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Theorem cnllycmp 25184

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 25009	. 2 ⊢ 𝐽 ∈ Top
3		cnxmet 24998	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
4	1	cnfldtopn 25007	. . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
5	4	mopni2 24719	. . . . 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 4899	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
10	9	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
11	1	cnfldtopon 25008	. . . . . . . . . . . 12 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	toponunii 23141	. . . . . . . . . . 11 ⊢ ℂ = ∪ 𝐽
13	10, 12	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ)
14		simplr 781	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥)
15	13, 14	sseldd 3932	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ)
16		rphalfcl 13071	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
17	16	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ⁺)
18	17	rpxrd 13087	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ^*)
19	4	blopn 24726	. . . . . . . . 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 24639	. . . . . . . . 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 23344	. . . . . . . 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 24644	. . . . . . . . 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 23281	. . . . . . . 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 596	. . . . . . 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 13052	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
30	29	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ^*)
31		rphalflt 13073	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
32	31	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) < 𝑟)
33	4	blsscls 24733	. . . . . . . . . 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 785	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
36	34, 35	sstrd 3941	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
37	36, 13	sstrd 3941	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ ℂ)
38	12	ssnei2 23341	. . . . . . 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 852	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ ((nei‘𝐽)‘{𝑦}))
40		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
41	40	elpw2 5299	. . . . . . 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 4146	. . . . 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 23272	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
45	7, 26, 44	syl2anc 596	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ (Clsd‘𝐽))
46	15	abscld 15526	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs‘𝑦) ∈ ℝ)
47	17	rpred 13086	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ)
48	46, 47	readdcld 11262	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((abs‘𝑦) + (𝑟 / 2)) ∈ ℝ)
49		eqid 2760	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} = {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)}
50	4, 49	blcls 24732	. . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
53	15	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
54	52, 53	abs2difd 15547	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
55	52	abscld 15526	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑧) ∈ ℝ)
56	46	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑦) ∈ ℝ)
57	55, 56	resubcld 11666	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ∈ ℝ)
58	52, 53	subcld 11593	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑦) ∈ ℂ)
59	58	abscld 15526	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
60	47	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑟 / 2) ∈ ℝ)
61		letr 11328	. . . . . . . . . . . . 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 708	. . . . . . . . . . 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 15543	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
65		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
66	65	cnmetdval 24996	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
67	15, 66	sylan 592	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
68	64, 67	eqtr4d 2798	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (𝑦(abs ∘ − )𝑧))
69	68	breq1d 5113	. . . . . . . . . . 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 11837	. . . . . . . . . . 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 3154	. . . . . . . . 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 7421	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑦(abs ∘ − )𝑤) = (𝑦(abs ∘ − )𝑧))
74	73	breq1d 5113	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2) ↔ (𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2)))
75	74	ralrab 3652	. . . . . . . . 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 4000	. . . . . . . 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 5165	. . . . . . 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 596	. . . . . 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 2760	. . . . . . . 8 ⊢ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
82	1, 81	cnheibor 25183	. . . . . . 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 726	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp)
85		oveq2 7421	. . . . . . 7 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → (𝐽 ↾_t 𝑢) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))))
86	85	eleq1d 2845	. . . . . 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 3576	. . . . 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 596	. . . 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 3169	. . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
90	89	rgen2 3202	. 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp
91		isnlly 23695	. 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 724	1 ⊢ 𝐽 ∈ 𝑛-Locally Comp

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 {crab 3412 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 {csn 4584 ∪ cuni 4867 class class class wbr 5103 ∘ ccom 5659 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 + caddc 11127 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11465 / cdiv 11895 2c2 12319 ℝ⁺crp 13042 abscabs 15321 ↾_t crest 17505 TopOpenctopn 17506 ∞Metcxmet 21570 ballcbl 21572 ℂ_fldccnfld 21585 Topctop 23118 Clsdccld 23241 clsccl 23243 neicnei 23322 Compccmp 23611 𝑛-Locally cnlly 23691

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-icc 13405 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-cls 23246 df-nei 23323 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-nlly 23693 df-tx 23788 df-hmeo 23981 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106

This theorem is used by: rellycmp 25185

W3C validator