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Theorem cnllycmp 25115

Description: The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)

**Hypothesis**
Ref	Expression
cnllycmp.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
cnllycmp	⊢ 𝐽 ∈ 𝑛-Locally Comp

Proof of Theorem cnllycmp Dummy variables 𝑠 𝑟 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnllycmp.1	. . 3 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
2	1	cnfldtop 24940	. 2 ⊢ 𝐽 ∈ Top
3		cnxmet 24929	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
4	1	cnfldtopn 24938	. . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
5	4	mopni2 24650	. . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
6	3, 5	mp3an1 1477	. . . 4 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
7	2	a1i 11	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝐽 ∈ Top)
8	3	a1i 11	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
9		elssuni 4904	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
10	9	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
11	1	cnfldtopon 24939	. . . . . . . . . . . 12 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	toponunii 23073	. . . . . . . . . . 11 ⊢ ℂ = ∪ 𝐽
13	10, 12	sseqtrrdi 3978	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ)
14		simplr 780	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥)
15	13, 14	sseldd 3938	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ)
16		rphalfcl 13040	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
17	16	ad2antrl 740	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ⁺)
18	17	rpxrd 13056	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ^*)
19	4	blopn 24657	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
20	8, 15, 18, 19	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
21		blcntr 24570	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
22	8, 15, 17, 21	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
23		opnneip 23276	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽 ∧ 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ ((nei‘𝐽)‘{𝑦}))
24	7, 20, 22, 23	syl3anc 1398	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ ((nei‘𝐽)‘{𝑦}))
25		blssm 24575	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
26	8, 15, 18, 25	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
27	12	sscls 23213	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
28	7, 26, 27	syl2anc 595	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
29		rpxr 13021	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
30	29	ad2antrl 740	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ^*)
31		rphalflt 13042	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
32	31	ad2antrl 740	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) < 𝑟)
33	4	blsscls 24664	. . . . . . . . . 10 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ) ∧ ((𝑟 / 2) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ (𝑟 / 2) < 𝑟)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ (𝑦(ball‘(abs ∘ − ))𝑟))
34	8, 15, 18, 30, 32, 33	syl23anc 1404	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ (𝑦(ball‘(abs ∘ − ))𝑟))
35		simprr 784	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
36	34, 35	sstrd 3947	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
37	36, 13	sstrd 3947	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ ℂ)
38	12	ssnei2 23273	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ ((nei‘𝐽)‘{𝑦})) ∧ ((𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∧ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ ℂ)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ ((nei‘𝐽)‘{𝑦}))
39	7, 24, 28, 37, 38	syl22anc 851	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ ((nei‘𝐽)‘{𝑦}))
40		vex 3459	. . . . . . . 8 ⊢ 𝑥 ∈ V
41	40	elpw2 5305	. . . . . . 7 ⊢ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
42	36, 41	sylibr 237	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ 𝒫 𝑥)
43	39, 42	elind 4153	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
44	12	clscld 23204	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
45	7, 26, 44	syl2anc 595	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
46	15	abscld 15486	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs‘𝑦) ∈ ℝ)
47	17	rpred 13055	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ)
48	46, 47	readdcld 11233	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((abs‘𝑦) + (𝑟 / 2)) ∈ ℝ)
49		eqid 2763	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} = {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)}
50	4, 49	blcls 24663	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)})
51	8, 15, 18, 50	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)})
52		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
53	15	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
54	52, 53	abs2difd 15507	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
55	52	abscld 15486	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑧) ∈ ℝ)
56	46	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑦) ∈ ℝ)
57	55, 56	resubcld 11637	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ∈ ℝ)
58	52, 53	subcld 11564	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑦) ∈ ℂ)
59	58	abscld 15486	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
60	47	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑟 / 2) ∈ ℝ)
61		letr 11299	. . . . . . . . . . . . 13 ⊢ ((((abs‘𝑧) − (abs‘𝑦)) ∈ ℝ ∧ (abs‘(𝑧 − 𝑦)) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ) → ((((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)) ∧ (abs‘(𝑧 − 𝑦)) ≤ (𝑟 / 2)) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (𝑟 / 2)))
62	57, 59, 60, 61	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)) ∧ (abs‘(𝑧 − 𝑦)) ≤ (𝑟 / 2)) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (𝑟 / 2)))
63	54, 62	mpand 707	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘(𝑧 − 𝑦)) ≤ (𝑟 / 2) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (𝑟 / 2)))
64	52, 53	abssubd 15503	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
65		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
66	65	cnmetdval 24927	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
67	15, 66	sylan 591	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
68	64, 67	eqtr4d 2801	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (𝑦(abs ∘ − )𝑧))
69	68	breq1d 5119	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘(𝑧 − 𝑦)) ≤ (𝑟 / 2) ↔ (𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2)))
70	55, 56, 60	lesubadd2d 11808	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (((abs‘𝑧) − (abs‘𝑦)) ≤ (𝑟 / 2) ↔ (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
71	63, 69, 70	3imtr3d 296	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2) → (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
72	71	ralrimiva 3157	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∀𝑧 ∈ ℂ ((𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2) → (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
73		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑦(abs ∘ − )𝑤) = (𝑦(abs ∘ − )𝑧))
74	73	breq1d 5119	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2) ↔ (𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2)))
75	74	ralrab 3657	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2)) ↔ ∀𝑧 ∈ ℂ ((𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2) → (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
76	72, 75	sylibr 237	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∀𝑧 ∈ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2)))
77		ssralv 4006	. . . . . . . 8 ⊢ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} → (∀𝑧 ∈ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2)) → ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
78	51, 76, 77	sylc 66	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2)))
79		brralrspcev 5171	. . . . . . 7 ⊢ ((((abs‘𝑦) + (𝑟 / 2)) ∈ ℝ ∧ ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ 𝑠)
80	48, 78, 79	syl2anc 595	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ 𝑠)
81		eqid 2763	. . . . . . . 8 ⊢ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
82	1, 81	cnheibor 25114	. . . . . . 7 ⊢ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ ℂ → ((𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp ↔ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽) ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ 𝑠)))
83	37, 82	syl 18	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp ↔ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽) ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ 𝑠)))
84	45, 80, 83	mpbir2and 725	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp)
85		oveq2 7418	. . . . . . 7 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → (𝐽 ↾_t 𝑢) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))))
86	85	eleq1d 2848	. . . . . 6 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → ((𝐽 ↾_t 𝑢) ∈ Comp ↔ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp))
87	86	rspcev 3581	. . . . 5 ⊢ ((((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
88	43, 84, 87	syl2anc 595	. . . 4 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
89	6, 88	rexlimddv 3172	. . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
90	89	rgen2 3205	. 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp
91		isnlly 23626	. 2 ⊢ (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp))
92	2, 90, 91	mpbir2an 723	1 ⊢ 𝐽 ∈ 𝑛-Locally Comp

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 {crab 3416 ∩ cin 3904 ⊆ wss 3905 𝒫 cpw 4562 {csn 4589 ∪ cuni 4872 class class class wbr 5109 ∘ ccom 5665 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 + caddc 11098 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 2c2 12290 ℝ⁺crp 13011 abscabs 15281 ↾_t crest 17468 TopOpenctopn 17469 ∞Metcxmet 21507 ballcbl 21509 ℂ_fldccnfld 21522 Topctop 23050 Clsdccld 23173 clsccl 23175 neicnei 23254 Compccmp 23543 𝑛-Locally cnlly 23622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-cls 23178 df-nei 23255 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-nlly 23624 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037

This theorem is referenced by: rellycmp 25116

W3C validator