Theorem ubthlem14 8486

Description: Lemma for ubthi 8488. The operator norms of the operators T` n have an upper bound.

Hypotheses

Ref	Expression
ubthlem10.1	|- X = (Base` U)
ubthlem10.2	|- Y = (Base` 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))
ubthlem14.7	|- U e. CBan

Assertion

Ref	Expression
ubthlem14	|- (A.x e. X E.c e. RR A.n e. NN (N` ((T` n)` x)) <_ c -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)

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

Proof of Theorem ubthlem14

Step	Hyp	Ref	Expression
1		ubthlem10.1	. . 3 |- X = (Base` U)
2		ubthlem10.2	. . 3 |- Y = (Base` W)
3		ubthlem10.3	. . 3 |- N = (norm` W)
4		ubthlem10.5	. . 3 |- B = (U BLnOp W)
5		ubthlem10.6	. . 3 |- T:NN-->B
6		ubthlem14.7	. . 3 |- U e. CBan
7		ubthlem10.8	. . 3 |- W e. NrmCVec
8		ubthlem10.9	. . 3 |- D = (IndMet` U)
9		eqid 1473	. . 3 |- (Open` D) = (Open` D)
10		ubthlem10.11	. . 3 |- A = {<.j, y>. | (j e. NN /\ y = {z e. X | A.h e. NN (N` ((T` h)` z)) <_ j})}
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ubthlem6 8478	. 2 |- (A.x e. X E.c e. RR A.n e. NN (N` ((T` n)` x)) <_ c -> E.k e. NN E.p e. X E.r e. RR (0 < r /\ (p( ball ` D)r) (_ (A` k)))
12		breq2 2618	. . . . . . . . . . 11 |- (d = ((2 / r) x. (2 x. k)) -> ((O` (T` n)) <_ d <-> (O` (T` n)) <_ ((2 / r) x. (2 x. k))))
13	12	ralbidv 1660	. . . . . . . . . 10 |- (d = ((2 / r) x. (2 x. k)) -> (A.n e. NN (O` (T` n)) <_ d <-> A.n e. NN (O` (T` n)) <_ ((2 / r) x. (2 x. k))))
14	13	rcla4ev 1873	. . . . . . . . 9 |- ((((2 / r) x. (2 x. k)) e. RR /\ A.n e. NN (O` (T` n)) <_ ((2 / r) x. (2 x. k))) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)
15		axmulrcl 5254	. . . . . . . . . . . . 13 |- (((2 / r) e. RR /\ (2 x. k) e. RR) -> ((2 / r) x. (2 x. k)) e. RR)
16		gt0ne0t 5600	. . . . . . . . . . . . . 14 |- ((r e. RR /\ 0 < r) -> r =/= 0)
17		2re 5934	. . . . . . . . . . . . . . 15 |- 2 e. RR
18		redivclt 5764	. . . . . . . . . . . . . . 15 |- ((2 e. RR /\ r e. RR /\ r =/= 0) -> (2 / r) e. RR)
19	17, 18	mp3an1 901	. . . . . . . . . . . . . 14 |- ((r e. RR /\ r =/= 0) -> (2 / r) e. RR)
20	16, 19	syldan 467	. . . . . . . . . . . . 13 |- ((r e. RR /\ 0 < r) -> (2 / r) e. RR)
21		nnret 5885	. . . . . . . . . . . . . 14 |- (k e. NN -> k e. RR)
22		axmulrcl 5254	. . . . . . . . . . . . . . 15 |- ((2 e. RR /\ k e. RR) -> (2 x. k) e. RR)
23	17, 22	mpan 694	. . . . . . . . . . . . . 14 |- (k e. RR -> (2 x. k) e. RR)
24	21, 23	syl 10	. . . . . . . . . . . . 13 |- (k e. NN -> (2 x. k) e. RR)
25	15, 20, 24	syl2an 454	. . . . . . . . . . . 12 |- (((r e. RR /\ 0 < r) /\ k e. NN) -> ((2 / r) x. (2 x. k)) e. RR)
26	25	ancoms 436	. . . . . . . . . . 11 |- ((k e. NN /\ (r e. RR /\ 0 < r)) -> ((2 / r) x. (2 x. k)) e. RR)
27	26	adantrl 394	. . . . . . . . . 10 |- ((k e. NN /\ (p e. X /\ (r e. RR /\ 0 < r))) -> ((2 / r) x. (2 x. k)) e. RR)
28	27	adantrr 395	. . . . . . . . 9 |- ((k e. NN /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k))) -> ((2 / r) x. (2 x. k)) e. RR)
29		ubthlem10.4	. . . . . . . . . . . . . 14 |- O = (UnormOpW)
30		ubthlem10.7	. . . . . . . . . . . . . 14 |- U e. NrmCVec
