Theorem bcthlem33 9309

Description: Lemma for bcth 9310. All members of reference sequence M cannot have an empty interior.

Hypotheses

Ref	Expression
bcthlem34.1	|- D e. CMet
bcthlem34.2	|- X =/= (/)
bcthlem34.3	|- X = dom dom D
bcthlem34.4	|- J = (Open` D)
bcthlem34.5	|- M:NN-->~PX

Assertion

Ref	Expression
bcthlem33	|- ((U.ran M = X /\ ran M C_ (Clsd` J)) -> E.k e. NN ((int` J)` (M` k)) =/= (/))

Distinct variable groups:   D,k   k,J   k,M   k,X

Proof of Theorem bcthlem33

Step	Hyp	Ref	Expression
1		1nn 7117	. . . . . . . 8 |- 1 e. NN
2		fveq2 4681	. . . . . . . . . . . 12 |- (k = 1 -> (M` k) = (M` 1))
3	2	fveq2d 4685	. . . . . . . . . . 11 |- (k = 1 -> ((cls` J)` (M` k)) = ((cls` J)` (M` 1)))
4	3	fveq2d 4685	. . . . . . . . . 10 |- (k = 1 -> ((int` J)` ((cls` J)` (M` k))) = ((int` J)` ((cls` J)` (M` 1))))
5	4	eqeq1d 1892	. . . . . . . . 9 |- (k = 1 -> (((int` J)` ((cls` J)` (M` k))) = (/) <-> ((int` J)` ((cls` J)` (M` 1))) = (/)))
6	5	rcla4v 2376	. . . . . . . 8 |- (1 e. NN -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> ((int` J)` ((cls` J)` (M` 1))) = (/)))
7	1, 6	ax-mp 7	. . . . . . 7 |- (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> ((int` J)` ((cls` J)` (M` 1))) = (/))
8		bcthlem34.5	. . . . . . . . . . 11 |- M:NN-->~PX
9	8	ffvelrni 4788	. . . . . . . . . 10 |- (1 e. NN -> (M` 1) e. ~PX)
10	1, 9	ax-mp 7	. . . . . . . . 9 |- (M` 1) e. ~PX
11		elpwi 3039	. . . . . . . . 9 |- ((M` 1) e. ~PX -> (M` 1) C_ X)
12	10, 11	ax-mp 7	. . . . . . . 8 |- (M` 1) C_ X
13		bcthlem34.1	. . . . . . . . 9 |- D e. CMet
14		bcthlem34.3	. . . . . . . . 9 |- X = dom dom D
15		bcthlem34.4	. . . . . . . . 9 |- J = (Open` D)
16		bcthlem34.2	. . . . . . . . 9 |- X =/= (/)
17	13, 14, 15, 16	bcthlem10 9286	. . . . . . . 8 |- (((M` 1) C_ X /\ ((int` J)` ((cls` J)` (M` 1))) = (/)) -> ((X \ ((cls` J)` (M` 1))) =/= (/) /\ (X \ ((cls` J)` (M` 1))) e. J))
18	12, 17	mpan 759	. . . . . . 7 |- (((int` J)` ((cls` J)` (M` 1))) = (/) -> ((X \ ((cls` J)` (M` 1))) =/= (/) /\ (X \ ((cls` J)` (M` 1))) e. J))
19	13, 14, 15	bcthlem8 9284	. . . . . . . 8 |- (((X \ ((cls` J)` (M` 1))) =/= (/) /\ (X \ ((cls` J)` (M` 1))) e. J /\ 1 e. NN) -> E.q e. (X \ ((cls` J)` (M` 1)))E.s e. RR (0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))))
20	1, 19	mp3an3 1180	. . . . . . 7 |- (((X \ ((cls` J)` (M` 1))) =/= (/) /\ (X \ ((cls` J)` (M` 1))) e. J) -> E.q e. (X \ ((cls` J)` (M` 1)))E.s e. RR (0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))))
21	7, 18, 20	3syl 24	. . . . . 6 |- (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> E.q e. (X \ ((cls` J)` (M` 1)))E.s e. RR (0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))))
22	13, 14, 15, 8	bcthlem32 9308	. . . . . . . . . . . . . . 15 |- (((s < (1 / 2) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))) /\ (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) /\ (q e. X /\ (s e. RR /\ 0 < s)))) -> -. U.ran M = X)
