Theorem ubthlem13 9885

Description: Lemma for ubthi 9889. Upper bound for the operator norm of any operator T` n.

Hypotheses

Ref	Expression
ubthlem10.1	|- X = (BaseSet` U)
ubthlem10.2	|- Y = (BaseSet` W)
ubthlem10.3	|- N = (norm` W)
ubthlem10.4	|- O = (UnormOpW)
ubthlem10.5	|- B = (U BLnOp W)
ubthlem10.6	|- T:NN-->B
ubthlem10.7	|- U e. NrmCVec
ubthlem10.8	|- W e. NrmCVec
ubthlem10.9	|- D = (IndMet` U)
ubthlem10.n	|- L = (norm` U)
ubthlem10.g	|- G = (+v` U)
ubthlem10.m	|- M = (-v` U)
ubthlem10.r	|- R = (.s` U)
ubthlem10.z	|- Z = (0v` U)
ubthlem10.11	|- A = {<.j, y>. | (j e. NN /\ y = {z e. X | A.h e. NN (N` ((T` h)` z)) <_ j})}
ubthlem10.q	|- Q = (pG(((r / 2) x. (1 / (L` x)))Rx))

Assertion

Ref	Expression
ubthlem13	|- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> (O` (T` n)) <_ ((2 / r) x. (2 x. k)))

Distinct variable groups:   k,n,p,r,x,A   D,k,n,p,r,x   x,L   h,j,k,n,x,y,z,N   k,O,p,r   Q,h,n,z   h,p,r,T,j,k,n,y,z,x   x,U   x,W   j,X,k,n,r,x,y,z

