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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 24970 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) | |
2 | uzssz 12265 | . . . 4 ⊢ (ℤ≥‘𝑛) ⊆ ℤ | |
3 | ovex 7189 | . . . . 5 ⊢ (ℂ ↑m 𝑆) ∈ V | |
4 | zex 11991 | . . . . 5 ⊢ ℤ ∈ V | |
5 | elpm2r 8424 | . . . . 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 700 | . . . 4 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)) |
7 | 2, 6 | mpan2 689 | . . 3 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)) |
8 | 7 | rexlimivw 3282 | . 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 398 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ↑pm cpm 8407 ℂcc 10535 ℤcz 11982 ℤ≥cuz 12244 ⇝𝑢culm 24964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-pm 8409 df-neg 10873 df-z 11983 df-uz 12245 df-ulm 24965 |
This theorem is referenced by: ulmf2 24972 |
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