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Theorem cnllycmp 24319

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 24147	. 2 ⊢ 𝐽 ∈ Top
3		cnxmet 24136	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
4	1	cnfldtopn 24145	. . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
5	4	mopni2 23849	. . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)
6	3, 5	mp3an1 1448	. . . 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 4898	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
10	9	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
11	1	cnfldtopon 24146	. . . . . . . . . . . 12 ⊢ 𝐽 ∈ (TopOn‘ℂ)
12	11	toponunii 22265	. . . . . . . . . . 11 ⊢ ℂ = ∪ 𝐽
13	10, 12	sseqtrrdi 3995	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ)
14		simplr 767	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥)
15	13, 14	sseldd 3945	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ)
16		rphalfcl 12942	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
17	16	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ⁺)
18	17	rpxrd 12958	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ^*)
19	4	blopn 23856	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
20	8, 15, 18, 19	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ 𝐽)
21		blcntr 23766	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
22	8, 15, 17, 21	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))
23		opnneip 22470	. . . . . . . 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 1371	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ∈ ((nei‘𝐽)‘{𝑦}))
25		blssm 23771	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
26	8, 15, 18, 25	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))(𝑟 / 2)) ⊆ ℂ)
27	12	sscls 22407	. . . . . . . 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 12924	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
30	29	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ^*)
31		rphalflt 12944	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
32	31	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) < 𝑟)
33	4	blsscls 23863	. . . . . . . . . 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 1377	. . . . . . . . 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 771	. . . . . . . . 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 22467	. . . . . . 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 837	. . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ ((nei‘𝐽)‘{𝑦}))
40		vex 3449	. . . . . . . 8 ⊢ 𝑥 ∈ V
41	40	elpw2 5302	. . . . . . 7 ⊢ (((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) ⊆ 𝑥)
42	36, 41	sylibr 233	. . . . . 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 4154	. . . . 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 22398	. . . . . . 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 15321	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs‘𝑦) ∈ ℝ)
47	17	rpred 12957	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑟 / 2) ∈ ℝ)
48	46, 47	readdcld 11184	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ((abs‘𝑦) + (𝑟 / 2)) ∈ ℝ)
49		eqid 2736	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)} = {𝑤 ∈ ℂ ∣ (𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2)}
50	4, 49	blcls 23862	. . . . . . . . 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 1371	. . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
53	15	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → 𝑦 ∈ ℂ)
54	52, 53	abs2difd 15342	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ≤ (abs‘(𝑧 − 𝑦)))
55	52	abscld 15321	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑧) ∈ ℝ)
56	46	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘𝑦) ∈ ℝ)
57	55, 56	resubcld 11583	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → ((abs‘𝑧) − (abs‘𝑦)) ∈ ℝ)
58	52, 53	subcld 11512	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑦) ∈ ℂ)
59	58	abscld 15321	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) ∈ ℝ)
60	47	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (𝑟 / 2) ∈ ℝ)
61		letr 11249	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 15338	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (abs‘(𝑦 − 𝑧)))
65		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
66	65	cnmetdval 24134	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) ∧ 𝑧 ∈ ℂ) → (abs‘(𝑧 − 𝑦)) = (𝑦(abs ∘ − )𝑧))
69	68	breq1d 5115	. . . . . . . . . . 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 11754	. . . . . . . . . . 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 292	. . . . . . . . . 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 3143	. . . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑦(abs ∘ − )𝑤) = (𝑦(abs ∘ − )𝑧))
74	73	breq1d 5115	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((𝑦(abs ∘ − )𝑤) ≤ (𝑟 / 2) ↔ (𝑦(abs ∘ − )𝑧) ≤ (𝑟 / 2)))
75	74	ralrab 3651	. . . . . . . . 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 233	. . . . . . . 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 4010	. . . . . . . 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 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 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 2736	. . . . . . . 8 ⊢ (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))))
82	1, 81	cnheibor 24318	. . . . . . 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 711	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))) ∈ Comp)
85		oveq2 7365	. . . . . . 7 ⊢ (𝑢 = ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2))) → (𝐽 ↾_t 𝑢) = (𝐽 ↾_t ((cls‘𝐽)‘(𝑦(ball‘(abs ∘ − ))(𝑟 / 2)))))
86	85	eleq1d 2822	. . . . . 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 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 3158	. . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp)
90	89	rgen2 3194	. 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾_t 𝑢) ∈ Comp
91		isnlly 22820	. 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 709	1 ⊢ 𝐽 ∈ 𝑛-Locally Comp

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 {crab 3407 ∩ cin 3909 ⊆ wss 3910 𝒫 cpw 4560 {csn 4586 ∪ cuni 4865 class class class wbr 5105 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 2c2 12208 ℝ⁺crp 12915 abscabs 15119 ↾_t crest 17302 TopOpenctopn 17303 ∞Metcxmet 20781 ballcbl 20783 ℂ_fldccnfld 20796 Topctop 22242 Clsdccld 22367 clsccl 22369 neicnei 22448 Compccmp 22737 𝑛-Locally cnlly 22816

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-cls 22372 df-nei 22449 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-nlly 22818 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241

This theorem is referenced by: rellycmp 24320

W3C validator