Theorem heiborlem7 33616

Description: Lemma for heibor 33620. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	|- J = ( MetOpen ` D )
heibor.3	|- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v }
heibor.4	|- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
heibor.5	|- B = ( z e. X , m e. NN0 |-> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
heibor.6	|- ( ph -> D e. ( CMet ` X ) )
heibor.7	|- ( ph -> F : NN0 --> ( ~P X i^i Fin ) )
heibor.8	|- ( ph -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) )
heibor.9	|- ( ph -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
heibor.10	|- ( ph -> C G 0 )
heibor.11	|- S = seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) )
heibor.12	|- M = ( n e. NN |-> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. )

Assertion

Ref	Expression
heiborlem7	|- A. r e. RR+ E. k e. NN ( 2nd ` ( M ` k ) ) < r

Distinct variable groups:   x, n, y, k, r, u, F   k, G, x   ph, k, r, x   k, m, v, z, D, n, r, u, x, y   k, M, m, r, u, x, y, z   T, m, n, x, y, z   B, n, u, v, y   k, J, m, n, r, u, v, x, y, z   U, n, u, v, x, y, z   S, k, m, n, u, v, x, y, z   k, X, m, n, r, u, v, x, y, z   C, m, n, u, v, y   n, K, x, y, z   x, B

Allowed substitution hints:   ph( y, z, v, u, m, n)   B( z, k, m, r)   C( x, z, k, r)   S( r)   T( v, u, k, r)   U( k, m, r)   F( z, v, m)   G( y, z, v, u, m, n, r)   K( v, u, k, m, r)   M( v, n)

