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Theorem ulmpm 26319

Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

**Assertion**
Ref	Expression
ulmpm	⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))

Proof of Theorem ulmpm Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmf 26318	. 2 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆))
2		uzssz 12753	. . . 4 ⊢ (ℤ_≥‘𝑛) ⊆ ℤ
3		ovex 7379	. . . . 5 ⊢ (ℂ ↑_m 𝑆) ∈ V
4		zex 12477	. . . . 5 ⊢ ℤ ∈ V
5		elpm2r 8769	. . . . 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 702	. . . 4 ⊢ ((𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ (ℤ_≥‘𝑛) ⊆ ℤ) → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
7	2, 6	mpan2 691	. . 3 ⊢ (𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
8	7	rexlimivw 3129	. 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 395 ∈ wcel 2111 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 class class class wbr 5089 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 ↑_pm cpm 8751 ℂcc 11004 ℤcz 12468 ℤ_≥cuz 12732 ⇝_𝑢culm 26312

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-map 8752 df-pm 8753 df-neg 11347 df-z 12469 df-uz 12733 df-ulm 26313

This theorem is referenced by: ulmf2 26320

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