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Theorem zcld 24320

Description: The integers are a closed set in the topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)

**Hypothesis**
Ref	Expression
zcld.1	⊢ 𝐽 = (topGen‘ran (,))

**Assertion**
Ref	Expression
zcld	⊢ ℤ ∈ (Clsd‘𝐽)

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

Step	Hyp	Ref	Expression
1		eliun 5000	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) ↔ ∃𝑥 ∈ ℤ 𝑦 ∈ (𝑥(,)(𝑥 + 1)))
2		elioore 13350	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥(,)(𝑥 + 1)) → 𝑦 ∈ ℝ)
3	2	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ (𝑥(,)(𝑥 + 1))) → 𝑦 ∈ ℝ)
4		eliooord 13379	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥(,)(𝑥 + 1)) → (𝑥 < 𝑦 ∧ 𝑦 < (𝑥 + 1)))
5		btwnnz 12634	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝑥 < 𝑦 ∧ 𝑦 < (𝑥 + 1)) → ¬ 𝑦 ∈ ℤ)
6	5	3expb 1120	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ (𝑥 < 𝑦 ∧ 𝑦 < (𝑥 + 1))) → ¬ 𝑦 ∈ ℤ)
7	4, 6	sylan2 593	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ (𝑥(,)(𝑥 + 1))) → ¬ 𝑦 ∈ ℤ)
8	3, 7	eldifd 3958	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ (𝑥(,)(𝑥 + 1))) → 𝑦 ∈ (ℝ ∖ ℤ))
9	8	rexlimiva 3147	. . . . . 6 ⊢ (∃𝑥 ∈ ℤ 𝑦 ∈ (𝑥(,)(𝑥 + 1)) → 𝑦 ∈ (ℝ ∖ ℤ))
10		eldifi 4125	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → 𝑦 ∈ ℝ)
11	10	flcld 13759	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (⌊‘𝑦) ∈ ℤ)
12	11	zred 12662	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (⌊‘𝑦) ∈ ℝ)
13		flle 13760	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (⌊‘𝑦) ≤ 𝑦)
14	10, 13	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (⌊‘𝑦) ≤ 𝑦)
15		eldifn 4126	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → ¬ 𝑦 ∈ ℤ)
16		nelne2 3040	. . . . . . . . . . 11 ⊢ (((⌊‘𝑦) ∈ ℤ ∧ ¬ 𝑦 ∈ ℤ) → (⌊‘𝑦) ≠ 𝑦)
17	11, 15, 16	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (⌊‘𝑦) ≠ 𝑦)
18	17	necomd 2996	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → 𝑦 ≠ (⌊‘𝑦))
19	12, 10, 14, 18	leneltd 11364	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (⌊‘𝑦) < 𝑦)
20		flltp1 13761	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 𝑦 < ((⌊‘𝑦) + 1))
21	10, 20	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → 𝑦 < ((⌊‘𝑦) + 1))
22	12	rexrd 11260	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (⌊‘𝑦) ∈ ℝ^*)
23		peano2re 11383	. . . . . . . . . . 11 ⊢ ((⌊‘𝑦) ∈ ℝ → ((⌊‘𝑦) + 1) ∈ ℝ)
24	12, 23	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → ((⌊‘𝑦) + 1) ∈ ℝ)
25	24	rexrd 11260	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → ((⌊‘𝑦) + 1) ∈ ℝ^*)
26		elioo2 13361	. . . . . . . . 9 ⊢ (((⌊‘𝑦) ∈ ℝ^* ∧ ((⌊‘𝑦) + 1) ∈ ℝ^*) → (𝑦 ∈ ((⌊‘𝑦)(,)((⌊‘𝑦) + 1)) ↔ (𝑦 ∈ ℝ ∧ (⌊‘𝑦) < 𝑦 ∧ 𝑦 < ((⌊‘𝑦) + 1))))
