Theorem bcthlem9 7969

Description: Lemma for bcth 7994. If M is rare in X, the intersection of the complement of its closure with any ball is nonempty and open. (Use bcthlem8 7968 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 7967	. . . 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 7924	. . . . . . . . . . 11 |- D e. Met
6	2	blssm 7812	. . . . . . . . . . 11 |- (((D e. Met /\ P e. X) /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) (_ X)
7	5, 6	mpanl1 705	. . . . . . . . . 10 |- ((P e. X /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) (_ X)
8	7	3impb 828	. . . . . . . . 9 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (P( ball ` D)(R / 2)) (_ X)
9		df-ss 2050	. . . . . . . . 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 2085	. . . . . . 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 610	. . . . . 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 2330	. . . . . . . 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 2205	. . . . . . . . . 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 1493	. . . . . . . . 9 |- O = ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M)))
17	16	eqeq1i 1480	. . . . . . . 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 1595	. . . . . 6 |- (-. ((P( ball ` D)(R / 2)) i^i X) (_ ((cls` J)` M) <-> O =/= (/))
20	12, 19	syl5bbr 533	. . . . 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 7831	. . . . . . 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 902	. . . . . 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 1550	. . . . 5 |- (((X \ ((cls` J)` M)) e. J /\ (P( ball ` D)(R / 2)) e. J) -> O e. J)
26	1, 2, 3	bcthlem6 7966	. . . . . 6 |- J e. Top
27	1, 2, 3	bcthlem5 7965	. . . . . . 7 |- X = U.J
28	27	cmclsopn 7653	. . . . . 6 |- ((J e. Top /\ M (_ X) -> (X \ ((cls` J)` M)) e. J)
29	26, 28	mpan 694	. . . . 5 |- (M (_ X -> (X \ ((cls` J)` M)) e. J)
30	2, 3	blopn 7838	. . . . . . 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 705	. . . . . 6 |- ((P e. X /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) e. J)
32	31	3impb 828	. . . . 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 787	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> P e. X)
37		rehalfclt 5991	. . . 4 |- (R e. RR -> (R / 2) e. RR)
38	37	3ad2ant2 800	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> (R / 2) e. RR)
39		halfpos2t 5994	. . . . 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 796	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> 0 < (R / 2))
42	36, 38, 41	3jca 818	. 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 774   = wceq 955   e. wcel 957   =/= wne 1583   \ cdif 2041   i^i cin 2043   (_ wss 2044  (/)c0 2277   class class class wbr 2615  dom cdm 3166  ` cfv 3178  (class class class)co 3958  RRcr 5216  0cc0 5217   / cdiv 5277   < clt 5469  2c2 5918  Topctop 7548  intcnt 7621  clsccl 7622  Metcme 7749   ball cbl 7751  Opencopn 7752  CMetcms 7883

This theorem is referenced by:  bcthlem14 7974

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-iin 2565  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-2 5927  df-top 7552  df-cld 7623  df-ntr 7624  df-cls 7625  df-met 7753  df-bl 7755  df-opn 7756  df-cmet 7886

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