Proof of Theorem heiborlem7

Step	Hyp	Ref	Expression
1		3re 11094	. . . . . . 7 |- 3 e. RR
2		3pos 11114	. . . . . . 7 |- 0 < 3
3	1, 2	elrpii 11835	. . . . . 6 |- 3 e. RR+
4		rpdivcl 11856	. . . . . 6 |- ( ( r e. RR+ /\ 3 e. RR+ ) -> ( r / 3 ) e. RR+ )
5	3, 4	mpan2 707	. . . . 5 |- ( r e. RR+ -> ( r / 3 ) e. RR+ )
6		2re 11090	. . . . . 6 |- 2 e. RR
7		1lt2 11194	. . . . . 6 |- 1 < 2
8		expnlbnd 12994	. . . . . 6 |- ( ( ( r / 3 ) e. RR+ /\ 2 e. RR /\ 1 < 2 ) -> E. k e. NN ( 1 / ( 2 ^ k ) ) < ( r / 3 ) )
9	6, 7, 8	mp3an23 1416	. . . . 5 |- ( ( r / 3 ) e. RR+ -> E. k e. NN ( 1 / ( 2 ^ k ) ) < ( r / 3 ) )
10	5, 9	syl 17	. . . 4 |- ( r e. RR+ -> E. k e. NN ( 1 / ( 2 ^ k ) ) < ( r / 3 ) )
11		2nn 11185	. . . . . . . . . . 11 |- 2 e. NN
12		nnnn0 11299	. . . . . . . . . . 11 |- ( k e. NN -> k e. NN0 )
13		nnexpcl 12873	. . . . . . . . . . 11 |- ( ( 2 e. NN /\ k e. NN0 ) -> ( 2 ^ k ) e. NN )
14	11, 12, 13	sylancr 695	. . . . . . . . . 10 |- ( k e. NN -> ( 2 ^ k ) e. NN )
15	14	nnrpd 11870	. . . . . . . . 9 |- ( k e. NN -> ( 2 ^ k ) e. RR+ )
16		rpcn 11841	. . . . . . . . . 10 |- ( ( 2 ^ k ) e. RR+ -> ( 2 ^ k ) e. CC )
17		rpne0 11848	. . . . . . . . . 10 |- ( ( 2 ^ k ) e. RR+ -> ( 2 ^ k ) =/= 0 )
18		3cn 11095	. . . . . . . . . . 11 |- 3 e. CC
19		divrec 10701	. . . . . . . . . . 11 |- ( ( 3 e. CC /\ ( 2 ^ k ) e. CC /\ ( 2 ^ k ) =/= 0 ) -> ( 3 / ( 2 ^ k ) ) = ( 3 x. ( 1 / ( 2 ^ k ) ) ) )
20	18, 19	mp3an1 1411	. . . . . . . . . 10 |- ( ( ( 2 ^ k ) e. CC /\ ( 2 ^ k ) =/= 0 ) -> ( 3 / ( 2 ^ k ) ) = ( 3 x. ( 1 / ( 2 ^ k ) ) ) )
21	16, 17, 20	syl2anc 693	. . . . . . . . 9 |- ( ( 2 ^ k ) e. RR+ -> ( 3 / ( 2 ^ k ) ) = ( 3 x. ( 1 / ( 2 ^ k ) ) ) )
22	15, 21	syl 17	. . . . . . . 8 |- ( k e. NN -> ( 3 / ( 2 ^ k ) ) = ( 3 x. ( 1 / ( 2 ^ k ) ) ) )
23	22	adantl 482	. . . . . . 7 |- ( ( r e. RR+ /\ k e. NN ) -> ( 3 / ( 2 ^ k ) ) = ( 3 x. ( 1 / ( 2 ^ k ) ) ) )
24	23	breq1d 4663	. . . . . 6 |- ( ( r e. RR+ /\ k e. NN ) -> ( ( 3 / ( 2 ^ k ) ) < r <-> ( 3 x. ( 1 / ( 2 ^ k ) ) ) < r ) )
25	14	nnrecred 11066	. . . . . . 7 |- ( k e. NN -> ( 1 / ( 2 ^ k ) ) e. RR )
26		rpre 11839	. . . . . . 7 |- ( r e. RR+ -> r e. RR )
27	1, 2	pm3.2i 471	. . . . . . . 8 |- ( 3 e. RR /\ 0 < 3 )
28		ltmuldiv2 10897	. . . . . . . 8 |- ( ( ( 1 / ( 2 ^ k ) ) e. RR /\ r e. RR /\ ( 3 e. RR /\ 0 < 3 ) ) -> ( ( 3 x. ( 1 / ( 2 ^ k ) ) ) < r <-> ( 1 / ( 2 ^ k ) ) < ( r / 3 ) ) )
29	27, 28	mp3an3 1413	. . . . . . 7 |- ( ( ( 1 / ( 2 ^ k ) ) e. RR /\ r e. RR ) -> ( ( 3 x. ( 1 / ( 2 ^ k ) ) ) < r <-> ( 1 / ( 2 ^ k ) ) < ( r / 3 ) ) )
30	25, 26, 29	syl2anr 495	. . . . . 6 |- ( ( r e. RR+ /\ k e. NN ) -> ( ( 3 x. ( 1 / ( 2 ^ k ) ) ) < r <-> ( 1 / ( 2 ^ k ) ) < ( r / 3 ) ) )
31	24, 30	bitrd 268	. . . . 5 |- ( ( r e. RR+ /\ k e. NN ) -> ( ( 3 / ( 2 ^ k ) ) < r <-> ( 1 / ( 2 ^ k ) ) < ( r / 3 ) ) )
32	31	rexbidva 3049	. . . 4 |- ( r e. RR+ -> ( E. k e. NN ( 3 / ( 2 ^ k ) ) < r <-> E. k e. NN ( 1 / ( 2 ^ k ) ) < ( r / 3 ) ) )
33	10, 32	mpbird 247	. . 3 |- ( r e. RR+ -> E. k e. NN ( 3 / ( 2 ^ k ) ) < r )
34		fveq2 6191	. . . . . . . . 9 |- ( n = k -> ( S ` n ) = ( S ` k ) )
35		oveq2 6658	. . . . . . . . . 10 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
36	35	oveq2d 6666	. . . . . . . . 9 |- ( n = k -> ( 3 / ( 2 ^ n ) ) = ( 3 / ( 2 ^ k ) ) )
37	34, 36	opeq12d 4410	. . . . . . . 8 |- ( n = k -> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. = <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. )
38		heibor.12	. . . . . . . 8 |- M = ( n e. NN |-> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. )
39		opex 4932	. . . . . . . 8 |- <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. e. _V
40	37, 38, 39	fvmpt 6282	. . . . . . 7 |- ( k e. NN -> ( M ` k ) = <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. )
41	40	fveq2d 6195	. . . . . 6 |- ( k e. NN -> ( 2nd ` ( M ` k ) ) = ( 2nd ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) )
42		fvex 6201	. . . . . . 7 |- ( S ` k ) e. _V
43		ovex 6678	. . . . . . 7 |- ( 3 / ( 2 ^ k ) ) e. _V
44	42, 43	op2nd 7177	. . . . . 6 |- ( 2nd ` <. ( S ` k ) , ( 3 / ( 2 ^ k ) ) >. ) = ( 3 / ( 2 ^ k ) )
45	41, 44	syl6eq 2672	. . . . 5 |- ( k e. NN -> ( 2nd ` ( M ` k ) ) = ( 3 / ( 2 ^ k ) ) )
46	45	breq1d 4663	. . . 4 |- ( k e. NN -> ( ( 2nd ` ( M ` k ) ) < r <-> ( 3 / ( 2 ^ k ) ) < r ) )
47	46	rexbiia 3040	. . 3 |- ( E. k e. NN ( 2nd ` ( M ` k ) ) < r <-> E. k e. NN ( 3 / ( 2 ^ k ) ) < r )
48	33, 47	sylibr 224	. 2 |- ( r e. RR+ -> E. k e. NN ( 2nd ` ( M ` k ) ) < r )
49	48	rgen 2922	1 |- A. r e. RR+ E. k e. NN ( 2nd ` ( M ` k ) ) < r

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ifcif 4086   ~Pcpw 4158   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   RR+crp 11832   seqcseq 12801   ^cexp 12860   ballcbl 19733   MetOpencmopn 19736   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-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-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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  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-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-seq 12802  df-exp 12861

This theorem is referenced by:  heiborlem8  33617  heiborlem9  33618