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Theorem ulmpm 26292

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 26291	. 2 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆))
2		uzssz 12814	. . . 4 ⊢ (ℤ_≥‘𝑛) ⊆ ℤ
3		ovex 7420	. . . . 5 ⊢ (ℂ ↑_m 𝑆) ∈ V
4		zex 12538	. . . . 5 ⊢ ℤ ∈ V
5		elpm2r 8818	. . . . 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 3130	. 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 2109 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 class class class wbr 5107 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 ↑_pm cpm 8800 ℂcc 11066 ℤcz 12529 ℤ_≥cuz 12793 ⇝_𝑢culm 26285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-map 8801 df-pm 8802 df-neg 11408 df-z 12530 df-uz 12794 df-ulm 26286

This theorem is referenced by: ulmf2 26293

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