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Theorem cnllycmp 22563

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 22397	. 2 ⊢ 𝐽 ∈ Top
3		cnxmet 22386	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
4	1	cnfldtopn 22395	. . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
5	4	mopni2 22108	. . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
6	3, 5	mp3an1 1403	. . . 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 4403	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
10	9	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
11	1	cnfldtopon 22396	. . . . . . . . . . . 12 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	toponunii 20547	. . . . . . . . . . 11 ⊢ ℂ = ∪ 𝐽
13	10, 12	syl6sseqr 3615	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ)
14		simplr 788	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥)
15	13, 14	sseldd 3569	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ)
16		rphalfcl 11734	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
17	16	ad2antrl 760	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ⁺)
18	17	rpxrd 11749	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ^*)
19	4	blopn 22115	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
20	8, 15, 18, 19	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
21		blcntr 22028	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
22	8, 15, 17, 21	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
23		opnneip 20733	. . . . . . . 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 1318	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ ((nei‘𝐽)‘{𝑦}))
25		blssm 22033	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
26	8, 15, 18, 25	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
27	12	sscls 20670	. . . . . . . 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 691	. . . . . . 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 11716	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
30	29	ad2antrl 760	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ^*)
31		rphalflt 11736	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
32	31	ad2antrl 760	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) < 𝑟)
33	4	blsscls 22122	. . . . . . . . . 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 1325	. . . . . . . . 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 792	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
36	34, 35	sstrd 3578	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
37	36, 13	sstrd 3578	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ ℂ)
38	12	ssnei2 20730	. . . . . . 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 1319	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ ((nei‘𝐽)‘{𝑦}))
40		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
41	40	elpw2 4755	. . . . . . 7 ⊢ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
42	36, 41	sylibr 223	. . . . . 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 3760	. . . . 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 20661	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
45	7, 26, 44	syl2anc 691	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
46	15	abscld 14023	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs‘𝑦) ∈ ℝ)
47	17	rpred 11748	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ)
48	46, 47	readdcld 9948	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((abs‘𝑦) + (𝑟 / 2)) ∈ ℝ)
49		eqid 2610	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} = {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)}
50	4, 49	blcls 22121	. . . . . . . . 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 1318	. . . . . . . 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 476	. . . . . . . . . . . . 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 14044	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
55	52	abscld 14023	. . . . . . . . . . . . . 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 10337	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ∈ ℝ)
58	52, 53	subcld 10271	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑦) ∈ ℂ)
59	58	abscld 14023	. . . . . . . . . . . . 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 10010	. . . . . . . . . . . . 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 1318	. . . . . . . . . . . 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 14040	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
65		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
66	65	cnmetdval 22384	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
67	15, 66	sylan 487	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
68	64, 67	eqtr4d 2647	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (𝑦(abs ∘ − )𝑧))
69	68	breq1d 4593	. . . . . . . . . . 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 10505	. . . . . . . . . . 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 281	. . . . . . . . . 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 2949	. . . . . . . . 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 6557	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑦(abs ∘ − )𝑤) = (𝑦(abs ∘ − )𝑧))
74	73	breq1d 4593	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2) ↔ (𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2)))
75	74	ralrab 3335	. . . . . . . . 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 223	. . . . . . . 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 3629	. . . . . . . 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 63	. . . . . . 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		breq2 4587	. . . . . . . . 9 ⊢ (𝑠 = ((abs‘𝑦) + (𝑟 / 2)) → ((abs‘𝑧) ≤ 𝑠 ↔ (abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
80	79	ralbidv 2969	. . . . . . . 8 ⊢ (𝑠 = ((abs‘𝑦) + (𝑟 / 2)) → (∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ 𝑠 ↔ ∀𝑧 ∈ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))(abs‘𝑧) ≤ ((abs‘𝑦) + (𝑟 / 2))))
81	80	rspcev 3282	. . . . . . 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)
82	48, 78, 81	syl2anc 691	. . . . . 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)
83		eqid 2610	. . . . . . . 8 ⊢ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
84	1, 83	cnheibor 22562	. . . . . . 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)))
85	37, 84	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)))
86	45, 82, 85	mpbir2and 959	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp)
87		oveq2 6557	. . . . . . 7 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → (𝐽 ↾_t 𝑢) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))))
88	87	eleq1d 2672	. . . . . 6 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → ((𝐽 ↾_t 𝑢) ∈ Comp ↔ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp))
89	88	rspcev 3282	. . . . 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)
90	43, 86, 89	syl2anc 691	. . . 4 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
91	6, 90	rexlimddv 3017	. . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
92	91	rgen2 2958	. 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp
93		isnlly 21082	. 2 ⊢ (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp))
94	2, 92, 93	mpbir2an 957	1 ⊢ 𝐽 ∈ 𝑛-Locally Comp

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 + caddc 9818 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 abscabs 13822 ↾_t crest 15904 TopOpenctopn 15905 ∞Metcxmt 19552 ballcbl 19554 ℂ_fldccnfld 19567 Topctop 20517 Clsdccld 20630 clsccl 20632 neicnei 20711 Compccmp 20999 𝑛-Locally cnlly 21078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-nlly 21080 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489

This theorem is referenced by: rellycmp 22564

W3C validator