blcld - Metamath Proof Explorer

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Theorem blcld 21598

Description: A "closed ball" in a metric space is actually closed. (Contributed by Mario Carneiro, 31-Dec-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)

**Hypotheses**
Ref	Expression
mopni.1
blcld.3

**Assertion**
Ref	Expression
blcld	$/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

Proof of Theorem blcld Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mopni.1	. . . . . 6
2	1	mopnuni 21534	. . . . 5
3	2	3ad2ant1 1051	. . . 4 $/\$ $/\$
4	3	difeq1d 3539	. . 3 $/\$ $/\$ $\$ $\$
5		difssd 3550	. . . 4 $/\$ $/\$ $\$
6		simpl3 1035	. . . . . . 7 $/\$ $/\$ $/\$ $\$
7		simpl1 1033	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
8		simpl2 1034	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
9		eldifi 3544	. . . . . . . . 9 $\$
10	9	adantl 473	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
11		xmetcl 21424	. . . . . . . 8 $/\$ $/\$
12	7, 8, 10, 11	syl3anc 1292	. . . . . . 7 $/\$ $/\$ $/\$ $\$
13		eldif 3400	. . . . . . . . . 10 $\$ $/\$
14		oveq2 6316	. . . . . . . . . . . . . 14
15	14	breq1d 4405	. . . . . . . . . . . . 13
16		blcld.3	. . . . . . . . . . . . 13
17	15, 16	elrab2 3186	. . . . . . . . . . . 12 $/\$
18	17	simplbi2 637	. . . . . . . . . . 11
19	18	con3dimp 448	. . . . . . . . . 10 $/\$
20	13, 19	sylbi 200	. . . . . . . . 9 $\$
21	20	adantl 473	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
22		xrltnle 9719	. . . . . . . . 9 $/\$
23	6, 12, 22	syl2anc 673	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
24	21, 23	mpbird 240	. . . . . . 7 $/\$ $/\$ $/\$ $\$
25		qbtwnxr 11516	. . . . . . 7 $/\$ $/\$ $/\$
26	6, 12, 24, 25	syl3anc 1292	. . . . . 6 $/\$ $/\$ $/\$ $\$ $/\$
27		qre 11292	. . . . . . . 8
28	7	adantr 472	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
29	10	adantr 472	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
30	12	adantr 472	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
31		rexr 9704	. . . . . . . . . . . . . 14
32	31	ad2antrl 742	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
33	32	xnegcld 11611	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
34	30, 33	xaddcld 11612	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
35		blelrn 21510	. . . . . . . . . . 11 $/\$ $/\$
36	28, 29, 34, 35	syl3anc 1292	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
37		simprrr 783	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
38		xposdif 11573	. . . . . . . . . . . . 13 $/\$
39	32, 30, 38	syl2anc 673	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
40	37, 39	mpbid 215	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
41		xblcntr 21504	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42	28, 29, 34, 40, 41	syl112anc 1296	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
43		incom 3616	. . . . . . . . . . . . 13
44	8	adantr 472	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
45		xaddcom 11555	. . . . . . . . . . . . . . . . 17 $/\$
46	32, 34, 45	syl2anc 673	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
47		simprl 772	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
48		xnpcan 11563	. . . . . . . . . . . . . . . . 17 $/\$
49	30, 47, 48	syl2anc 673	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
50	46, 49	eqtrd 2505	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
51		xrleid 11472	. . . . . . . . . . . . . . . 16
52	30, 51	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
53	50, 52	eqbrtrd 4416	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
54		bldisj 21491	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
55	28, 44, 29, 32, 34, 53, 54	syl33anc 1307	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
56	43, 55	syl5eq 2517	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
57		blssm 21511	. . . . . . . . . . . . . 14 $/\$ $/\$
58	28, 29, 34, 57	syl3anc 1292	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
59		reldisj 3812	. . . . . . . . . . . . 13 $\$
60	58, 59	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
61	56, 60	mpbid 215	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
62	6	adantr 472	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
63		simprrl 782	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
64	1, 16	blsscls2 21597	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
65	28, 44, 62, 32, 63, 64	syl23anc 1299	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
66	65	sscond 3559	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$ $\$
67	61, 66	sstrd 3428	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
68		eleq2 2538	. . . . . . . . . . . 12
69		sseq1 3439	. . . . . . . . . . . 12 $\$ $\$
70	68, 69	anbi12d 725	. . . . . . . . . . 11 $/\$ $\$ $/\$ $\$
71	70	rspcev 3136	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ $\$
72	36, 42, 67, 71	syl12anc 1290	. . . . . . . . 9 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $/\$ $\$
73	72	expr 626	. . . . . . . 8 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
74	27, 73	sylan2 482	. . . . . . 7 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
75	74	rexlimdva 2871	. . . . . 6 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
76	26, 75	mpd 15	. . . . 5 $/\$ $/\$ $/\$ $\$ $/\$ $\$
77	76	ralrimiva 2809	. . . 4 $/\$ $/\$ $\$ $/\$ $\$
78	1	elmopn 21535	. . . . 5 $\$ $\$ $/\$ $\$ $/\$ $\$
79	78	3ad2ant1 1051	. . . 4 $/\$ $/\$ $\$ $\$ $/\$ $\$ $/\$ $\$
80	5, 77, 79	mpbir2and 936	. . 3 $/\$ $/\$ $\$
81	4, 80	eqeltrrd 2550	. 2 $/\$ $/\$ $\$
82	1	mopntop 21533	. . . 4
83	82	3ad2ant1 1051	. . 3 $/\$ $/\$
84		ssrab2 3500	. . . . 5
85	16, 84	eqsstri 3448	. . . 4
86	85, 3	syl5sseq 3466	. . 3 $/\$ $/\$
87		eqid 2471	. . . 4
88	87	iscld2 20120	. . 3 $/\$ $\$
89	83, 86, 88	syl2anc 673	. 2 $/\$ $/\$ $\$
90	81, 89	mpbird 240	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $/\$ wa 376 $/\$ w3a 1007

wceq 1452

wcel 1904

wral 2756

wrex 2757

crab 2760 $\$ cdif 3387

cin 3389

wss 3390

c0 3722

cuni 4190 class class class wbr 4395

crn 4840

cfv 5589 (class class class)co 6308

cr 9556

cc0 9557

cxr 9692

clt 9693

cle 9694

cq 11287

cxne 11429

cxad 11430

cxmt 19032

cbl 19034

cmopn 19037

ctop 19994

ccld 20108

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 ax-pre-sup 9635

This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-1st 6812 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-er 7381 df-map 7492 df-en 7588 df-dom 7589 df-sdom 7590 df-sup 7974 df-inf 7975 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-div 10292 df-nn 10632 df-2 10690 df-n0 10894 df-z 10962 df-uz 11183 df-q 11288 df-rp 11326 df-xneg 11432 df-xadd 11433 df-xmul 11434 df-topgen 15420 df-psmet 19039 df-xmet 19040 df-bl 19042 df-mopn 19043 df-top 19998 df-bases 19999 df-topon 20000 df-cld 20111

This theorem is referenced by: blcls 21599 lmle 22349 minveclem4 22452 minveclem4OLD 22464 lhop1lem 23044 ftalem3 24078 ubthlem1 26593

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