class class wbr 4786  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793   ↑𝑚 cmap 8009  ℂcc 10136   < clt 10276   − cmin 10468  ℤcz 11579  ℤ≥cuz 11888  ℝ+crp 12035  abscabs 14182  ⇝𝑢culm 24350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-pm 8012  df-neg 10471  df-z 11580  df-uz 11889  df-ulm 24351 This theorem is referenced by:  ulmpm  24357  ulmuni  24366  ulmss  24371
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