Theorem bcthlem2 23122

Description: Lemma for bcth 23126. The balls in the sequence form an inclusion chain. (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
bcthlem2	|- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )

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

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

Proof of Theorem bcthlem2

Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . 5 |- ( ph -> A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) )
2		oveq1 6657	. . . . . . . 8 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
3	2	fveq2d 6195	. . . . . . 7 |- ( k = n -> ( g ` ( k + 1 ) ) = ( g ` ( n + 1 ) ) )
4		id 22	. . . . . . . 8 |- ( k = n -> k = n )
5		fveq2 6191	. . . . . . . 8 |- ( k = n -> ( g ` k ) = ( g ` n ) )
6	4, 5	oveq12d 6668	. . . . . . 7 |- ( k = n -> ( k F ( g ` k ) ) = ( n F ( g ` n ) ) )
7	3, 6	eleq12d 2695	. . . . . 6 |- ( k = n -> ( ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) <-> ( g ` ( n + 1 ) ) e. ( n F ( g ` n ) ) ) )
8	7	rspccva 3308	. . . . 5 |- ( ( A. k e. NN ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) /\ n e. NN ) -> ( g ` ( n + 1 ) ) e. ( n F ( g ` n ) ) )
9	1, 8	sylan 488	. . . 4 |- ( ( ph /\ n e. NN ) -> ( g ` ( n + 1 ) ) e. ( n F ( g ` n ) ) )
10		bcthlem.9	. . . . . 6 |- ( ph -> g : NN --> ( X X. RR+ ) )
11	10	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( g ` n ) e. ( X X. RR+ ) )
12		bcth.2	. . . . . . 7 |- J = ( MetOpen ` D )
13		bcthlem.4	. . . . . . 7 |- ( ph -> D e. ( CMet ` X ) )
14		bcthlem.5	. . . . . . 7 |- 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	. . . . . 6 |- ( ( ph /\ ( n e. NN /\ ( g ` n ) e. ( X X. RR+ ) ) ) -> ( ( g ` ( n + 1 ) ) e. ( n F ( g ` n ) ) <-> ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) ) ) )
16	15	expr 643	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( ( g ` n ) e. ( X X. RR+ ) -> ( ( g ` ( n + 1 ) ) e. ( n F ( g ` n ) ) <-> ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) ) ) ) )
17	11, 16	mpd 15	. . . 4 |- ( ( ph /\ n e. NN ) -> ( ( g ` ( n + 1 ) ) e. ( n F ( g ` n ) ) <-> ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) ) ) )
18	9, 17	mpbid 222	. . 3 |- ( ( ph /\ n e. NN ) -> ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) ) )
19		cmetmet 23084	. . . . . . . . . . . . 13 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
20	13, 19	syl 17	. . . . . . . . . . . 12 |- ( ph -> D e. ( Met ` X ) )
21		metxmet 22139	. . . . . . . . . . . 12 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
22	20, 21	syl 17	. . . . . . . . . . 11 |- ( ph -> D e. ( *Met ` X ) )
23	12	mopntop 22245	. . . . . . . . . . 11 |- ( D e. ( *Met ` X ) -> J e. Top )
24	22, 23	syl 17	. . . . . . . . . 10 |- ( ph -> J e. Top )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> J e. Top )
26		xp1st 7198	. . . . . . . . . . . 12 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( 1st ` ( g ` ( n + 1 ) ) ) e. X )
27		xp2nd 7199	. . . . . . . . . . . . 13 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( 2nd ` ( g ` ( n + 1 ) ) ) e. RR+ )
28	27	rpxrd 11873	. . . . . . . . . . . 12 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( 2nd ` ( g ` ( n + 1 ) ) ) e. RR* )
29	26, 28	jca 554	. . . . . . . . . . 11 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( ( 1st ` ( g ` ( n + 1 ) ) ) e. X /\ ( 2nd ` ( g ` ( n + 1 ) ) ) e. RR* ) )
30		blssm 22223	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ ( 1st ` ( g ` ( n + 1 ) ) ) e. X /\ ( 2nd ` ( g ` ( n + 1 ) ) ) e. RR* ) -> ( ( 1st ` ( g ` ( n + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( n + 1 ) ) ) ) C_ X )
31	30	3expb 1266	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ ( ( 1st ` ( g ` ( n + 1 ) ) ) e. X /\ ( 2nd ` ( g ` ( n + 1 ) ) ) e. RR* ) ) -> ( ( 1st ` ( g ` ( n + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( n + 1 ) ) ) ) C_ X )
