Theorem heiborlem5 33614

Description: Lemma for heibor 33620. The function M is a set of point-and-radius pairs suitable for application to caubl 23106. (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
heiborlem5	|- ( ph -> M : NN --> ( X X. RR+ ) )

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   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)   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 heiborlem5

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 11299	. . . . . 6 |- ( k e. NN -> k e. NN0 )
2		inss1 3833	. . . . . . . . 9 |- ( ~P X i^i Fin ) C_ ~P X
3		heibor.7	. . . . . . . . . 10 |- ( ph -> F : NN0 --> ( ~P X i^i Fin ) )
4	3	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. ( ~P X i^i Fin ) )
5	2, 4	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. ~P X )
6	5	elpwid 4170	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) C_ X )
7		heibor.1	. . . . . . . . 9 |- J = ( MetOpen ` D )
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.6	. . . . . . . . 9 |- ( ph -> D e. ( CMet ` X ) )
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	7, 8, 9, 10, 11, 3, 12, 13, 14, 15	heiborlem4 33613	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( S ` k ) G k )
17		fvex 6201	. . . . . . . . . 10 |- ( S ` k ) e. _V
18		vex 3203	. . . . . . . . . 10 |- k e. _V
19	7, 8, 9, 17, 18	heiborlem2 33611	. . . . . . . . 9 |- ( ( S ` k ) G k <-> ( k e. NN0 /\ ( S ` k ) e. ( F ` k ) /\ ( ( S ` k ) B k ) e. K ) )
20	19	simp2bi 1077	. . . . . . . 8 |- ( ( S ` k ) G k -> ( S ` k ) e. ( F ` k ) )
21	16, 20	syl 17	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( S ` k ) e. ( F ` k ) )
22	6, 21	sseldd 3604	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( S ` k ) e. X )
23	1, 22	sylan2 491	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( S ` k ) e. X )
24	23	ralrimiva 2966	. . . 4 |- ( ph -> A. k e. NN ( S ` k ) e. X )
25		fveq2 6191	. . . . . 6 |- ( k = n -> ( S ` k ) = ( S ` n ) )
26	25	eleq1d 2686	. . . . 5 |- ( k = n -> ( ( S ` k ) e. X <-> ( S ` n ) e. X ) )
27	26	cbvralv 3171	. . . 4 |- ( A. k e. NN ( S ` k ) e. X <-> A. n e. NN ( S ` n ) e. X )
28	24, 27	sylib 208	. . 3 |- ( ph -> A. n e. NN ( S ` n ) e. X )
29		3re 11094	. . . . . . 7 |- 3 e. RR
30		3pos 11114	. . . . . . 7 |- 0 < 3
31	29, 30	elrpii 11835	. . . . . 6 |- 3 e. RR+
32		2nn 11185	. . . . . . . 8 |- 2 e. NN
33		nnnn0 11299	. . . . . . . 8 |- ( n e. NN -> n e. NN0 )
34		nnexpcl 12873	. . . . . . . 8 |- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN )
35	32, 33, 34	sylancr 695	. . . . . . 7 |- ( n e. NN -> ( 2 ^ n ) e. NN )
36	35	nnrpd 11870	. . . . . 6 |- ( n e. NN -> ( 2 ^ n ) e. RR+ )
37		rpdivcl 11856	. . . . . 6 |- ( ( 3 e. RR+ /\ ( 2 ^ n ) e. RR+ ) -> ( 3 / ( 2 ^ n ) ) e. RR+ )
38	31, 36, 37	sylancr 695	. . . . 5 |- ( n e. NN -> ( 3 / ( 2 ^ n ) ) e. RR+ )
39		opelxpi 5148	. . . . . 6 |- ( ( ( S ` n ) e. X /\ ( 3 / ( 2 ^ n ) ) e. RR+ ) -> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. e. ( X X. RR+ ) )
40	39	expcom 451	. . . . 5 |- ( ( 3 / ( 2 ^ n ) ) e. RR+ -> ( ( S ` n ) e. X -> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. e. ( X X. RR+ ) ) )
41	38, 40	syl 17	. . . 4 |- ( n e. NN -> ( ( S ` n ) e. X -> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. e. ( X X. RR+ ) ) )
42	41	ralimia 2950	. . 3 |- ( A. n e. NN ( S ` n ) e. X -> A. n e. NN <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. e. ( X X. RR+ ) )
43	28, 42	syl 17	. 2 |- ( ph -> A. n e. NN <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. e. ( X X. RR+ ) )
44		heibor.12	. . 3 |- M = ( n e. NN |-> <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. )
45	44	fmpt 6381	. 2 |- ( A. n e. NN <. ( S ` n ) , ( 3 / ( 2 ^ n ) ) >. e. ( X X. RR+ ) <-> M : NN --> ( X X. RR+ ) )
46	43, 45	sylib 208	1 |- ( ph -> M : NN --> ( X X. RR+ ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   - 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

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-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-seq 12802  df-exp 12861

This theorem is referenced by:  heiborlem8  33617  heiborlem9  33618