metdseq0 - Metamath Proof Explorer

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Theorem metdseq0 22657

Description: The distance from a point to a set is zero iff the point is in the closure set. (Contributed by Mario Carneiro, 14-Feb-2015.)

**Hypotheses**
Ref	Expression
metdscn.f	inf
metdscn.j

**Assertion**
Ref	Expression
metdseq0	$/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

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Proof of Theorem metdseq0 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll1 1100	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
2		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
3		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
4		metdscn.j	. . . . . . . 8
5	4	mopni2 22298	. . . . . . 7 $/\$ $/\$
6	1, 2, 3, 5	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
7		simprr 796	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		ssrin 3838	. . . . . . . 8
9	7, 8	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		rpgt0 11844	. . . . . . . . . 10
11		0re 10040	. . . . . . . . . . 11
12		rpre 11839	. . . . . . . . . . 11
13		ltnle 10117	. . . . . . . . . . 11 $/\$
14	11, 12, 13	sylancr 695	. . . . . . . . . 10
15	10, 14	mpbid 222	. . . . . . . . 9
16	15	ad2antrl 764	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		simpllr 799	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	17	breq2d 4665	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	1	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simpl2 1065	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
21	20	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simpl3 1066	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
23	22	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		rpxr 11840	. . . . . . . . . . . . 13
25	24	ad2antrl 764	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		metdscn.f	. . . . . . . . . . . . 13 inf
27	26	metdsge 22652	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28	19, 21, 23, 25, 27	syl31anc 1329	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	18, 28	bitr3d 270	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		incom 3805	. . . . . . . . . . 11
31	30	eqeq1i 2627	. . . . . . . . . 10
32	29, 31	syl6bb 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	necon3bbid 2831	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	16, 33	mpbid 222	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		ssn0 3976	. . . . . . 7 $/\$
36	9, 34, 35	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 36	rexlimddv 3035	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38	37	expr 643	. . . 4 $/\$ $/\$ $/\$ $/\$
39	38	ralrimiva 2966	. . 3 $/\$ $/\$ $/\$
40	4	mopntopon 22244	. . . . . . 7 TopOn
41	40	3ad2ant1 1082	. . . . . 6 $/\$ $/\$ TopOn
42	41	adantr 481	. . . . 5 $/\$ $/\$ $/\$ TopOn
43		topontop 20718	. . . . 5 TopOn
44	42, 43	syl 17	. . . 4 $/\$ $/\$ $/\$
45		toponuni 20719	. . . . . 6 TopOn
46	42, 45	syl 17	. . . . 5 $/\$ $/\$ $/\$
47	20, 46	sseqtrd 3641	. . . 4 $/\$ $/\$ $/\$
48	22, 46	eleqtrd 2703	. . . 4 $/\$ $/\$ $/\$
49		eqid 2622	. . . . 5
50	49	elcls 20877	. . . 4 $/\$ $/\$
51	44, 47, 48, 50	syl3anc 1326	. . 3 $/\$ $/\$ $/\$
52	39, 51	mpbird 247	. 2 $/\$ $/\$ $/\$
53		incom 3805	. . . . . . 7
54	26	metdsf 22651	. . . . . . . . . . . 12 $/\$
55	54	ffvelrnda 6359	. . . . . . . . . . 11 $/\$ $/\$
56	55	3impa 1259	. . . . . . . . . 10 $/\$ $/\$
57		elxrge0 12281	. . . . . . . . . . 11 $/\$
58	57	simplbi 476	. . . . . . . . . 10
59	56, 58	syl 17	. . . . . . . . 9 $/\$ $/\$
60		xrleid 11983	. . . . . . . . 9
61	59, 60	syl 17	. . . . . . . 8 $/\$ $/\$
62	26	metdsge 22652	. . . . . . . . 9 $/\$ $/\$ $/\$
63	59, 62	mpdan 702	. . . . . . . 8 $/\$ $/\$
64	61, 63	mpbid 222	. . . . . . 7 $/\$ $/\$
65	53, 64	syl5eq 2668	. . . . . 6 $/\$ $/\$
66	65	adantr 481	. . . . 5 $/\$ $/\$ $/\$
67	41	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ TopOn
68	67, 43	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
69		simpll2 1101	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
70	67, 45	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
71	69, 70	sseqtrd 3641	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
72		simplr 792	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
73		simpll1 1100	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
74		simpll3 1102	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
75	59	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
76	4	blopn 22305	. . . . . . . . 9 $/\$ $/\$
77	73, 74, 75, 76	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
78		simpr 477	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
79		xblcntr 22216	. . . . . . . . 9 $/\$ $/\$ $/\$
80	73, 74, 75, 78, 79	syl112anc 1330	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
81	49	clsndisj 20879	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
82	68, 71, 72, 77, 80, 81	syl32anc 1334	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
83	82	ex 450	. . . . . 6 $/\$ $/\$ $/\$
84	83	necon2bd 2810	. . . . 5 $/\$ $/\$ $/\$
85	66, 84	mpd 15	. . . 4 $/\$ $/\$ $/\$
86	57	simprbi 480	. . . . . . . 8
87	56, 86	syl 17	. . . . . . 7 $/\$ $/\$
88		0xr 10086	. . . . . . . 8
89		xrleloe 11977	. . . . . . . 8 $/\$ $\/$
90	88, 59, 89	sylancr 695	. . . . . . 7 $/\$ $/\$ $\/$
91	87, 90	mpbid 222	. . . . . 6 $/\$ $/\$ $\/$
92	91	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $\/$
93	92	ord 392	. . . 4 $/\$ $/\$ $/\$
94	85, 93	mpd 15	. . 3 $/\$ $/\$ $/\$
95	94	eqcomd 2628	. 2 $/\$ $/\$ $/\$
96	52, 95	impbida 877	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

cin 3573

wss 3574

c0 3915

cuni 4436 class class class wbr 4653

cmpt 4729

crn 5115

cfv 5888 (class class class)co 6650 infcinf 8347

cr 9935

cc0 9936

cpnf 10071

cxr 10073

clt 10074

cle 10075

crp 11832

cicc 12178

cxmt 19731

cbl 19733

cmopn 19736

ctop 20698 TopOnctopon 20715

ccl 20822

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-rep 4771 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-int 4476 df-iun 4522 df-iin 4523 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-icc 12182 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 df-ntr 20824 df-cls 20825

This theorem is referenced by: metnrmlem1a 22661 lebnumlem1 22760

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