32	22, 29, 31	syl2an 494	. . . . . . . . . 10 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> ( ( 1st ` ( g ` ( n + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( n + 1 ) ) ) ) C_ X )
33		1st2nd2 7205	. . . . . . . . . . . . 13 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( g ` ( n + 1 ) ) = <. ( 1st ` ( g ` ( n + 1 ) ) ) , ( 2nd ` ( g ` ( n + 1 ) ) ) >. )
34	33	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( n + 1 ) ) ) , ( 2nd ` ( g ` ( n + 1 ) ) ) >. ) )
35		df-ov 6653	. . . . . . . . . . . 12 |- ( ( 1st ` ( g ` ( n + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( n + 1 ) ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( g ` ( n + 1 ) ) ) , ( 2nd ` ( g ` ( n + 1 ) ) ) >. )
36	34, 35	syl6reqr 2675	. . . . . . . . . . 11 |- ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( ( 1st ` ( g ` ( n + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( n + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) )
37	36	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> ( ( 1st ` ( g ` ( n + 1 ) ) ) ( ball ` D ) ( 2nd ` ( g ` ( n + 1 ) ) ) ) = ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) )
38	12	mopnuni 22246	. . . . . . . . . . . 12 |- ( D e. ( *Met ` X ) -> X = U. J )
39	22, 38	syl 17	. . . . . . . . . . 11 |- ( ph -> X = U. J )
40	39	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> X = U. J )
41	32, 37, 40	3sstr3d 3647	. . . . . . . . 9 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ U. J )
42		eqid 2622	. . . . . . . . . 10 |- U. J = U. J
43	42	sscls 20860	. . . . . . . . 9 |- ( ( J e. Top /\ ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ U. J ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) )
44	25, 41, 43	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) )
45		difss2 3739	. . . . . . . 8 |- ( ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) -> ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
46		sstr2 3610	. . . . . . . 8 |- ( ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) -> ( ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) ) )
47	44, 45, 46	syl2im 40	. . . . . . 7 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> ( ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) ) )
48	47	a1d 25	. . . . . 6 |- ( ( ph /\ ( g ` ( n + 1 ) ) e. ( X X. RR+ ) ) -> ( ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) -> ( ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) ) ) )
49	48	ex 450	. . . . 5 |- ( ph -> ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) -> ( ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) -> ( ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) ) ) ) )
50	49	3impd 1281	. . . 4 |- ( ph -> ( ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) ) )
51	50	adantr 481	. . 3 |- ( ( ph /\ n e. NN ) -> ( ( ( g ` ( n + 1 ) ) e. ( X X. RR+ ) /\ ( 2nd ` ( g ` ( n + 1 ) ) ) < ( 1 / n ) /\ ( ( cls ` J ) ` ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) ) C_ ( ( ( ball ` D ) ` ( g ` n ) ) \ ( M ` n ) ) ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) ) )
52	18, 51	mpd 15	. 2 |- ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )
53	52	ralrimiva 2966	1 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( g ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( g ` n ) ) )

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   U.cuni 4436   class class class wbr 4653   {copab 4712   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   / cdiv 10684   NNcn 11020   RR+crp 11832   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736   Topctop 20698   Clsdccld 20820   clsccl 20822   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-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-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-cls 20825  df-cmet 23055

This theorem is referenced by:  bcthlem3  23123  bcthlem4  23124