Theorem bcthlem29 9305

Description: Lemma for bcth 9310. Therefore the union of all members of reference sequence M does not occupy the entire metric space X. Also, use metric space completeness (via bcthlem23 9299) to eliminate the limit point q from the antecedents.

Hypotheses

Ref	Expression
bcthlem29.1	|- D e. CMet
bcthlem29.3	|- X = dom dom D
bcthlem29.4	|- J = (Open` D)
bcthlem29.5	|- M:NN-->~PX
bcthlem29.6	|- F = {<.<.j, y>., z>. | ((j e. NN /\ y e. A) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) C_ O))})}
bcthlem29.7	|- A = (X X. {x e. RR | 0 < x})
bcthlem29.8	|- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))

Assertion

Ref	Expression
bcthlem29	|- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1)))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> -. U.ran M = X)

Distinct variable groups:   g,j,k,p,y,z,A   g,r,x,D,j,k,p,y,z   g,F,k   g,J,j,p,r,x,y,z   g,M,j,p,r,x,y,z   O,p,r,z   Q,g   g,X,j,k,p,r,x,y,z

Proof of Theorem bcthlem29

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . 7 |- ((g` 1) = Q -> (2nd` (g` 1)) = (2nd` Q))
2	1	breq1d 3348	. . . . . 6 |- ((g` 1) = Q -> ((2nd` (g` 1)) < (1 / 2) <-> (2nd` Q) < (1 / 2)))
3		bcthlem29.1	. . . . . . . . 9 |- D e. CMet
4		bcthlem29.3	. . . . . . . . 9 |- X = dom dom D
5		bcthlem29.6	. . . . . . . . 9 |- F = {<.<.j, y>., z>. | ((j e. NN /\ y e. A) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) C_ O))})}
6		bcthlem29.7	. . . . . . . . 9 |- A = (X X. {x e. RR | 0 < x})
7		bcthlem29.8	. . . . . . . . 9 |- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))
8	3, 4, 5, 6, 7	bcthlem23 9299	. . . . . . . 8 |- ((g:NN-->A /\ ((2nd` (g` 1)) < (1 / 2) /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))) -> E.q e. X (1st o. g)(~~>m` D)q)
9	8	exp32 408	. . . . . . 7 |- (g:NN-->A -> ((2nd` (g` 1)) < (1 / 2) -> (A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)) -> E.q e. X (1st o. g)(~~>m` D)q)))
10	9	com12 14	. . . . . 6 |- ((2nd` (g` 1)) < (1 / 2) -> (g:NN-->A -> (A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)) -> E.q e. X (1st o. g)(~~>m` D)q)))
11	2, 10	syl6bir 232	. . . . 5 |- ((g` 1) = Q -> ((2nd` Q) < (1 / 2) -> (g:NN-->A -> (A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)) -> E.q e. X (1st o. g)(~~>m` D)q))))
12	11	com3l 38	. . . 4 |- ((2nd` Q) < (1 / 2) -> (g:NN-->A -> ((g` 1) = Q -> (A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)) -> E.q e. X (1st o. g)(~~>m` D)q))))
13	12	imp45 399	. . 3 |- (((2nd` Q) < (1 / 2) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> E.q e. X (1st o. g)(~~>m` D)q)
14	13	adantlr 429	. 2 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1)))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> E.q e. X (1st o. g)(~~>m` D)q)
15		bcthlem29.4	. . . . . . . . . 10 |- J = (Open` D)
16		bcthlem29.5	. . . . . . . . . 10 |- M:NN-->~PX
17	3, 4, 15, 16, 5, 6, 7	bcthlem28 9304	. . . . . . . . 9 |- ((((q e. X /\ ((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1)))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) /\ (1st o. g)(~~>m` D)q) -> -. E.m e. NN q e. (M` m))
18	17	exp41 413	. . . . . . . 8 |- (q e. X -> (((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1))) -> ((g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))) -> ((1st o. g)(~~>m` D)q -> -. E.m e. NN q e. (M` m)))))
19	18	imp4c 393	. . . . . . 7 |- (q e. X -> (((((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) /\ (1st o. g)(~~>m` D)q) -> -. E.m e. NN q e. (M` m)))
20		eleq2 1958	. . . . . . . . . 10 |- (U.ran M = X -> (q e. U.ran M <-> q e. X))
21	20	biimpar 461	. . . . . . . . 9 |- ((U.ran M = X /\ q e. X) -> q e. U.ran M)
22		ffun 4565	. . . . . . . . . . . 12 |- (M:NN-->~PX -> Fun M)
23	16, 22	ax-mp 7	. . . . . . . . . . 11 |- Fun M
24		elunirn 4844	. . . . . . . . . . 11 |- (Fun M -> (q e. U.ran M <-> E.m e. dom M q e. (M` m)))
25	23, 24	ax-mp 7	. . . . . . . . . 10 |- (q e. U.ran M <-> E.m e. dom M q e. (M` m))
26	16	fdmi 4568	. . . . . . . . . . 11 |- dom M = NN
27		rexeq 2267	. . . . . . . . . . 11 |- (dom M = NN -> (E.m e. dom M q e. (M` m) <-> E.m e. NN q e. (M` m)))
28	26, 27	ax-mp 7	. . . . . . . . . 10 |- (E.m e. dom M q e. (M` m) <-> E.m e. NN q e. (M` m))
29	25, 28	bitri 190	. . . . . . . . 9 |- (q e. U.ran M <-> E.m e. NN q e. (M` m))
30	21, 29	sylib 215	. . . . . . . 8 |- ((U.ran M = X /\ q e. X) -> E.m e. NN q e. (M` m))
31	30	expcom 403	. . . . . . 7 |- (q e. X -> (U.ran M = X -> E.m e. NN q e. (M` m)))
32	19, 31	nsyld 132	. . . . . 6 |- (q e. X -> (((((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) /\ (1st o. g)(~~>m` D)q) -> -. U.ran M = X))
33	32	exp3a 405	. . . . 5 |- (q e. X -> ((((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> ((1st o. g)(~~>m` D)q -> -. U.ran M = X)))
34	33	com12 14	. . . 4 |- ((((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> (q e. X -> ((1st o. g)(~~>m` D)q -> -. U.ran M = X)))
35	34	r19.23adv 2215	. . 3 |- ((((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> (E.q e. X (1st o. g)(~~>m` D)q -> -. U.ran M = X))
36	35	adantll 428	. 2 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1)))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> (E.q e. X (1st o. g)(~~>m` D)q -> -. U.ran M = X))
37	14, 36	mpd 29	1 |- ((((2nd` Q) < (1 / 2) /\ ((1st` Q)( ball ` D)(2nd` Q)) C_ (X \ ((cls` J)` (M` 1)))) /\ (g:NN-->A /\ ((g` 1) = Q /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))) -> -. U.ran M = X)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  {crab 2108   \ cdif 2590   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  U.cuni 3177   class class class wbr 3338  {copab 3395   X. cxp 3984  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   / cdiv 6447  NNcn 6449   < clt 6653  2c2 7145  clsccl 8938   ball cbl 9068  Opencopn 9069  ~~>mclm 9197  CMetcms 9199

This theorem is referenced by:  bcthlem31 9307

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-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-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-cls 8941  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-cau 9201  df-cmet 9202

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