metdseq0 - Metamath Proof Explorer

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Theorem metdseq0 18837

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
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 996	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
2		simprl 733	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
3		simprr 734	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
4		metdscn.j	. . . . . . . 8
5	4	mopni2 18476	. . . . . . 7 $/\$ $/\$
6	1, 2, 3, 5	syl3anc 1184	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
7		simprr 734	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		ssrin 3526	. . . . . . . 8
9	7, 8	syl 16	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		rpgt0 10579	. . . . . . . . . 10
11		0re 9047	. . . . . . . . . . 11
12		rpre 10574	. . . . . . . . . . 11
13		ltnle 9111	. . . . . . . . . . 11 $/\$
14	11, 12, 13	sylancr 645	. . . . . . . . . 10
15	10, 14	mpbid 202	. . . . . . . . 9
16	15	ad2antrl 709	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		simpllr 736	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	17	breq2d 4184	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	1	adantr 452	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simpl2 961	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
21	20	ad2antrr 707	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simpl3 962	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
23	22	ad2antrr 707	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		rpxr 10575	. . . . . . . . . . . . 13
25	24	ad2antrl 709	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		metdscn.f	. . . . . . . . . . . . 13
27	26	metdsge 18832	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28	19, 21, 23, 25, 27	syl31anc 1187	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	18, 28	bitr3d 247	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		incom 3493	. . . . . . . . . . 11
31	30	eqeq1i 2411	. . . . . . . . . 10
32	29, 31	syl6bb 253	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	necon3bbid 2601	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	16, 33	mpbid 202	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		ssn0 3620	. . . . . . 7 $/\$
36	9, 34, 35	syl2anc 643	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 36	rexlimddv 2794	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38	37	expr 599	. . . 4 $/\$ $/\$ $/\$ $/\$
39	38	ralrimiva 2749	. . 3 $/\$ $/\$ $/\$
40	4	mopntopon 18422	. . . . . . 7 TopOn
41	40	3ad2ant1 978	. . . . . 6 $/\$ $/\$ TopOn
42	41	adantr 452	. . . . 5 $/\$ $/\$ $/\$ TopOn
43		topontop 16946	. . . . 5 TopOn
44	42, 43	syl 16	. . . 4 $/\$ $/\$ $/\$
45		toponuni 16947	. . . . . 6 TopOn
46	42, 45	syl 16	. . . . 5 $/\$ $/\$ $/\$
47	20, 46	sseqtrd 3344	. . . 4 $/\$ $/\$ $/\$
48	22, 46	eleqtrd 2480	. . . 4 $/\$ $/\$ $/\$
49		eqid 2404	. . . . 5
50	49	elcls 17092	. . . 4 $/\$ $/\$
51	44, 47, 48, 50	syl3anc 1184	. . 3 $/\$ $/\$ $/\$
52	39, 51	mpbird 224	. 2 $/\$ $/\$ $/\$
53		incom 3493	. . . . . . 7
54	26	metdsf 18831	. . . . . . . . . . . 12 $/\$
55	54	ffvelrnda 5829	. . . . . . . . . . 11 $/\$ $/\$
56	55	3impa 1148	. . . . . . . . . 10 $/\$ $/\$
57		elxrge0 10964	. . . . . . . . . . 11 $/\$
58	57	simplbi 447	. . . . . . . . . 10
59	56, 58	syl 16	. . . . . . . . 9 $/\$ $/\$
60		xrleid 10699	. . . . . . . . 9
61	59, 60	syl 16	. . . . . . . 8 $/\$ $/\$
62	26	metdsge 18832	. . . . . . . . 9 $/\$ $/\$ $/\$
63	59, 62	mpdan 650	. . . . . . . 8 $/\$ $/\$
64	61, 63	mpbid 202	. . . . . . 7 $/\$ $/\$
65	53, 64	syl5eq 2448	. . . . . 6 $/\$ $/\$
66	65	adantr 452	. . . . 5 $/\$ $/\$ $/\$
67	41	ad2antrr 707	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ TopOn
68	67, 43	syl 16	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
69		simpll2 997	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
70	67, 45	syl 16	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
71	69, 70	sseqtrd 3344	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
72		simplr 732	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
73		simpll1 996	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
74		simpll3 998	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
75	59	ad2antrr 707	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
76	4	blopn 18483	. . . . . . . . 9 $/\$ $/\$
77	73, 74, 75, 76	syl3anc 1184	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
78		simpr 448	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
79		xblcntr 18394	. . . . . . . . 9 $/\$ $/\$ $/\$
80	73, 74, 75, 78, 79	syl112anc 1188	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
81	49	clsndisj 17094	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
82	68, 71, 72, 77, 80, 81	syl32anc 1192	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
83	82	ex 424	. . . . . 6 $/\$ $/\$ $/\$
84	83	necon2bd 2616	. . . . 5 $/\$ $/\$ $/\$
85	66, 84	mpd 15	. . . 4 $/\$ $/\$ $/\$
86	57	simprbi 451	. . . . . . . 8
87	56, 86	syl 16	. . . . . . 7 $/\$ $/\$
88		0xr 9087	. . . . . . . 8
89		xrleloe 10693	. . . . . . . 8 $/\$ $\/$
90	88, 59, 89	sylancr 645	. . . . . . 7 $/\$ $/\$ $\/$
91	87, 90	mpbid 202	. . . . . 6 $/\$ $/\$ $\/$
92	91	adantr 452	. . . . 5 $/\$ $/\$ $/\$ $\/$
93	92	ord 367	. . . 4 $/\$ $/\$ $/\$
94	85, 93	mpd 15	. . 3 $/\$ $/\$ $/\$
95	94	eqcomd 2409	. 2 $/\$ $/\$ $/\$
96	52, 95	impbida 806	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 177 $\/$ wo 358 $/\$ wa 359 $/\$ w3a 936

wceq 1649

wcel 1721

wne 2567

wral 2666

wrex 2667

cin 3279

wss 3280

c0 3588

cuni 3975 class class class wbr 4172

cmpt 4226

ccnv 4836

crn 4838

cfv 5413 (class class class)co 6040

csup 7403

cr 8945

cc0 8946

cpnf 9073

cxr 9075

clt 9076

cle 9077

crp 10568

cicc 10875

cxmt 16641

cbl 16643

cmopn 16646

ctop 16913 TopOnctopon 16914

ccl 17037

This theorem is referenced by: metnrmlem1a 18841 lebnumlem1 18939

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-cnex 9002 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 ax-pre-sup 9024

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-pss 3296 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-tp 3782 df-op 3783 df-uni 3976 df-int 4011 df-iun 4055 df-iin 4056 df-br 4173 df-opab 4227 df-mpt 4228 df-tr 4263 df-eprel 4454 df-id 4458 df-po 4463 df-so 4464 df-fr 4501 df-we 4503 df-ord 4544 df-on 4545 df-lim 4546 df-suc 4547 df-om 4805 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-riota 6508 df-recs 6592 df-rdg 6627 df-er 6864 df-map 6979 df-en 7069 df-dom 7070 df-sdom 7071 df-sup 7404 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 df-div 9634 df-nn 9957 df-2 10014 df-n0 10178 df-z 10239 df-uz 10445 df-q 10531 df-rp 10569 df-xneg 10666 df-xadd 10667 df-xmul 10668 df-icc 10879 df-topgen 13622 df-psmet 16649 df-xmet 16650 df-bl 16652 df-mopn 16653 df-top 16918 df-bases 16920 df-topon 16921 df-cld 17038 df-ntr 17039 df-cls 17040

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