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Theorem cnllycmp 24888

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 24704	. 2 ⊢ 𝐽 ∈ Top
3		cnxmet 24693	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
4	1	cnfldtopn 24702	. . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
5	4	mopni2 24414	. . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
6	3, 5	mp3an1 1450	. . . 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 4897	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
10	9	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
11	1	cnfldtopon 24703	. . . . . . . . . . . 12 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	toponunii 22836	. . . . . . . . . . 11 ⊢ ℂ = ∪ 𝐽
13	10, 12	sseqtrrdi 3985	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ)
14		simplr 768	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥)
15	13, 14	sseldd 3944	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ)
16		rphalfcl 12956	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
17	16	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ⁺)
18	17	rpxrd 12972	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ^*)
19	4	blopn 24421	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
20	8, 15, 18, 19	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
21		blcntr 24334	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
22	8, 15, 17, 21	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
23		opnneip 23039	. . . . . . . 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 1373	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ ((nei‘𝐽)‘{𝑦}))
25		blssm 24339	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
26	8, 15, 18, 25	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
27	12	sscls 22976	. . . . . . . 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 584	. . . . . . 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 12937	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
30	29	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ^*)
31		rphalflt 12958	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
32	31	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) < 𝑟)
33	4	blsscls 24428	. . . . . . . . . 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 1379	. . . . . . . . 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 772	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
36	34, 35	sstrd 3954	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
37	36, 13	sstrd 3954	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ ℂ)
38	12	ssnei2 23036	. . . . . . 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 838	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ ((nei‘𝐽)‘{𝑦}))
40		vex 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
41	40	elpw2 5284	. . . . . . 7 ⊢ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
42	36, 41	sylibr 234	. . . . . 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 4159	. . . . 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 22967	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
45	7, 26, 44	syl2anc 584	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
46	15	abscld 15381	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs‘𝑦) ∈ ℝ)
47	17	rpred 12971	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ)
48	46, 47	readdcld 11179	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((abs‘𝑦) + (𝑟 / 2)) ∈ ℝ)
49		eqid 2729	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} = {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)}
50	4, 49	blcls 24427	. . . . . . . . 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 1373	. . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
53	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
54	52, 53	abs2difd 15402	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
55	52	abscld 15381	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑧) ∈ ℝ)
56	46	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑦) ∈ ℝ)
57	55, 56	resubcld 11582	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ∈ ℝ)
58	52, 53	subcld 11509	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑦) ∈ ℂ)
59	58	abscld 15381	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
60	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑟 / 2) ∈ ℝ)
61		letr 11244	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 695	. . . . . . . . . . 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 15398	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
65		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
66	65	cnmetdval 24691	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
67	15, 66	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
68	64, 67	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (𝑦(abs ∘ − )𝑧))
69	68	breq1d 5112	. . . . . . . . . . 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 11753	. . . . . . . . . . 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 293	. . . . . . . . . 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 3125	. . . . . . . . 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 7377	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑦(abs ∘ − )𝑤) = (𝑦(abs ∘ − )𝑧))
74	73	breq1d 5112	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2) ↔ (𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2)))
75	74	ralrab 3662	. . . . . . . . 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 234	. . . . . . . 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 4012	. . . . . . . 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 65	. . . . . . 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 5162	. . . . . . 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 584	. . . . . 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 2729	. . . . . . . 8 ⊢ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
82	1, 81	cnheibor 24887	. . . . . . 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 17	. . . . . 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 713	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp)
85		oveq2 7377	. . . . . . 7 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → (𝐽 ↾_t 𝑢) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))))
86	85	eleq1d 2813	. . . . . 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 3585	. . . . 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 584	. . . 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 3140	. . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
90	89	rgen2 3175	. 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp
91		isnlly 23389	. 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 711	1 ⊢ 𝐽 ∈ 𝑛-Locally Comp

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3402 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4559 {csn 4585 ∪ cuni 4867 class class class wbr 5102 ∘ ccom 5635 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 + caddc 11047 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 ℝ⁺crp 12927 abscabs 15176 ↾_t crest 17359 TopOpenctopn 17360 ∞Metcxmet 21281 ballcbl 21283 ℂ_fldccnfld 21296 Topctop 22813 Clsdccld 22936 clsccl 22938 neicnei 23017 Compccmp 23306 𝑛-Locally cnlly 23385

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-cls 22941 df-nei 23018 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-nlly 23387 df-tx 23482 df-hmeo 23675 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804

This theorem is referenced by: rellycmp 24889

W3C validator