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Theorem zdis 24712

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 17401	. 2 ⊢ (𝐽 ↾_t ℤ) ⊆ 𝒫 ℤ
2		elpwi 4573	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℤ → 𝑥 ⊆ ℤ)
3	2	sselda 3949	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℤ)
4	3	zcnd 12646	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℂ)
5		cnxmet 24667	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
6		1xr 11240	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
7		recld2.1	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
8	7	cnfldtopn 24676	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
9	8	blopn 24395	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽)
10	5, 6, 9	mp3an13 1454	. . . . . . 7 ⊢ (𝑦 ∈ ℂ → (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽)
11	7	cnfldtop 24678	. . . . . . . 8 ⊢ 𝐽 ∈ Top
12		zex 12545	. . . . . . . 8 ⊢ ℤ ∈ V
13		elrestr 17398	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ℤ ∈ V ∧ (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
14	11, 12, 13	mp3an12 1453	. . . . . . 7 ⊢ ((𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽 → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
15	4, 10, 14	3syl 18	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
16		1rp 12962	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
17		blcntr 24308	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
18	5, 16, 17	mp3an13 1454	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
19	4, 18	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
20	19, 3	elind 4166	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ))
21	4	adantr 480	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ ℂ)
22		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ))
23	22	elin2d 4171	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ℤ)
24	23	zcnd 12646	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ℂ)
25	3	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ ℤ)
26	25, 23	zsubcld 12650	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) ∈ ℤ)
27	26	zcnd 12646	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) ∈ ℂ)
28		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
29	28	cnmetdval 24665	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
30	21, 24, 29	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
31	22	elin1d 4170	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1))
32		elbl2 24285	. . . . . . . . . . . . . . . 16 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
33	5, 6, 32	mpanl12 702	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
34	21, 24, 33	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
35	31, 34	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦(abs ∘ − )𝑧) < 1)
36	30, 35	eqbrtrrd 5134	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (abs‘(𝑦 − 𝑧)) < 1)
37		nn0abscl 15285	. . . . . . . . . . . . 13 ⊢ ((𝑦 − 𝑧) ∈ ℤ → (abs‘(𝑦 − 𝑧)) ∈ ℕ₀)
38		nn0lt10b 12603	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝑦 − 𝑧)) ∈ ℕ₀ → ((abs‘(𝑦 − 𝑧)) < 1 ↔ (abs‘(𝑦 − 𝑧)) = 0))
39	26, 37, 38	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → ((abs‘(𝑦 − 𝑧)) < 1 ↔ (abs‘(𝑦 − 𝑧)) = 0))
40	36, 39	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (abs‘(𝑦 − 𝑧)) = 0)
41	27, 40	abs00d 15422	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) = 0)
42	21, 24, 41	subeq0d 11548	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 = 𝑧)
43		simplr 768	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ 𝑥)
44	42, 43	eqeltrrd 2830	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ 𝑥)
45	44	ex 412	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → (𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → 𝑧 ∈ 𝑥))
46	45	ssrdv 3955	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥)
47		eleq2 2818	. . . . . . . 8 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)))
48		sseq1 3975	. . . . . . . 8 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → (𝑧 ⊆ 𝑥 ↔ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥))
49	47, 48	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∧ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥)))
50	49	rspcev 3591	. . . . . 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))
51	15, 20, 46, 50	syl12anc 836	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
52	51	ralrimiva 3126	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℤ → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
53		resttop 23054	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ℤ ∈ V) → (𝐽 ↾_t ℤ) ∈ Top)
54	11, 12, 53	mp2an 692	. . . . 5 ⊢ (𝐽 ↾_t ℤ) ∈ Top
55		eltop2 22869	. . . . 5 ⊢ ((𝐽 ↾_t ℤ) ∈ Top → (𝑥 ∈ (𝐽 ↾_t ℤ) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
56	54, 55	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ (𝐽 ↾_t ℤ) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
57	52, 56	sylibr 234	. . 3 ⊢ (𝑥 ∈ 𝒫 ℤ → 𝑥 ∈ (𝐽 ↾_t ℤ))
58	57	ssriv 3953	. 2 ⊢ 𝒫 ℤ ⊆ (𝐽 ↾_t ℤ)
59	1, 58	eqssi 3966	1 ⊢ (𝐽 ↾_t ℤ) = 𝒫 ℤ

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 𝒫 cpw 4566 class class class wbr 5110 ∘ ccom 5645 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 ℝ^*cxr 11214 < clt 11215 − cmin 11412 ℕ₀cn0 12449 ℤcz 12536 ℝ⁺crp 12958 abscabs 15207 ↾_t crest 17390 TopOpenctopn 17391 ∞Metcxmet 21256 ballcbl 21258 ℂ_fldccnfld 21271 Topctop 22787

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17392 df-topn 17393 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-xms 24215 df-ms 24216

This theorem is referenced by: sszcld 24713

W3C validator