Theorem bcthlem9 8007

Description: Lemma for bcth 8032. If M is rare in X, the intersection of the complement of its closure with any ball is nonempty and open. (Use bcthlem8 8006 for existence of an included ball.)

Hypotheses

Ref	Expression
bcthlem6.1	|- D e. CMet
bcthlem6.3	|- X = dom dom D
bcthlem6.4	|- J = (Open` D)
bcthlem9.5	|- O = ((X \ ((cls` J)` M)) i^i (P( ball ` D)(R / 2)))

Assertion

Ref	Expression
bcthlem9	|- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ R e. RR /\ 0 < R)) -> (O =/= (/) /\ O e. J))

Proof of Theorem bcthlem9

Step	Hyp	Ref	Expression
1		bcthlem6.1	. . . . 5 |- D e. CMet
2		bcthlem6.3	. . . . 5 |- X = dom dom D
3		bcthlem6.4	. . . . 5 |- J = (Open` D)
4	1, 2, 3	bcthlem7 8005	. . . 4 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> -. (P( ball ` D)(R / 2)) (_ ((cls` J)` M))
5	1	cmsmeti 7962	. . . . . . . . . . 11 |- D e. Met
6	2	blssm 7850	. . . . . . . . . . 11 |- (((D e. Met /\ P e. X) /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) (_ X)
7	5, 6	mpanl1 706	. . . . . . . . . 10 |- ((P e. X /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) (_ X)
8	7	3impb 829	. . . . . . . . 9 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (P( ball ` D)(R / 2)) (_ X)
9		df-ss 2053	. . . . . . . . 9 |- ((P( ball ` D)(R / 2)) (_ X <-> ((P( ball ` D)(R / 2)) i^i X) = (P( ball ` D)(R / 2)))
10	8, 9	sylib 198	. . . . . . . 8 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> ((P( ball ` D)(R / 2)) i^i X) = (P( ball ` D)(R / 2)))
11	10	sseq1d 2088	. . . . . . 7 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (((P( ball ` D)(R / 2)) i^i X) (_ ((cls` J)` M) <-> (P( ball ` D)(R / 2)) (_ ((cls` J)` M)))
12	11	negbid 611	. . . . . 6 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (-. ((P( ball ` D)(R / 2)) i^i X) (_ ((cls` J)` M) <-> -. (P( ball ` D)(R / 2)) (_ ((cls` J)` M)))
13		inssdif0 2333	. . . . . . . 8 |- (((P( ball ` D)(R / 2)) i^i X) (_ ((cls` J)` M) <-> ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M))) = (/))
14		bcthlem9.5	. . . . . . . . . 10 |- O = ((X \ ((cls` J)` M)) i^i (P( ball ` D)(R / 2)))
15		incom 2208	. . . . . . . . . 10 |- ((X \ ((cls` J)` M)) i^i (P( ball ` D)(R / 2))) = ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M)))
16	14, 15	eqtr 1495	. . . . . . . . 9 |- O = ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M)))
17	16	eqeq1i 1482	. . . . . . . 8 |- (O = (/) <-> ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M))) = (/))
18	13, 17	bitr4 176	. . . . . . 7 |- (((P( ball ` D)(R / 2)) i^i X) (_ ((cls` J)` M) <-> O = (/))
19	18	necon3bbii 1597	. . . . . 6 |- (-. ((P( ball ` D)(R / 2)) i^i X) (_ ((cls` J)` M) <-> O =/= (/))
20	12, 19	syl5bbr 534	. . . . 5 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (O =/= (/) <-> -. (P( ball ` D)(R / 2)) (_ ((cls` J)` M)))
21	20	adantl 388	. . . 4 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> (O =/= (/) <-> -. (P( ball ` D)(R / 2)) (_ ((cls` J)` M)))
22	4, 21	mpbird 196	. . 3 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> O =/= (/))
23	3	opnin 7869	. . . . . . 7 |- ((D e. Met /\ (X \ ((cls` J)` M)) e. J /\ (P( ball ` D)(R / 2)) e. J) -> ((X \ ((cls` J)` M)) i^i (P( ball ` D)(R / 2))) e. J)
24	5, 23	mp3an1 903	. . . . . 6 |- (((X \ ((cls` J)` M)) e. J /\ (P( ball ` D)(R / 2)) e. J) -> ((X \ ((cls` J)` M)) i^i (P( ball ` D)(R / 2))) e. J)
25	24, 14	syl5eqel 1552	. . . . 5 |- (((X \ ((cls` J)` M)) e. J /\ (P( ball ` D)(R / 2)) e. J) -> O e. J)
26	1, 2, 3	bcthlem6 8004	. . . . . 6 |- J e. Top
27	1, 2, 3	bcthlem5 8003	. . . . . . 7 |- X = U.J
28	27	cmclsopn 7693	. . . . . 6 |- ((J e. Top /\ M (_ X) -> (X \ ((cls` J)` M)) e. J)
29	26, 28	mpan 695	. . . . 5 |- (M (_ X -> (X \ ((cls` J)` M)) e. J)
30	2, 3	blopn 7876	. . . . . . 7 |- (((D e. Met /\ P e. X) /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) e. J)
31	5, 30	mpanl1 706	. . . . . 6 |- ((P e. X /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) e. J)
32	31	3impb 829	. . . . 5 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (P( ball ` D)(R / 2)) e. J)
33	25, 29, 32	syl2an 454	. . . 4 |- ((M (_ X /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> O e. J)
34	33	adantlr 393	. . 3 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> O e. J)
35	22, 34	jca 288	. 2 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> (O =/= (/) /\ O e. J))
36		3simp1 788	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> P e. X)
37		rehalfclt 6034	. . . 4 |- (R e. RR -> (R / 2) e. RR)
38	37	3ad2ant2 801	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> (R / 2) e. RR)
39		halfpos2t 6037	. . . . 5 |- (R e. RR -> (0 < R <-> 0 < (R / 2)))
40	39	biimpa 416	. . . 4 |- ((R e. RR /\ 0 < R) -> 0 < (R / 2))
41	40	3adant1 797	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> 0 < (R / 2))
42	36, 38, 41	3jca 819	. 2 |- ((P e. X /\ R e. RR /\ 0 < R) -> (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)))
43	35, 42	sylan2 451	1 |- (((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ R e. RR /\ 0 < R)) -> (O =/= (/) /\ O e. J))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280   class class class wbr 2619  dom cdm 3170  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234   / cdiv 5294   < clt 5486  2c2 5961  Topctop 7588  intcnt 7661  clsccl 7662  Metcme 7789   ball cbl 7791  Opencopn 7792  CMetcms 7921

This theorem is referenced by:  bcthlem14 8012

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-div 5703  df-2 5970  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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