blcld - Metamath Proof Explorer

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Theorem blcld 22310

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 22246	. . . . 5
3	2	3ad2ant1 1082	. . . 4 $/\$ $/\$
4	3	difeq1d 3727	. . 3 $/\$ $/\$ $\$ $\$
5		difssd 3738	. . . 4 $/\$ $/\$ $\$
6		simpl3 1066	. . . . . . 7 $/\$ $/\$ $/\$ $\$
7		simpl1 1064	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
8		simpl2 1065	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
9		eldifi 3732	. . . . . . . . 9 $\$
10	9	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
11		xmetcl 22136	. . . . . . . 8 $/\$ $/\$
12	7, 8, 10, 11	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $\$
13		eldif 3584	. . . . . . . . . 10 $\$ $/\$
14		oveq2 6658	. . . . . . . . . . . . . 14
15	14	breq1d 4663	. . . . . . . . . . . . 13
16		blcld.3	. . . . . . . . . . . . 13
17	15, 16	elrab2 3366	. . . . . . . . . . . 12 $/\$
18	17	simplbi2 655	. . . . . . . . . . 11
19	18	con3dimp 457	. . . . . . . . . 10 $/\$
20	13, 19	sylbi 207	. . . . . . . . 9 $\$
21	20	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
22		xrltnle 10105	. . . . . . . . 9 $/\$
23	6, 12, 22	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $\$
24	21, 23	mpbird 247	. . . . . . 7 $/\$ $/\$ $/\$ $\$
25		qbtwnxr 12031	. . . . . . 7 $/\$ $/\$ $/\$
26	6, 12, 24, 25	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $\$ $/\$
27		qre 11793	. . . . . . . 8
28	7	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
29	10	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
30	12	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
31		rexr 10085	. . . . . . . . . . . . . 14
32	31	ad2antrl 764	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
33	32	xnegcld 12130	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
34	30, 33	xaddcld 12131	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
35		blelrn 22222	. . . . . . . . . . 11 $/\$ $/\$
36	28, 29, 34, 35	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
37		simprrr 805	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
38		xposdif 12092	. . . . . . . . . . . . 13 $/\$
39	32, 30, 38	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
40	37, 39	mpbid 222	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
41		xblcntr 22216	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42	28, 29, 34, 40, 41	syl112anc 1330	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
43		incom 3805	. . . . . . . . . . . . 13
44	8	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
45		xaddcom 12071	. . . . . . . . . . . . . . . . 17 $/\$
46	32, 34, 45	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
47		simprl 794	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
48		xnpcan 12082	. . . . . . . . . . . . . . . . 17 $/\$
49	30, 47, 48	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
50	46, 49	eqtrd 2656	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
51		xrleid 11983	. . . . . . . . . . . . . . . 16
52	30, 51	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
53	50, 52	eqbrtrd 4675	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
54		bldisj 22203	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
55	28, 44, 29, 32, 34, 53, 54	syl33anc 1341	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
56	43, 55	syl5eq 2668	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
57		blssm 22223	. . . . . . . . . . . . . 14 $/\$ $/\$
58	28, 29, 34, 57	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
59		reldisj 4020	. . . . . . . . . . . . 13 $\$
60	58, 59	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
61	56, 60	mpbid 222	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
62	6	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
63		simprrl 804	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
64	1, 16	blsscls2 22309	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
65	28, 44, 62, 32, 63, 64	syl23anc 1333	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$
66	65	sscond 3747	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$ $\$
67	61, 66	sstrd 3613	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
68		eleq2 2690	. . . . . . . . . . . 12
69		sseq1 3626	. . . . . . . . . . . 12 $\$ $\$
70	68, 69	anbi12d 747	. . . . . . . . . . 11 $/\$ $\$ $/\$ $\$
71	70	rspcev 3309	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ $\$
72	36, 42, 67, 71	syl12anc 1324	. . . . . . . . 9 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $/\$ $\$
73	72	expr 643	. . . . . . . 8 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
74	27, 73	sylan2 491	. . . . . . 7 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
75	74	rexlimdva 3031	. . . . . 6 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
76	26, 75	mpd 15	. . . . 5 $/\$ $/\$ $/\$ $\$ $/\$ $\$
77	76	ralrimiva 2966	. . . 4 $/\$ $/\$ $\$ $/\$ $\$
78	1	elmopn 22247	. . . . 5 $\$ $\$ $/\$ $\$ $/\$ $\$
79	78	3ad2ant1 1082	. . . 4 $/\$ $/\$ $\$ $\$ $/\$ $\$ $/\$ $\$
80	5, 77, 79	mpbir2and 957	. . 3 $/\$ $/\$ $\$
81	4, 80	eqeltrrd 2702	. 2 $/\$ $/\$ $\$
82	1	mopntop 22245	. . . 4
83	82	3ad2ant1 1082	. . 3 $/\$ $/\$
84		ssrab2 3687	. . . . 5
85	16, 84	eqsstri 3635	. . . 4
86	85, 3	syl5sseq 3653	. . 3 $/\$ $/\$
87		eqid 2622	. . . 4
88	87	iscld2 20832	. . 3 $/\$ $\$
89	83, 86, 88	syl2anc 693	. 2 $/\$ $/\$ $\$
90	81, 89	mpbird 247	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913

crab 2916 $\$ cdif 3571

cin 3573

wss 3574

c0 3915

cuni 4436 class class class wbr 4653

crn 5115

cfv 5888 (class class class)co 6650

cr 9935

cc0 9936

cxr 10073

clt 10074

cle 10075

cq 11788

cxne 11943

cxad 11944

cxmt 19731

cbl 19733

cmopn 19736

ctop 20698

ccld 20820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823

This theorem is referenced by: blcls 22311 lmle 23099 minveclem4 23203 lhop1lem 23776 ftalem3 24801 ubthlem1 27726

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