27	22, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → (𝑦 ∈ ((⌊‘𝑦)(,)((⌊‘𝑦) + 1)) ↔ (𝑦 ∈ ℝ ∧ (⌊‘𝑦) < 𝑦 ∧ 𝑦 < ((⌊‘𝑦) + 1))))
28	10, 19, 21, 27	mpbir3and 1342	. . . . . . 7 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → 𝑦 ∈ ((⌊‘𝑦)(,)((⌊‘𝑦) + 1)))
29		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (⌊‘𝑦) → 𝑥 = (⌊‘𝑦))
30		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑥 = (⌊‘𝑦) → (𝑥 + 1) = ((⌊‘𝑦) + 1))
31	29, 30	oveq12d 7423	. . . . . . . . 9 ⊢ (𝑥 = (⌊‘𝑦) → (𝑥(,)(𝑥 + 1)) = ((⌊‘𝑦)(,)((⌊‘𝑦) + 1)))
32	31	eleq2d 2819	. . . . . . . 8 ⊢ (𝑥 = (⌊‘𝑦) → (𝑦 ∈ (𝑥(,)(𝑥 + 1)) ↔ 𝑦 ∈ ((⌊‘𝑦)(,)((⌊‘𝑦) + 1))))
33	32	rspcev 3612	. . . . . . 7 ⊢ (((⌊‘𝑦) ∈ ℤ ∧ 𝑦 ∈ ((⌊‘𝑦)(,)((⌊‘𝑦) + 1))) → ∃𝑥 ∈ ℤ 𝑦 ∈ (𝑥(,)(𝑥 + 1)))
34	11, 28, 33	syl2anc 584	. . . . . 6 ⊢ (𝑦 ∈ (ℝ ∖ ℤ) → ∃𝑥 ∈ ℤ 𝑦 ∈ (𝑥(,)(𝑥 + 1)))
35	9, 34	impbii 208	. . . . 5 ⊢ (∃𝑥 ∈ ℤ 𝑦 ∈ (𝑥(,)(𝑥 + 1)) ↔ 𝑦 ∈ (ℝ ∖ ℤ))
36	1, 35	bitri 274	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) ↔ 𝑦 ∈ (ℝ ∖ ℤ))
37	36	eqriv 2729	. . 3 ⊢ ∪ 𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) = (ℝ ∖ ℤ)
38		zcld.1	. . . . 5 ⊢ 𝐽 = (topGen‘ran (,))
39		retop 24269	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
40	38, 39	eqeltri 2829	. . . 4 ⊢ 𝐽 ∈ Top
41		iooretop 24273	. . . . . 6 ⊢ (𝑥(,)(𝑥 + 1)) ∈ (topGen‘ran (,))
42	41, 38	eleqtrri 2832	. . . . 5 ⊢ (𝑥(,)(𝑥 + 1)) ∈ 𝐽
43	42	rgenw 3065	. . . 4 ⊢ ∀𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) ∈ 𝐽
44		iunopn 22391	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) ∈ 𝐽) → ∪ 𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) ∈ 𝐽)
45	40, 43, 44	mp2an 690	. . 3 ⊢ ∪ 𝑥 ∈ ℤ (𝑥(,)(𝑥 + 1)) ∈ 𝐽
46	37, 45	eqeltrri 2830	. 2 ⊢ (ℝ ∖ ℤ) ∈ 𝐽
47		zssre 12561	. . 3 ⊢ ℤ ⊆ ℝ
48		uniretop 24270	. . . . 5 ⊢ ℝ = ∪ (topGen‘ran (,))
49	38	unieqi 4920	. . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
50	48, 49	eqtr4i 2763	. . . 4 ⊢ ℝ = ∪ 𝐽
51	50	iscld2 22523	. . 3 ⊢ ((𝐽 ∈ Top ∧ ℤ ⊆ ℝ) → (ℤ ∈ (Clsd‘𝐽) ↔ (ℝ ∖ ℤ) ∈ 𝐽))
52	40, 47, 51	mp2an 690	. 2 ⊢ (ℤ ∈ (Clsd‘𝐽) ↔ (ℝ ∖ ℤ) ∈ 𝐽)
53	46, 52	mpbir 230	1 ⊢ ℤ ∈ (Clsd‘𝐽)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3944 ⊆ wss 3947 ∪ cuni 4907 ∪ ciun 4996 class class class wbr 5147 ran crn 5676 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 1c1 11107 + caddc 11109 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℤcz 12554 (,)cioo 13320 ⌊cfl 13751 topGenctg 17379 Topctop 22386 Clsdccld 22511

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-ioo 13324 df-fl 13753 df-topgen 17385 df-top 22387 df-bases 22440 df-cld 22514

This theorem is referenced by: zcld2 24322

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