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Theorem ulmpm 26616

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 26615	. 2 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆))
2		uzssz 12911	. . . 4 ⊢ (ℤ_≥‘𝑛) ⊆ ℤ
3		ovex 7449	. . . . 5 ⊢ (ℂ ↑_m 𝑆) ∈ V
4		zex 12627	. . . . 5 ⊢ ℤ ∈ V
5		elpm2r 8847	. . . . 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 715	. . . 4 ⊢ ((𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ (ℤ_≥‘𝑛) ⊆ ℤ) → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
7	2, 6	mpan2 704	. . 3 ⊢ (𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
8	7	rexlimivw 3161	. 2 ⊢ (∃𝑛 ∈ ℤ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
9	1, 8	syl 18	1 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2145 ∃wrex 3088 Vcvv 3453 ⊆ wss 3902 class class class wbr 5107 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ↑_m cmap 8829 ↑_pm cpm 8830 ℂcc 11125 ℤcz 12618 ℤ_≥cuz 12890 ⇝_𝑢culm 26609

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8831 df-pm 8832 df-neg 11471 df-z 12619 df-uz 12891 df-ulm 26610

This theorem is used by: ulmf2 26617

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