Theorem bcthlem9 9285

Description: Lemma for bcth 9310. If M is rare in X, the intersection of the complement of its closure with any ball is nonempty and open. (Use bcthlem8 9284 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 C_ 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 9283	. . . 4 |- (((M C_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> -. (P( ball ` D)(R / 2)) C_ ((cls` J)` M))
5	1	cmsmeti 9240	. . . . . . . . . . 11 |- D e. Met
6	2	blssm 9127	. . . . . . . . . . 11 |- (((D e. Met /\ P e. X) /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) C_ X)
7	5, 6	mpanl1 770	. . . . . . . . . 10 |- ((P e. X /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) C_ X)
8	7	3impb 1063	. . . . . . . . 9 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (P( ball ` D)(R / 2)) C_ X)
9		df-ss 2605	. . . . . . . . 9 |- ((P( ball ` D)(R / 2)) C_ X <-> ((P( ball ` D)(R / 2)) i^i X) = (P( ball ` D)(R / 2)))
10	8, 9	sylib 215	. . . . . . . 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 2644	. . . . . . 7 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (((P( ball ` D)(R / 2)) i^i X) C_ ((cls` J)` M) <-> (P( ball ` D)(R / 2)) C_ ((cls` J)` M)))
12	11	notbid 673	. . . . . 6 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (-. ((P( ball ` D)(R / 2)) i^i X) C_ ((cls` J)` M) <-> -. (P( ball ` D)(R / 2)) C_ ((cls` J)` M)))
13		inssdif0 2940	. . . . . . . 8 |- (((P( ball ` D)(R / 2)) i^i X) C_ ((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 2787	. . . . . . . . . 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	eqtri 1908	. . . . . . . . 9 |- O = ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M)))
17	16	eqeq1i 1891	. . . . . . . 8 |- (O = (/) <-> ((P( ball ` D)(R / 2)) i^i (X \ ((cls` J)` M))) = (/))
18	13, 17	bitr4i 193	. . . . . . 7 |- (((P( ball ` D)(R / 2)) i^i X) C_ ((cls` J)` M) <-> O = (/))
19	18	necon3bbii 2031	. . . . . 6 |- (-. ((P( ball ` D)(R / 2)) i^i X) C_ ((cls` J)` M) <-> O =/= (/))
20	12, 19	syl5bbr 593	. . . . 5 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (O =/= (/) <-> -. (P( ball ` D)(R / 2)) C_ ((cls` J)` M)))
21	20	adantl 424	. . . 4 |- (((M C_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> (O =/= (/) <-> -. (P( ball ` D)(R / 2)) C_ ((cls` J)` M)))
22	4, 21	mpbird 213	. . 3 |- (((M C_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> O =/= (/))
23	3	opnin 9146	. . . . . . 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 1178	. . . . . 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 1975	. . . . 5 |- (((X \ ((cls` J)` M)) e. J /\ (P( ball ` D)(R / 2)) e. J) -> O e. J)
26	1, 2, 3	bcthlem6 9282	. . . . . 6 |- J e. Top
27	1, 2, 3	bcthlem5 9281	. . . . . . 7 |- X = U.J
28	27	cmclsopn 8969	. . . . . 6 |- ((J e. Top /\ M C_ X) -> (X \ ((cls` J)` M)) e. J)
29	26, 28	mpan 759	. . . . 5 |- (M C_ X -> (X \ ((cls` J)` M)) e. J)
30	2, 3	blopn 9153	. . . . . . 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 770	. . . . . 6 |- ((P e. X /\ ((R / 2) e. RR /\ 0 < (R / 2))) -> (P( ball ` D)(R / 2)) e. J)
32	31	3impb 1063	. . . . 5 |- ((P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)) -> (P( ball ` D)(R / 2)) e. J)
33	25, 29, 32	syl2an 503	. . . 4 |- ((M C_ X /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> O e. J)
34	33	adantlr 429	. . 3 |- (((M C_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> O e. J)
35	22, 34	jca 310	. 2 |- (((M C_ X /\ ((int` J)` ((cls` J)` M)) = (/)) /\ (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2))) -> (O =/= (/) /\ O e. J))
36		simp1 876	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> P e. X)
37		rehalfcl 7220	. . . 4 |- (R e. RR -> (R / 2) e. RR)
38	37	3ad2ant2 898	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> (R / 2) e. RR)
39		halfpos2 7223	. . . . 5 |- (R e. RR -> (0 < R <-> 0 < (R / 2)))
40	39	biimpa 460	. . . 4 |- ((R e. RR /\ 0 < R) -> 0 < (R / 2))
41	40	3adant1 894	. . 3 |- ((P e. X /\ R e. RR /\ 0 < R) -> 0 < (R / 2))
42	36, 38, 41	3jca 1050	. 2 |- ((P e. X /\ R e. RR /\ 0 < R) -> (P e. X /\ (R / 2) e. RR /\ 0 < (R / 2)))
43	35, 42	sylan2 500	1 |- (((M C_ 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 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875   class class class wbr 3338  dom cdm 3986  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   / cdiv 6447   < clt 6653  2c2 7145  Topctop 8857  intcnt 8937  clsccl 8938  Metcme 9066   ball cbl 9068  Opencopn 9069  CMetcms 9199

This theorem is referenced by:  bcthlem14 9290

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-2 7154  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-met 9070  df-bl 9072  df-opn 9073  df-cmet 9202

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