31		ubthlem10.n	. . . . . . . . . . . . . 14 |- L = (norm` U)
32		ubthlem10.g	. . . . . . . . . . . . . 14 |- G = (+v` U)
33		ubthlem10.m	. . . . . . . . . . . . . 14 |- M = (-v` U)
34		ubthlem10.r	. . . . . . . . . . . . . 14 |- R = (.s` U)
35		ubthlem10.z	. . . . . . . . . . . . . 14 |- Z = (0v` U)
36		ubthlem10.q	. . . . . . . . . . . . . 14 |- Q = (pG(((r / 2) x. (1 / (L` x)))Rx))
37	1, 2, 3, 29, 4, 5, 30, 7, 8, 31, 32, 33, 34, 35, 10, 36	ubthlem13 8485	. . . . . . . . . . . . 13 |- (((k e. NN /\ n e. NN) /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k))) -> (O` (T` n)) <_ ((2 / r) x. (2 x. k)))
38	37	exp31 376	. . . . . . . . . . . 12 |- (k e. NN -> (n e. NN -> (((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k)) -> (O` (T` n)) <_ ((2 / r) x. (2 x. k)))))
39	38	com23 32	. . . . . . . . . . 11 |- (k e. NN -> (((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k)) -> (n e. NN -> (O` (T` n)) <_ ((2 / r) x. (2 x. k)))))
40	39	imp 350	. . . . . . . . . 10 |- ((k e. NN /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k))) -> (n e. NN -> (O` (T` n)) <_ ((2 / r) x. (2 x. k))))
41	40	r19.21aiv 1710	. . . . . . . . 9 |- ((k e. NN /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k))) -> A.n e. NN (O` (T` n)) <_ ((2 / r) x. (2 x. k)))
42	14, 28, 41	sylanc 471	. . . . . . . 8 |- ((k e. NN /\ ((p e. X /\ (r e. RR /\ 0 < r)) /\ (p( ball ` D)r) (_ (A` k))) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)
43	42	exp44 385	. . . . . . 7 |- (k e. NN -> (p e. X -> ((r e. RR /\ 0 < r) -> ((p( ball ` D)r) (_ (A` k) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d))))
44	43	imp 350	. . . . . 6 |- ((k e. NN /\ p e. X) -> ((r e. RR /\ 0 < r) -> ((p( ball ` D)r) (_ (A` k) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)))
45	44	exp3a 375	. . . . 5 |- ((k e. NN /\ p e. X) -> (r e. RR -> (0 < r -> ((p( ball ` D)r) (_ (A` k) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d))))
46	45	imp4a 364	. . . 4 |- ((k e. NN /\ p e. X) -> (r e. RR -> ((0 < r /\ (p( ball ` D)r) (_ (A` k)) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)))
47	46	r19.23adv 1743	. . 3 |- ((k e. NN /\ p e. X) -> (E.r e. RR (0 < r /\ (p( ball ` D)r) (_ (A` k)) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d))
48	47	r19.23aivv 1745	. 2 |- (E.k e. NN E.p e. X E.r e. RR (0 < r /\ (p( ball ` D)r) (_ (A` k)) -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)
49	11, 48	syl 10	1 |- (A.x e. X E.c e. RR A.n e. NN (N` ((T` n)` x)) <_ c -> E.d e. RR A.n e. NN (O` (T` n)) <_ d)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582  A.wral 1642  E.wrex 1643  {crab 1645   (_ wss 2043   class class class wbr 2614  {copab 2661  -->wf 3173  ` cfv 3177  (class class class)co 3954  RRcr 5213  0cc0 5214  1c1 5215   x. cmul 5219   / cdiv 5274   <_ cle 5275  NNcn 5276   < clt 5466  2c2 5916   ball cbl 7741  Opencopn 7742  NrmCVeccnv 8155  +vcpv 8156  Basecba 8157  .scns 8158  0vcn0v 8159  -vcnsb 8160  normcnm 8161  IndMetcims 8162  normOpcnmo 8349   BLnOp cblo 8350  CBancbn 8466

This theorem is referenced by:  ubthii 8487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-iin 2564  df-br 26