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Mirrors > Home > MPE Home > Th. List > ulmcl | Structured version Visualization version GIF version |
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
ulmcl | ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmscl 26436 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | |
2 | ulmval 26437 | . . . 4 ⊢ (𝑆 ∈ V → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
4 | 3 | ibi 267 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
5 | simp2 1136 | . . 3 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → 𝐺:𝑆⟶ℂ) | |
6 | 5 | rexlimivw 3148 | . 2 ⊢ (∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → 𝐺:𝑆⟶ℂ) |
7 | 4, 6 | syl 17 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 Vcvv 3477 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 ℂcc 11150 < clt 11292 − cmin 11489 ℤcz 12610 ℤ≥cuz 12875 ℝ+crp 13031 abscabs 15269 ⇝𝑢culm 26433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-pm 8867 df-neg 11492 df-z 12611 df-uz 12876 df-ulm 26434 |
This theorem is referenced by: ulmi 26443 ulmclm 26444 ulmres 26445 ulmshftlem 26446 ulmuni 26449 ulmcau 26452 ulmss 26454 ulmbdd 26455 ulmcn 26456 ulmdvlem1 26457 ulmdvlem3 26459 ulmdv 26460 mbfulm 26463 iblulm 26464 itgulm 26465 itgulm2 26466 pserulm 26479 lgamgulmlem6 27091 lgamgulm2 27093 knoppcnlem9 36483 |
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