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Theorem ulmcl 26330

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

**Assertion**
Ref	Expression
ulmcl	⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)

Proof of Theorem ulmcl Dummy variables 𝑗 𝑘 𝑛 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmscl 26328	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝑆 ∈ V)
2		ulmval 26329	. . . 4 ⊢ (𝑆 ∈ V → (𝐹(⇝_𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
3	1, 2	syl 17	. . 3 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → (𝐹(⇝_𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
4	3	ibi 267	. 2 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ (𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
5		simp2 1135	. . 3 ⊢ ((𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → 𝐺:𝑆⟶ℂ)
6	5	rexlimivw 3148	. 2 ⊢ (∃𝑛 ∈ ℤ (𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → 𝐺:𝑆⟶ℂ)
7	4, 6	syl 17	1 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 Vcvv 3471 class class class wbr 5148 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ↑_m cmap 8845 ℂcc 11137 < clt 11279 − cmin 11475 ℤcz 12589 ℤ_≥cuz 12853 ℝ⁺crp 13007 abscabs 15214 ⇝_𝑢culm 26325

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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-pm 8848 df-neg 11478 df-z 12590 df-uz 12854 df-ulm 26326

This theorem is referenced by: ulmi 26335 ulmclm 26336 ulmres 26337 ulmshftlem 26338 ulmuni 26341 ulmcau 26344 ulmss 26346 ulmbdd 26347 ulmcn 26348 ulmdvlem1 26349 ulmdvlem3 26351 ulmdv 26352 mbfulm 26355 iblulm 26356 itgulm 26357 itgulm2 26358 pserulm 26371 lgamgulmlem6 26979 lgamgulm2 26981 knoppcnlem9 35976

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