Theorem bcthlem3 23123

Description: Lemma for bcth 23126. The limit point of the centers in the sequence is in the intersection of every ball in the sequence. (Contributed by Mario Carneiro, 7-Jan-2014.)

Hypotheses

Ref	Expression
bcth.2	|- J = ( MetOpen ` D )
bcthlem.4	|- ( ph -> D e. ( CMet ` X ) )
bcthlem.5	|- F = ( k e. NN , z e. ( X X. RR+ ) |-> { <. x , r >. | ( ( x e. X /\ r e. RR+ ) /\ ( r < ( 1 / k ) /\ ( ( cls ` J ) ` ( x ( ball ` D ) r ) ) C_ ( ( ( ball ` D ) ` z ) \ ( M ` k ) ) ) ) } )
bcthlem.6	|- ( ph -> M : NN --> ( Clsd ` J ) )
bcthlem.7	|- ( ph -> R e. RR+ )
bcthlem.8	|- ( ph -> C e. X )
bcthlem.9	|- ( ph -> g : NN --> ( X X. RR+ ) )
bcthlem.10	|- ( ph -> ( g ` 1 ) = <. C , R >. )
bcthlem.11	|- ( ph -> A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) )

Assertion

Ref	Expression
bcthlem3	|- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ A e. NN ) -> x e. ( ( ball ` D ) ` ( g ` A ) ) )

Distinct variable groups:   k, r, x, z, A   C, r, x   g, k, r, x, z, D   g, F, k, r, x, z   g, J, k, r, x, z   g, M, k, r, x, z   ph, k, r, x, z   x, R   g, X, k, r, x, z

Allowed substitution hints:   ph( g)   A( g)   C( z, g, k)   R( z, g, k, r)

Proof of Theorem bcthlem3

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . . . 7 |- ( ph -> A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) )
2		oveq1 6657	. . . . . . . . . 10 |- ( k = A -> ( k + 1 ) = ( A + 1 ) )
3	2	fveq2d 6195	. . . . . . . . 9 |- ( k = A -> ( g ` ( k + 1 ) ) = ( g ` ( A + 1 ) ) )
4		id 22	. . . . . . . . . 10 |- ( k = A -> k = A )
5		fveq2 6191	. . . . . . . . . 10 |- ( k = A -> ( g ` k ) = ( g ` A ) )
6	4, 5	oveq12d 6668	. . . . . . . . 9 |- ( k = A -> ( k F ( g ` k ) ) = ( A F ( g ` A ) ) )
7	3, 6	eleq12d 2695	. . . . . . . 8 |- ( k = A -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( A + 1 ) ) e. ( A F ( g ` A ) ) ) )
8	7	rspccva 3308	. . . . . . 7 |- ( ( A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) /\ A e. NN ) -> ( g ` ( A + 1 ) ) e. ( A F ( g ` A ) ) )
9	1, 8	sylan 488	. . . . . 6 |- ( ( ph /\ A e. NN ) -> ( g ` ( A + 1 ) ) e. ( A F ( g ` A ) ) )
10		bcthlem.9	. . . . . . . 8 |- ( ph -> g : NN --> ( X X. RR+ ) )
11	10	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ A e. NN ) -> ( g ` A ) e. ( X X. RR+ ) )
12		bcth.2	. . . . . . . . 9 |- J = ( MetOpen ` D )
13		bcthlem.4	. . . . . . . . 9 |- ( ph -> D e. ( CMet ` X ) )
14		bcthlem.5	. . . . . . . . 9 |- F = ( k e. NN , z e. ( X X. RR+ ) |-> { <. x , r >. | ( ( x e. X /\ r e. RR+ ) /\ ( r < ( 1 / k ) /\ ( ( cls ` J ) ` ( x ( ball ` D ) r ) ) C_ ( ( ( ball ` D ) ` z ) \ ( M ` k ) ) ) ) } )
15	12, 13, 14	bcthlem1 23121	. . . . . . . 8 |- ( ( ph /\ ( A e. NN /\ ( g ` A ) e. ( X X. RR+ ) ) ) -> ( ( g ` ( A + 1 ) ) e. ( A F ( g ` A ) ) <-> ( ( g ` ( A + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( A + 1 ) ) ) < ( 1 / A ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` A ) ) \ ( M ` A ) ) ) ) )
16	15	expr 643	. . . . . . 7 |- ( ( ph /\ A e. NN ) -> ( ( g ` A ) e. ( X X. RR+ ) -> ( ( g ` ( A + 1 ) ) e. ( A F ( g ` A ) ) <-> ( ( g ` ( A + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( A + 1 ) ) ) < ( 1 / A ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` A ) ) \ ( M ` A ) ) ) ) ) )
17	11, 16	mpd 15	. . . . . 6 |- ( ( ph /\ A e. NN ) -> ( ( g ` ( A + 1 ) ) e. ( A F ( g ` A ) ) <-> ( ( g ` ( A + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( A + 1 ) ) ) < ( 1 / A ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` A ) ) \ ( M ` A ) ) ) ) )
18	9, 17	mpbid 222	. . . . 5 |- ( ( ph /\ A e. NN ) -> ( ( g ` ( A + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( A + 1 ) ) ) < ( 1 / A ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` A ) ) \ ( M ` A ) ) ) )
19	18	simp3d 1075	. . . 4 |- ( ( ph /\ A e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` A ) ) \ ( M ` A ) ) )
20	19	difss2d 3740	. . 3 |- ( ( ph /\ A e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ball ` D ) ` ( g ` A ) ) )
21	20	3adant2 1080	. 2 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ A e. NN ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) C_ ( ( ball ` D ) ` ( g ` A ) ) )
22		peano2nn 11032	. . 3 |- ( A e. NN -> ( A + 1 ) e. NN )
23		cmetmet 23084	. . . . 5 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
24		metxmet 22139	. . . . 5 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
25	13, 23, 24	3syl 18	. . . 4 |- ( ph -> D e. ( *Met ` X ) )
26		bcthlem.6	. . . . 5 |- ( ph -> M : NN --> ( Clsd ` J ) )
27		bcthlem.7	. . . . 5 |- ( ph -> R e. RR+ )
28		bcthlem.8	. . . . 5 |- ( ph -> C e. X )
29		bcthlem.10	. . . . 5 |- ( ph -> ( g ` 1 ) = <. C , R >. )
30	12, 13, 14, 26, 27, 28, 10, 29, 1	bcthlem2 23122	. . . 4 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
31	25, 10, 30, 12	caublcls 23107	. . 3 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ ( A + 1 ) e. NN ) -> x e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) )
32	22, 31	syl3an3 1361	. 2 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ A e. NN ) -> x e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( A + 1 ) ) ) ) )
33	21, 32	sseldd 3604	1 |- ( ( ph /\ ( 1st o. g ) ( ~~> t ` J ) x /\ A e. NN ) -> x e. ( ( ball ` D ) ` ( g ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   1c1 9937   + caddc 9939   < clt 10074   / cdiv 10684   NNcn 11020   RR+crp 11832   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736   Clsdccld 20820   clsccl 20822   ~~> tclm 21030   CMetcms 23052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033  df-cmet 23055

This theorem is referenced by:  bcthlem4  23124