Theorem bcthlem7 8005

Description: Lemma for bcth 8032. If M is rare in X, i.e. the interior of its closure is empty, then its closure does not include any ball.

Hypotheses

Ref	Expression
bcthlem6.1	|- D e. CMet
bcthlem6.3	|- X = dom dom D
bcthlem6.4	|- J = (Open` D)

Assertion

Ref	Expression
bcthlem7	|- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ R e. RR /\ 0 < R)) -> -. (P( ball ` D)R) (_ ((cls` J)` M))

Proof of Theorem bcthlem7

Step	Hyp	Ref	Expression
1		bcthlem6.1	. . . . . . . 8 |- D e. CMet
2		bcthlem6.3	. . . . . . . 8 |- X = dom dom D
3		bcthlem6.4	. . . . . . . 8 |- J = (Open` D)
4	1, 2, 3	bcthlem6 8004	. . . . . . 7 |- J e. Top
5	1, 2, 3	bcthlem5 8003	. . . . . . . 8 |- X = U.J
6	5	clsss3 7691	. . . . . . 7 |- ((J e. Top /\ M (_ X) -> ((cls` J)` M) (_ X)
7	4, 6	mpan 695	. . . . . 6 |- (M (_ X -> ((cls` J)` M) (_ X)
8	5	ntreq0 7708	. . . . . . 7 |- ((J e. Top /\ ((cls` J)` M) (_ X) -> (((int` J)` ((cls` J)` M)) = (/) <-> A.x e. J (x (_ ((cls` J)` M) -> x = (/))))
9	4, 8	mpan 695	. . . . . 6 |- (((cls` J)` M) (_ X -> (((int` J)` ((cls` J)` M)) = (/) <-> A.x e. J (x (_ ((cls` J)` M) -> x = (/))))
10	7, 9	syl 10	. . . . 5 |- (M (_ X -> (((int` J)` ((cls` J)` M)) = (/) <-> A.x e. J (x (_ ((cls` J)` M) -> x = (/))))
11	10	biimpa 416	. . . 4 |- ((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) -> A.x e. J (x (_ ((cls` J)` M) -> x = (/)))
12		nne 1589	. . . . . . . 8 |- (-. x =/= (/) <-> x = (/))
13	12	imbi2i 185	. . . . . . 7 |- ((x (_ ((cls` J)` M) -> -. x =/= (/)) <-> (x (_ ((cls` J)` M) -> x = (/)))
14		imnan 242	. . . . . . 7 |- ((x (_ ((cls` J)` M) -> -. x =/= (/)) <-> -. (x (_ ((cls` J)` M) /\ x =/= (/)))
15	13, 14	bitr3 175	. . . . . 6 |- ((x (_ ((cls` J)` M) -> x = (/)) <-> -. (x (_ ((cls` J)` M) /\ x =/= (/)))
16	15	ralbii 1667	. . . . 5 |- (A.x e. J (x (_ ((cls` J)` M) -> x = (/)) <-> A.x e. J -. (x (_ ((cls` J)` M) /\ x =/= (/)))
17		ralnex 1653	. . . . 5 |- (A.x e. J -. (x (_ ((cls` J)` M) /\ x =/= (/)) <-> -. E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)))
18	16, 17	bitr 173	. . . 4 |- (A.x e. J (x (_ ((cls` J)` M) -> x = (/)) <-> -. E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)))
19	11, 18	sylib 198	. . 3 |- ((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) -> -. E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)))
20	19	adantr 389	. 2 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ R e. RR /\ 0 < R)) -> -. E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)))
21	1	cmsmeti 7962	. . . . . 6 |- D e. Met
22		sseq1 2082	. . . . . . . . . 10 |- (x = (P( ball ` D)R) -> (x (_ ((cls` J)` M) <-> (P( ball ` D)R) (_ ((cls` J)` M)))
23		neeq1 1590	. . . . . . . . . 10 |- (x = (P( ball ` D)R) -> (x =/= (/) <-> (P( ball ` D)R) =/= (/)))
24	22, 23	anbi12d 628	. . . . . . . . 9 |- (x = (P( ball ` D)R) -> ((x (_ ((cls` J)` M) /\ x =/= (/)) <-> ((P( ball ` D)R) (_ ((cls` J)` M) /\ (P( ball ` D)R) =/= (/))))
25	24	rcla4ev 1877	. . . . . . . 8 |- (((P( ball ` D)R) e. J /\ ((P( ball ` D)R) (_ ((cls` J)` M) /\ (P( ball ` D)R) =/= (/))) -> E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)))
26	2, 3	blopn 7876	. . . . . . . . 9 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R)) -> (P( ball ` D)R) e. J)
27	26	adantl 388	. . . . . . . 8 |- (((P( ball ` D)R) (_ ((cls` J)` M) /\ ((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R))) -> (P( ball ` D)R) e. J)
28	2	blne0 7846	. . . . . . . . 9 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R)) -> (P( ball ` D)R) =/= (/))
29	28	anim2i 335	. . . . . . . 8 |- (((P( ball ` D)R) (_ ((cls` J)` M) /\ ((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R))) -> ((P( ball ` D)R) (_ ((cls` J)` M) /\ (P( ball ` D)R) =/= (/)))
30	25, 27, 29	sylanc 471	. . . . . . 7 |- (((P( ball ` D)R) (_ ((cls` J)` M) /\ ((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R))) -> E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)))
31	30	expcom 374	. . . . . 6 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R)) -> ((P( ball ` D)R) (_ ((cls` J)` M) -> E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/))))
32	21, 31	mpanl1 706	. . . . 5 |- ((P e. X /\ (R e. RR /\ 0 < R)) -> ((P( ball ` D)R) (_ ((cls` J)` M) -> E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/))))
33	32	3impb 829	. . . 4 |- ((P e. X /\ R e. RR /\ 0 < R) -> ((P( ball ` D)R) (_ ((cls` J)` M) -> E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/))))
34	33	con3d 95	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> (-. E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)) -> -. (P( ball ` D)R) (_ ((cls` J)` M)))
35	34	adantl 388	. 2 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ R e. RR /\ 0 < R)) -> (-. E.x e. J (x (_ ((cls` J)` M) /\ x =/= (/)) -> -. (P( ball ` D)R) (_ ((cls` J)` M)))
36	20, 35	mpd 26	1 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ R e. RR /\ 0 < R)) -> -. (P( ball ` D)R) (_ ((cls` J)` M))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646   (_ wss 2047  (/)c0 2280   class class class wbr 2619  dom cdm 3170  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234   < clt 5486  Topctop 7588  intcnt 7661  clsccl 7662  Metcme 7789   ball cbl 7791  Opencopn 7792  CMetcms 7921

This theorem is referenced by:  bcthlem9 8007

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665  df-met 7793  df-bl 7795  df-opn 7796  df-cmet 7924

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