Theorem heiborlem9 33618

Description: Lemma for heibor 33620. Discharge the hypotheses of heiborlem8 33617 by applying caubl 23106 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-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 ) ) >. )
heibor.13	|- ( ph -> U C_ J )
heiborlem9.14	|- ( ph -> U. U = X )

Assertion

Ref	Expression
heiborlem9	|- ( ph -> ps )

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

Proof of Theorem heiborlem9

Dummy variables t k r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.6	. . . . . . 7 |- ( ph -> D e. ( CMet ` X ) )
2		cmetmet 23084	. . . . . . 7 |- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) )
3		metxmet 22139	. . . . . . 7 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
4	1, 2, 3	3syl 18	. . . . . 6 |- ( ph -> D e. ( *Met ` X ) )
5		heibor.1	. . . . . . 7 |- J = ( MetOpen ` D )
6	5	mopntopon 22244	. . . . . 6 |- ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) )
7	4, 6	syl 17	. . . . 5 |- ( ph -> J e. (TopOn ` X ) )
8		heibor.3	. . . . . . . . 9 |- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v }
9		heibor.4	. . . . . . . . 9 |- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) }
10		heibor.5	. . . . . . . . 9 |- B = ( z e. X , m e. NN0 |-> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) )
11		heibor.7	. . . . . . . . 9 |- ( ph -> F : NN0 --> ( ~P X i^i Fin ) )
12		heibor.8	. . . . . . . . 9 |- ( ph -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) )
13		heibor.9	. . . . . . . . 9 |- ( ph -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
14		heibor.10	. . . . . . . . 9 |- ( ph -> C G 0 )
15		heibor.11	. . . . . . . . 9 |- S = seq 0 ( T , ( m e. NN0 |-> if ( m = 0 , C , ( m - 1 ) ) ) )
16		heibor.12	. . . . . . . . 9 |- M = ( n e. NN |-> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. )
17	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	heiborlem5 33614	. . . . . . . 8 |- ( ph -> M : NN --> ( X X. RR+ ) )
18	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	heiborlem6 33615	. . . . . . . 8 |- ( ph -> A. k e. NN ( ( ball ` D ) ` ( M ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( M ` k ) ) )
19	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	heiborlem7 33616	. . . . . . . . 9 |- A. r e. RR+ E. k e. NN ( 2nd ` ( M ` k ) ) < r
20	19	a1i 11	. . . . . . . 8 |- ( ph -> A. r e. RR+ E. k e. NN ( 2nd ` ( M ` k ) ) < r )
21	4, 17, 18, 20	caubl 23106	. . . . . . 7 |- ( ph -> ( 1st o. M ) e. ( Cau ` D ) )
22	5	cmetcau 23087	. . . . . . 7 |- ( ( D e. ( CMet ` X ) /\ ( 1st o. M ) e. ( Cau ` D ) ) -> ( 1st o. M ) e. dom ( ~~> t ` J ) )
23	1, 21, 22	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st o. M ) e. dom ( ~~> t ` J ) )
24	5	methaus 22325	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> J e. Haus )
25	4, 24	syl 17	. . . . . . 7 |- ( ph -> J e. Haus )
26		lmfun 21185	. . . . . . 7 |- ( J e. Haus -> Fun ( ~~> t ` J ) )
27		funfvbrb 6330	. . . . . . 7 |- ( Fun ( ~~> t ` J ) -> ( ( 1st o. M ) e. dom ( ~~> t ` J ) <-> ( 1st o. M ) ( ~~> t ` J ) ( ( ~~> t ` J ) ` ( 1st o. M ) ) ) )
28	25, 26, 27	3syl 18	. . . . . 6 |- ( ph -> ( ( 1st o. M ) e. dom ( ~~> t ` J ) <-> ( 1st o. M ) ( ~~> t ` J ) ( ( ~~> t ` J ) ` ( 1st o. M ) ) ) )
29	23, 28	mpbid 222	. . . . 5 |- ( ph -> ( 1st o. M ) ( ~~> t ` J ) ( ( ~~> t ` J ) ` ( 1st o. M ) ) )
30		lmcl 21101	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ ( 1st o. M ) ( ~~> t ` J ) ( ( ~~> t ` J ) ` ( 1st o. M ) ) ) -> ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. X )
31	7, 29, 30	syl2anc 693	. . . 4 |- ( ph -> ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. X )
32		heiborlem9.14	. . . 4 |- ( ph -> U. U = X )
33	31, 32	eleqtrrd 2704	. . 3 |- ( ph -> ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. U. U )
34		eluni2 4440	. . 3 |- ( ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. U. U <-> E. t e. U ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t )
35	33, 34	sylib 208	. 2 |- ( ph -> E. t e. U ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t )
36	1	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> D e. ( CMet ` X ) )
37	11	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> F : NN0 --> ( ~P X i^i Fin ) )
38	12	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) )
39	13	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> A. x e. G ( ( T ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( T ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) )
40	14	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> C G 0 )
41		heibor.13	. . . 4 |- ( ph -> U C_ J )
42	41	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> U C_ J )
43		fvex 6201	. . 3 |- ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. _V
44		simprr 796	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t )
45		simprl 794	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> t e. U )
46	29	adantr 481	. . 3 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> ( 1st o. M ) ( ~~> t ` J ) ( ( ~~> t ` J ) ` ( 1st o. M ) ) )
47	5, 8, 9, 10, 36, 37, 38, 39, 40, 15, 16, 42, 43, 44, 45, 46	heiborlem8 33617	. 2 |- ( ( ph /\ ( t e. U /\ ( ( ~~> t ` J ) ` ( 1st o. M ) ) e. t ) ) -> ps )
48	35, 47	rexlimddv 3035	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   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   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   RR+crp 11832   seqcseq 12801   ^cexp 12860   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736  TopOnctopon 20715   ~~> tclm 21030   Hauscha 21112   Caucca 23051   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-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fl 12593  df-seq 12802  df-exp 12861  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lm 21033  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055

This theorem is referenced by:  heiborlem10  33619