23	22	exp43 415	. . . . . . . . . . . . . 14 |- (s < (1 / 2) -> ((q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1))) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> ((q e. X /\ (s e. RR /\ 0 < s)) -> -. U.ran M = X))))
24	23	com4r 45	. . . . . . . . . . . . 13 |- ((q e. X /\ (s e. RR /\ 0 < s)) -> (s < (1 / 2) -> ((q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1))) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> -. U.ran M = X))))
25	24	expr 418	. . . . . . . . . . . 12 |- ((q e. X /\ s e. RR) -> (0 < s -> (s < (1 / 2) -> ((q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1))) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> -. U.ran M = X)))))
26	25	com4l 43	. . . . . . . . . . 11 |- (0 < s -> (s < (1 / 2) -> ((q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1))) -> ((q e. X /\ s e. RR) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> -. U.ran M = X)))))
27		2cn 7164	. . . . . . . . . . . . . 14 |- 2 e. CC
28		exp1 7816	. . . . . . . . . . . . . 14 |- (2 e. CC -> (2^1) = 2)
29	27, 28	ax-mp 7	. . . . . . . . . . . . 13 |- (2^1) = 2
30	29	opreq2i 4893	. . . . . . . . . . . 12 |- (1 / (2^1)) = (1 / 2)
31	30	breq2i 3346	. . . . . . . . . . 11 |- (s < (1 / (2^1)) <-> s < (1 / 2))
32	26, 31	syl5ib 223	. . . . . . . . . 10 |- (0 < s -> (s < (1 / (2^1)) -> ((q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1))) -> ((q e. X /\ s e. RR) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> -. U.ran M = X)))))
33	32	3imp 1061	. . . . . . . . 9 |- ((0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))) -> ((q e. X /\ s e. RR) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> -. U.ran M = X)))
34	33	com13 37	. . . . . . . 8 |- (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> ((q e. X /\ s e. RR) -> ((0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))) -> -. U.ran M = X)))
35		eldifi 2730	. . . . . . . 8 |- (q e. (X \ ((cls` J)` (M` 1))) -> q e. X)
36	34, 35	sylani 513	. . . . . . 7 |- (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> ((q e. (X \ ((cls` J)` (M` 1))) /\ s e. RR) -> ((0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))) -> -. U.ran M = X)))
37	36	r19.23advv 2218	. . . . . 6 |- (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> (E.q e. (X \ ((cls` J)` (M` 1)))E.s e. RR (0 < s /\ s < (1 / (2^1)) /\ (q( ball ` D)s) C_ (X \ ((cls` J)` (M` 1)))) -> -. U.ran M = X))
38	21, 37	mpd 29	. . . . 5 |- (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) -> -. U.ran M = X)
39	38	con2i 113	. . . 4 |- (U.ran M = X -> -. A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/))
40	39	adantr 425	. . 3 |- ((U.ran M = X /\ ran M C_ (Clsd` J)) -> -. A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/))
41		cldcls 8958	. . . . . . . . 9 |- ((J e. Top /\ (M` k) e. (Clsd` J)) -> ((cls` J)` (M` k)) = (M` k))
42	13, 14, 15	bcthlem6 9282	. . . . . . . . 9 |- J e. Top