Proof of Theorem ubthlem13

Step	Hyp	Ref	Expression
1		ubthlem10.6	. . . . 5 |- T:NN-->B
2	1	ffvelrni 4788	. . . 4 |- (n e. NN -> (T` n) e. B)
3		ubthlem10.7	. . . . 5 |- U e. NrmCVec
4		ubthlem10.8	. . . . 5 |- W e. NrmCVec
5		eqid 1884	. . . . . 6 |- (U LnOp W) = (U LnOp W)
6		ubthlem10.5	. . . . . 6 |- B = (U BLnOp W)
7	5, 6	bloln 9784	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec /\ (T` n) e. B) -> (T` n) e. (U LnOp W))
8	3, 4, 7	mp3an12 1181	. . . 4 |- ((T` n) e. B -> (T` n) e. (U LnOp W))
9	2, 8	syl 12	. . 3 |- (n e. NN -> (T` n) e. (U LnOp W))
10	9	ad2antlr 441	. 2 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> (T` n) e. (U LnOp W))
11		remulcl 6457	. . . . . . 7 |- (((2 / r) e. RR /\ (2 x. k) e. RR) -> ((2 / r) x. (2 x. k)) e. RR)
12		gt0ne0 6800	. . . . . . . 8 |- ((r e. RR /\ 0 < r) -> r =/= 0)
13		2re 7163	. . . . . . . . 9 |- 2 e. RR
14		redivcl 6978	. . . . . . . . 9 |- ((2 e. RR /\ r e. RR /\ r =/= 0) -> (2 / r) e. RR)
15	13, 14	mp3an1 1178	. . . . . . . 8 |- ((r e. RR /\ r =/= 0) -> (2 / r) e. RR)
16	12, 15	syldan 516	. . . . . . 7 |- ((r e. RR /\ 0 < r) -> (2 / r) e. RR)
17		nnre 7112	. . . . . . . 8 |- (k e. NN -> k e. RR)
18		remulcl 6457	. . . . . . . . 9 |- ((2 e. RR /\ k e. RR) -> (2 x. k) e. RR)
19	13, 18	mpan 759	. . . . . . . 8 |- (k e. RR -> (2 x. k) e. RR)
20	17, 19	syl 12	. . . . . . 7 |- (k e. NN -> (2 x. k) e. RR)
21	11, 16, 20	syl2an 503	. . . . . 6 |- (((r e. RR /\ 0 < r) /\ k e. NN) -> ((2 / r) x. (2 x. k)) e. RR)
22		mulge0 6868	. . . . . . 7 |- ((((2 / r) e. RR /\ 0 <_ (2 / r)) /\ ((2 x. k) e. RR /\ 0 <_ (2 x. k))) -> 0 <_ ((2 / r) x. (2 x. k)))
23		0re 6603	. . . . . . . . . 10 |- 0 e. RR
24		2pos 7173	. . . . . . . . . 10 |- 0 < 2
25	23, 13, 24	ltleii 6756	. . . . . . . . 9 |- 0 <_ 2
26		divge0 7038	. . . . . . . . 9 |- (((2 e. RR /\ 0 <_ 2) /\ (r e. RR /\ 0 < r)) -> 0 <_ (2 / r))
27	13, 25, 26	mpanl12 773	. . . . . . . 8 |- ((r e. RR /\ 0 < r) -> 0 <_ (2 / r))
28	16, 27	jca 310	. . . . . . 7 |- ((r e. RR /\ 0 < r) -> ((2 / r) e. RR /\ 0 <_ (2 / r)))
29		nngt0 7129	. . . . . . . . . 10 |- (k e. NN -> 0 < k)
30		ltle 6690	. . . . . . . . . . 11 |- ((0 e. RR /\ k e. RR) -> (0 < k -> 0 <_ k))
31	23, 30	mpan 759	. . . . . . . . . 10 |- (k e. RR -> (0 < k -> 0 <_ k))
32	17, 29, 31	sylc 83	. . . . . . . . 9 |- (k e. NN -> 0 <_ k)
33		mulge0 6868	. . . . . . . . . 10 |- (((2 e. RR /\ 0 <_ 2) /\ (k e. RR /\ 0 <_ k)) -> 0 <_ (2 x. k))
34	13, 25, 33	mpanl12 773	. . . . . . . . 9 |- ((k e. RR /\ 0 <_ k) -> 0 <_ (2 x. k))
35	17, 32, 34	syl11anc 524	. . . . . . . 8 |- (k e. NN -> 0 <_ (2 x. k))
36	20, 35	jca 310	. . . . . . 7 |- (k e. NN -> ((2 x. k) e. RR /\ 0 <_ (2 x. k)))
37	22, 28, 36	syl2an 503	. . . . . 6 |- (((r e. RR /\ 0 < r) /\ k e. NN) -> 0 <_ ((2 / r) x. (2 x. k)))
38	21, 37	jca 310	. . . . 5 |- (((r e. RR /\ 0 < r) /\ k e. NN) -> (((2 / r) x. (2 x. k)) e. RR /\ 0 <_ ((2 / r) x. (2 x. k))))
39	38	ancoms 484	. . . 4 |- ((k e. NN /\ (r e. RR /\ 0 < r)) -> (((2 / r) x. (2 x. k)) e. RR /\ 0 <_ ((2 / r) x. (2 x. k))))
40	39	adantrl 430	. . 3 |- ((k e. NN /\ (p e. X /\ (r e. RR /\ 0 < r))) -> (((2 / r) x. (2 x. k)) e. RR /\ 0 <_ ((2 / r) x. (2 x. k))))
41	40	ad2ant2r 445	. 2 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> (((2 / r) x. (2 x. k)) e. RR /\ 0 <_ ((2 / r) x. (2 x. k))))
42		ubthlem10.1	. . . . . . . . . 10 |- X = (BaseSet` U)
43		ubthlem10.2	. . . . . . . . . 10 |- Y = (BaseSet` W)
44		ubthlem10.3	. . . . . . . . . 10 |- N = (norm` W)
45		ubthlem10.4	. . . . . . . . . 10 |- O = (UnormOpW)
46		ubthlem10.9	. . . . . . . . . 10 |- D = (IndMet` U)
47		ubthlem10.n	. . . . . . . . . 10 |- L = (norm` U)
48		ubthlem10.g	. . . . . . . . . 10 |- G = (+v` U)
49		ubthlem10.m	. . . . . . . . . 10 |- M = (-v` U)
50		ubthlem10.r	. . . . . . . . . 10 |- R = (.s` U)
51		ubthlem10.z	. . . . . . . . . 10 |- Z = (0v` U)
52		ubthlem10.11	. . . . . . . . . 10 |- A = {<.j, y>. | (j e. NN /\ y = {z e. X | A.h e. NN (N` ((T` h)` z)) <_ j})}
53		ubthlem10.q	. . . . . . . . . 10 |- Q = (pG(((r / 2) x. (1 / (L` x)))Rx))
54	42, 43, 44, 45, 6, 1, 3, 4, 46, 47, 48, 49, 50, 51, 52, 53	ubthlem12 9883	. . . . . . . . 9 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) /\ (p( ball ` D)r) C_ (A` k))) -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x)))
55	54	exp32 408	. . . . . . . 8 |- ((k e. NN /\ n e. NN) -> ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> ((p( ball ` D)r) C_ (A` k) -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x)))))
56	55	exp4d 412	. . . . . . 7 |- ((k e. NN /\ n e. NN) -> (p e. X -> ((r e. RR /\ 0 < r) -> ((x e. X /\ x =/= Z) -> ((p( ball ` D)r) C_ (A` k) -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x)))))))
57	56	imp3a 388	. . . . . 6 |- ((k e. NN /\ n e. NN) -> ((p e. X /\ (r e. RR /\ 0 < r)) -> ((x e. X /\ x =/= Z) -> ((p( ball ` D)r) C_ (A` k) -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x))))))
58	57	com34 40	. . . . 5 |- ((k e. NN /\ n e. NN) -> ((p e. X /\ (r e. RR /\ 0 < r)) -> ((p( ball ` D)r) C_ (A` k) -> ((x e. X /\ x =/= Z) -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x))))))
59	58	imp32 390	. . . 4 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> ((x e. X /\ x =/= Z) -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x))))
60	59	exp3a 405	. . 3 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> (x e. X -> (x =/= Z -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x)))))
61	60	r19.21aiv 2175	. 2 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> A.x e. X (x =/= Z -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x))))
62	42, 51, 47, 44, 45, 5, 3, 4	nmlnoubi 9796	. 2 |- (((T` n) e. (U LnOp W) /\ (((2 / r) x. (2 x. k)) e. RR /\ 0 <_ ((2 / r) x. (2 x. k))) /\ A.x e. X (x =/= Z -> (N` ((T` n)` x)) <_ (((2 / r) x. (2 x. k)) x. (L` x)))) -> (O` (T` n)) <_ ((2 / r) x. (2 x. k)))
63	10, 41, 61, 62	syl111anc 1100	1 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) C_ (A` k))) -> (O` (T` n)) <_ ((2 / r) x. (2 x. k)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  {crab 2108   C_ wss 2593   class class class wbr 3338  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447   <_ cle 6448  NNcn 6449   < clt 6653  2c2 7145   ball cbl 9068  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  0vcn0v 9539  -vcnsb 9540  normcnm 9541  IndMetcims 9542   LnOp clno 9740  normOpcnmo 9741   BLnOp cblo 9742

This theorem is referenced by:  ubthlem14 9887

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-met 9070  df-bl 9072  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-lno 9744  df-nmo 9745  df-blo 9746

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