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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 24962 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) | |
2 | uzssz 12256 | . . . 4 ⊢ (ℤ≥‘𝑛) ⊆ ℤ | |
3 | ovex 7181 | . . . . 5 ⊢ (ℂ ↑m 𝑆) ∈ V | |
4 | zex 11982 | . . . . 5 ⊢ ℤ ∈ V | |
5 | elpm2r 8416 | . . . . 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 3280 | . 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 2108 ∃wrex 3137 Vcvv 3493 ⊆ wss 3934 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ↑m cmap 8398 ↑pm cpm 8399 ℂcc 10527 ℤcz 11973 ℤ≥cuz 12235 ⇝𝑢culm 24956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-map 8400 df-pm 8401 df-neg 10865 df-z 11974 df-uz 12236 df-ulm 24957 |
This theorem is referenced by: ulmf2 24964 |
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