43		ffvelrn 4787	. . . . . . . . . 10 |- ((M:NN-->(Clsd` J) /\ k e. NN) -> (M` k) e. (Clsd` J))
44		ffn 4562	. . . . . . . . . . . . 13 |- (M:NN-->~PX -> M Fn NN)
45	8, 44	ax-mp 7	. . . . . . . . . . . 12 |- M Fn NN
46		dffn3 4570	. . . . . . . . . . . 12 |- (M Fn NN <-> M:NN-->ran M)
47	45, 46	mpbi 206	. . . . . . . . . . 11 |- M:NN-->ran M
48		fss 4571	. . . . . . . . . . 11 |- ((M:NN-->ran M /\ ran M C_ (Clsd` J)) -> M:NN-->(Clsd` J))
49	47, 48	mpan 759	. . . . . . . . . 10 |- (ran M C_ (Clsd` J) -> M:NN-->(Clsd` J))
50	43, 49	sylan 497	. . . . . . . . 9 |- ((ran M C_ (Clsd` J) /\ k e. NN) -> (M` k) e. (Clsd` J))
51	41, 42, 50	sylancr 526	. . . . . . . 8 |- ((ran M C_ (Clsd` J) /\ k e. NN) -> ((cls` J)` (M` k)) = (M` k))
52	51	fveq2d 4685	. . . . . . 7 |- ((ran M C_ (Clsd` J) /\ k e. NN) -> ((int` J)` ((cls` J)` (M` k))) = ((int` J)` (M` k)))
53	52	eqeq1d 1892	. . . . . 6 |- ((ran M C_ (Clsd` J) /\ k e. NN) -> (((int` J)` ((cls` J)` (M` k))) = (/) <-> ((int` J)` (M` k)) = (/)))
54	53	ralbidva 2119	. . . . 5 |- (ran M C_ (Clsd` J) -> (A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) <-> A.k e. NN ((int` J)` (M` k)) = (/)))
55	54	notbid 673	. . . 4 |- (ran M C_ (Clsd` J) -> (-. A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) <-> -. A.k e. NN ((int` J)` (M` k)) = (/)))
56	55	adantl 424	. . 3 |- ((U.ran M = X /\ ran M C_ (Clsd` J)) -> (-. A.k e. NN ((int` J)` ((cls` J)` (M` k))) = (/) <-> -. A.k e. NN ((int` J)` (M` k)) = (/)))
57	40, 56	mpbid 212	. 2 |- ((U.ran M = X /\ ran M C_ (Clsd` J)) -> -. A.k e. NN ((int` J)` (M` k)) = (/))
58		df-ne 2019	. . . 4 |- (((int` J)` (M` k)) =/= (/) <-> -. ((int` J)` (M` k)) = (/))
59	58	rexbii 2128	. . 3 |- (E.k e. NN ((int` J)` (M` k)) =/= (/) <-> E.k e. NN -. ((int` J)` (M` k)) = (/))
60		rexnal 2114	. . 3 |- (E.k e. NN -. ((int` J)` (M` k)) = (/) <-> -. A.k e. NN ((int` J)` (M` k)) = (/))
61	59, 60	bitr2i 191	. 2 |- (-. A.k e. NN ((int` J)` (M` k)) = (/) <-> E.k e. NN ((int` J)` (M` k)) =/= (/))
62	57, 61	sylib 215	1 |- ((U.ran M = X /\ ran M C_ (Clsd` J)) -> E.k e. NN ((int` J)` (M` k)) =/= (/))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   \ cdif 2590   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U.cuni 3177   class class class wbr 3338  dom cdm 3986  ran crn 3987   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   / cdiv 6447  NNcn 6449   < clt 6653  2c2 7145  ^cexp 7811  Topctop 8857  Clsdccld 8936  intcnt 8937  clsccl 8938   ball cbl 9068  Opencopn 9069  CMetcms 9199

This theorem is referenced by:  bcth 9310  ubthlem6 9877

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  ax-ac 5906

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-iso 4015  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-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-seq1 7721  df-exp 7812  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-cau 9201  df-cmet 9202

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