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Theorem zdis 24807

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 17392	. 2 ⊢ (𝐽 ↾_t ℤ) ⊆ 𝒫 ℤ
2		elpwi 4543	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℤ → 𝑥 ⊆ ℤ)
3	2	sselda 3922	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℤ)
4	3	zcnd 12632	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℂ)
5		cnxmet 24762	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
6		1xr 11202	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
7		recld2.1	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
8	7	cnfldtopn 24771	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
9	8	blopn 24490	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℝ^*) → (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽)
10	5, 6, 9	mp3an13 1460	. . . . . . 7 ⊢ (𝑦 ∈ ℂ → (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽)
11	7	cnfldtop 24773	. . . . . . . 8 ⊢ 𝐽 ∈ Top
12		zex 12531	. . . . . . . 8 ⊢ ℤ ∈ V
13		elrestr 17389	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ℤ ∈ V ∧ (𝑦(ball‘(abs ∘ − ))1) ∈ 𝐽) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ∈ (𝐽 ↾_t ℤ))
14	11, 12, 13	mp3an12 1459	. . . . . . 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 12944	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
17		blcntr 24403	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))1))
18	5, 16, 17	mp3an13 1460	. . . . . . . 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 4136	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ))
21	4	adantr 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ ℂ)
22		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ))
23	22	elin2d 4141	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ℤ)
24	23	zcnd 12632	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ ℂ)
25	3	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ ℤ)
26	25, 23	zsubcld 12636	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) ∈ ℤ)
27	26	zcnd 12632	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) ∈ ℂ)
28		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
29	28	cnmetdval 24760	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
30	21, 24, 29	syl2anc 590	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦(abs ∘ − )𝑧) = (abs‘(𝑦 − 𝑧)))
31	22	elin1d 4140	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1))
32		elbl2 24380	. . . . . . . . . . . . . . . 16 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
33	5, 6, 32	mpanl12 708	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 ∈ (𝑦(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )𝑧) < 1))
34	21, 24, 33	syl2anc 590	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦(abs ∘ − )𝑧) < 1)
36	30, 35	eqbrtrrd 5103	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (abs‘(𝑦 − 𝑧)) < 1)
37		nn0abscl 15272	. . . . . . . . . . . . 13 ⊢ ((𝑦 − 𝑧) ∈ ℤ → (abs‘(𝑦 − 𝑧)) ∈ ℕ₀)
38		nn0lt10b 12589	. . . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (abs‘(𝑦 − 𝑧)) = 0)
41	27, 40	abs00d 15409	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → (𝑦 − 𝑧) = 0)
42	21, 24, 41	subeq0d 11511	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 = 𝑧)
43		simplr 774	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑦 ∈ 𝑥)
44	42, 43	eqeltrrd 2841	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)) → 𝑧 ∈ 𝑥)
45	44	ex 413	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → (𝑧 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → 𝑧 ∈ 𝑥))
46	45	ssrdv 3928	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥)
47		eleq2 2829	. . . . . . . 8 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ)))
48		sseq1 3947	. . . . . . . 8 ⊢ (𝑧 = ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) → (𝑧 ⊆ 𝑥 ↔ ((𝑦(ball‘(abs ∘ − ))1) ∩ ℤ) ⊆ 𝑥))
49	47, 48	anbi12d 638	. . . . . . 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 3567	. . . . . 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 842	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 ℤ ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
52	51	ralrimiva 3132	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℤ → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐽 ↾_t ℤ)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
53		resttop 23150	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ℤ ∈ V) → (𝐽 ↾_t ℤ) ∈ Top)
54	11, 12, 53	mp2an 698	. . . . 5 ⊢ (𝐽 ↾_t ℤ) ∈ Top
55		eltop2 22965	. . . . 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 235	. . 3 ⊢ (𝑥 ∈ 𝒫 ℤ → 𝑥 ∈ (𝐽 ↾_t ℤ))
58	57	ssriv 3926	. 2 ⊢ 𝒫 ℤ ⊆ (𝐽 ↾_t ℤ)
59	1, 58	eqssi 3938	1 ⊢ (𝐽 ↾_t ℤ) = 𝒫 ℤ

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ∃wrex 3064 Vcvv 3432 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4536 class class class wbr 5079 ∘ ccom 5629 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 0cc0 11036 1c1 11037 ℝ^*cxr 11176 < clt 11177 − cmin 11375 ℕ₀cn0 12435 ℤcz 12522 ℝ⁺crp 12940 abscabs 15194 ↾_t crest 17381 TopOpenctopn 17382 ∞Metcxmet 21339 ballcbl 21341 ℂ_fldccnfld 21354 Topctop 22883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9321 df-sup 9352 df-inf 9353 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-fz 13460 df-seq 13962 df-exp 14022 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17178 df-plusg 17231 df-mulr 17232 df-starv 17233 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-rest 17383 df-topn 17384 df-topgen 17404 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-xms 24310 df-ms 24311

This theorem is referenced by: sszcld 24808

W3C validator