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Theorem zdis 23027

Description: The integers are a discrete set in the topology on ℂ. (Contributed by Mario Carneiro, 19-Sep-2015.)

**Hypothesis**
Ref	Expression
recld2.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
zdis	⊢ (𝐽 ↾_t ℤ) = 𝒫 ℤ

Proof of Theorem zdis Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		restsspw 16478	. 2 ⊢ (𝐽 ↾_t ℤ) ⊆ 𝒫 ℤ
2		elpwi 4388	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℤ → 𝑥 ⊆ ℤ)
3	2	sselda 3820	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℤ)
4	3	zcnd 11835	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℂ)
5		cnxmet 22984	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
6		1rp 12141	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
7		rpxr 12148	. . . . . . . . 9 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
9		recld2.1	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
10	9	cnfldtopn 22993	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
11	10	blopn 22713	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽)
12	5, 8, 11	mp3an13 1525	. . . . . . 7 ⊢ (𝑦 ∈ ℂ → (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽)
13	9	cnfldtop 22995	. . . . . . . 8 ⊢ 𝐽 ∈ Top
14		zex 11737	. . . . . . . 8 ⊢ ℤ ∈ V
15		elrestr 16475	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ℤ ∈ V ∧ (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
16	13, 14, 15	mp3an12 1524	. . . . . . 7 ⊢ ((𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽 → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
17	4, 12, 16	3syl 18	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
18		blcntr 22626	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
19	5, 6, 18	mp3an13 1525	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
20	4, 19	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
21	20, 3	elind 4020	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ))
22	4	adantr 474	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ ℂ)
23		inss2 4053	. . . . . . . . . . . 12 ⊢ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ ℤ
24		simpr 479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ))
25	23, 24	sseldi 3818	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ℤ)
26	25	zcnd 11835	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ℂ)
27	3	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ ℤ)
28	27, 25	zsubcld 11839	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) ∈ ℤ)
29	28	zcnd 11835	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) ∈ ℂ)
30		eqid 2777	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
31	30	cnmetdval 22982	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
32	22, 26, 31	syl2anc 579	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
33		inss1 4052	. . . . . . . . . . . . . . 15 ⊢ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ (𝑦(ball‘(abs ∘ − ))1)
34	33, 24	sseldi 3818	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1))
35		elbl2 22603	. . . . . . . . . . . . . . . 16 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
36	5, 8, 35	mpanl12 692	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
37	22, 26, 36	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
38	34, 37	mpbid 224	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦(abs ∘ − )𝑧) < 1)
39	32, 38	eqbrtrrd 4910	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (abs‘(𝑦 − 𝑧)) < 1)
40		nn0abscl 14459	. . . . . . . . . . . . 13 ⊢ ((𝑦 − 𝑧) ∈ ℤ → (abs‘(𝑦 − 𝑧)) ∈ ℕ₀)
41		nn0lt10b 11791	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝑦 − 𝑧)) ∈ ℕ₀ → ((abs‘(𝑦 − 𝑧)) < 1 ↔ (abs‘(𝑦 − 𝑧)) = 0))
42	28, 40, 41	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → ((abs‘(𝑦 − 𝑧)) < 1 ↔ (abs‘(𝑦 − 𝑧)) = 0))
43	39, 42	mpbid 224	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (abs‘(𝑦 − 𝑧)) = 0)
44	29, 43	abs00d 14593	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) = 0)
45	22, 26, 44	subeq0d 10742	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 = 𝑧)
46		simplr 759	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ 𝑥)
47	45, 46	eqeltrrd 2859	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ 𝑥)
48	47	ex 403	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → (𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → 𝑧 ∈ 𝑥))
49	48	ssrdv 3826	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥)
50		eleq2 2847	. . . . . . . 8 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)))
51		sseq1 3844	. . . . . . . 8 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → (𝑧 ⊆ 𝑥 ↔ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥))
52	50, 51	anbi12d 624	. . . . . . 7 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∧ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥)))
53	52	rspcev 3510	. . . . . 6 ⊢ ((((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ) ∧ (𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∧ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥)) → ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
54	17, 21, 49, 53	syl12anc 827	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
55	54	ralrimiva 3147	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℤ → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
56		resttop 21372	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ℤ ∈ V) → (𝐽 ↾_t ℤ) ∈ Top)
57	13, 14, 56	mp2an 682	. . . . 5 ⊢ (𝐽 ↾_t ℤ) ∈ Top
58		eltop2 21187	. . . . 5 ⊢ ((𝐽 ↾_t ℤ) ∈ Top → (𝑥 ∈ (𝐽 ↾_t ℤ) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
59	57, 58	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ (𝐽 ↾_t ℤ) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
60	55, 59	sylibr 226	. . 3 ⊢ (𝑥 ∈ 𝒫 ℤ → 𝑥 ∈ (𝐽 ↾_t ℤ))
61	60	ssriv 3824	. 2 ⊢ 𝒫 ℤ ⊆ (𝐽 ↾_t ℤ)
62	1, 61	eqssi 3836	1 ⊢ (𝐽 ↾_t ℤ) = 𝒫 ℤ

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 Vcvv 3397 ∩ cin 3790 ⊆ wss 3791 𝒫 cpw 4378 class class class wbr 4886 ∘ ccom 5359 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 ℝ^*cxr 10410 < clt 10411 − cmin 10606 ℕ₀cn0 11642 ℤcz 11728 ℝ⁺crp 12137 abscabs 14381 ↾_t crest 16467 TopOpenctopn 16468 ∞Metcxmet 20127 ballcbl 20129 ℂ_fldccnfld 20142 Topctop 21105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-xms 22533 df-ms 22534

This theorem is referenced by: sszcld 23028

W3C validator