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Mirrors > Home > MPE Home > Th. List > ulmpm | Structured version Visualization version GIF version |
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
ulmpm | ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmf 24477 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆)) | |
2 | uzssz 11950 | . . . 4 ⊢ (ℤ≥‘𝑛) ⊆ ℤ | |
3 | ovex 6910 | . . . . 5 ⊢ (ℂ ↑𝑚 𝑆) ∈ V | |
4 | zex 11675 | . . . . 5 ⊢ ℤ ∈ V | |
5 | elpm2r 8113 | . . . . 5 ⊢ ((((ℂ ↑𝑚 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ)) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) | |
6 | 3, 4, 5 | mpanl12 694 | . . . 4 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
7 | 2, 6 | mpan2 683 | . . 3 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
8 | 7 | rexlimivw 3210 | . 2 ⊢ (∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
9 | 1, 8 | syl 17 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ∃wrex 3090 Vcvv 3385 ⊆ wss 3769 class class class wbr 4843 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ↑𝑚 cmap 8095 ↑pm cpm 8096 ℂcc 10222 ℤcz 11666 ℤ≥cuz 11930 ⇝𝑢culm 24471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-map 8097 df-pm 8098 df-neg 10559 df-z 11667 df-uz 11931 df-ulm 24472 |
This theorem is referenced by: ulmf2 24479 |
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