Theorem ubthlem12 9883

Description: Lemma for ubthi 9889. Upper limit for the norm of an operator value at x.

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
ubthlem12	|- (((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)))

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 ubthlem12

Step	Hyp	Ref	Expression
1		ffvelrn 4787	. . . . . . . 8 |- (((T` n):X-->Y /\ (QMp) e. X) -> ((T` n)` (QMp)) e. Y)
2		ubthlem10.6	. . . . . . . . . 10 |- T:NN-->B
3	2	ffvelrni 4788	. . . . . . . . 9 |- (n e. NN -> (T` n) e. B)
4		ubthlem10.7	. . . . . . . . . 10 |- U e. NrmCVec
5		ubthlem10.8	. . . . . . . . . 10 |- W e. NrmCVec
6		ubthlem10.1	. . . . . . . . . . 11 |- X = (BaseSet` U)
7		ubthlem10.2	. . . . . . . . . . 11 |- Y = (BaseSet` W)
8		ubthlem10.5	. . . . . . . . . . 11 |- B = (U BLnOp W)
9	6, 7, 8	blof 9785	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. NrmCVec /\ (T` n) e. B) -> (T` n):X-->Y)
10	4, 5, 9	mp3an12 1181	. . . . . . . . 9 |- ((T` n) e. B -> (T` n):X-->Y)
11	3, 10	syl 12	. . . . . . . 8 |- (n e. NN -> (T` n):X-->Y)
12		ubthlem10.n	. . . . . . . . . . 11 |- L = (norm` U)
13		ubthlem10.g	. . . . . . . . . . 11 |- G = (+v` U)
14		ubthlem10.m	. . . . . . . . . . 11 |- M = (-v` U)
15		ubthlem10.r	. . . . . . . . . . 11 |- R = (.s` U)
16		ubthlem10.z	. . . . . . . . . . 11 |- Z = (0v` U)
17		ubthlem10.q	. . . . . . . . . . 11 |- Q = (pG(((r / 2) x. (1 / (L` x)))Rx))
18	6, 4, 12, 13, 14, 15, 16, 17	ubthlem7 9878	. . . . . . . . . 10 |- ((p e. X /\ (r e. RR /\ (x e. X /\ x =/= Z))) -> Q e. X)
19	18	adantrlr 437	. . . . . . . . 9 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> Q e. X)
20	6, 14	nvmcl 9599	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ Q e. X /\ p e. X) -> (QMp) e. X)
21	4, 20	mp3an1 1178	. . . . . . . . . 10 |- ((Q e. X /\ p e. X) -> (QMp) e. X)
22	21	ancoms 484	. . . . . . . . 9 |- ((p e. X /\ Q e. X) -> (QMp) e. X)
23	19, 22	syldan 516	. . . . . . . 8 |- ((p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> (QMp) e. X)
24	1, 11, 23	syl2an 503	. . . . . . 7 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((T` n)` (QMp)) e. Y)
25		ubthlem10.3	. . . . . . . . 9 |- N = (norm` W)
26	7, 25	nvcl 9619	. . . . . . . 8 |- ((W e. NrmCVec /\ ((T` n)` (QMp)) e. Y) -> (N` ((T` n)` (QMp))) e. RR)
27	5, 26	mpan 759	. . . . . . 7 |- (((T` n)` (QMp)) e. Y -> (N` ((T` n)` (QMp))) e. RR)
28	24, 27	syl 12	. . . . . 6 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (N` ((T` n)` (QMp))) e. RR)
29	28	adantll 428	. . . . 5 |- (((k e. NN /\ n e. NN) /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (N` ((T` n)` (QMp))) e. RR)
30		nnre 7112	. . . . . . 7 |- (k e. NN -> k e. RR)
31		2re 7163	. . . . . . . 8 |- 2 e. RR
32		remulcl 6457	. . . . . . . 8 |- ((2 e. RR /\ k e. RR) -> (2 x. k) e. RR)
33	31, 32	mpan 759	. . . . . . 7 |- (k e. RR -> (2 x. k) e. RR)
34	30, 33	syl 12	. . . . . 6 |- (k e. NN -> (2 x. k) e. RR)
35	34	ad2antrr 440	. . . . 5 |- (((k e. NN /\ n e. NN) /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (2 x. k) e. RR)
36		remulcl 6457	. . . . . . . . 9 |- (((2 / r) e. RR /\ (L` x) e. RR) -> ((2 / r) x. (L` x)) e. RR)
37		gt0ne0 6800	. . . . . . . . . 10 |- ((r e. RR /\ 0 < r) -> r =/= 0)
38		redivcl 6978	. . . . . . . . . . 11 |- ((2 e. RR /\ r e. RR /\ r =/= 0) -> (2 / r) e. RR)
39	31, 38	mp3an1 1178	. . . . . . . . . 10 |- ((r e. RR /\ r =/= 0) -> (2 / r) e. RR)
40	37, 39	syldan 516	. . . . . . . . 9 |- ((r e. RR /\ 0 < r) -> (2 / r) e. RR)
41	6, 12	nvcl 9619	. . . . . . . . . 10 |- ((U e. NrmCVec /\ x e. X) -> (L` x) e. RR)
42	4, 41	mpan 759	. . . . . . . . 9 |- (x e. X -> (L` x) e. RR)
43	36, 40, 42	syl2an 503	. . . . . . . 8 |- (((r e. RR /\ 0 < r) /\ x e. X) -> ((2 / r) x. (L` x)) e. RR)
44		mulge0 6868	. . . . . . . . 9 |- ((((2 / r) e. RR /\ 0 <_ (2 / r)) /\ ((L` x) e. RR /\ 0 <_ (L` x))) -> 0 <_ ((2 / r) x. (L` x)))
45		0re 6603	. . . . . . . . . . . 12 |- 0 e. RR
46		2pos 7173	. . . . . . . . . . . 12 |- 0 < 2
47	45, 31, 46	ltleii 6756	. . . . . . . . . . 11 |- 0 <_ 2
48		divge0 7038	. . . . . . . . . . 11 |- (((2 e. RR /\ 0 <_ 2) /\ (r e. RR /\ 0 < r)) -> 0 <_ (2 / r))
49	31, 47, 48	mpanl12 773	. . . . . . . . . 10 |- ((r e. RR /\ 0 < r) -> 0 <_ (2 / r))
50	40, 49	jca 310	. . . . . . . . 9 |- ((r e. RR /\ 0 < r) -> ((2 / r) e. RR /\ 0 <_ (2 / r)))
51	6, 12	nvge0 9634	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ x e. X) -> 0 <_ (L` x))
52	4, 51	mpan 759	. . . . . . . . . 10 |- (x e. X -> 0 <_ (L` x))
53	42, 52	jca 310	. . . . . . . . 9 |- (x e. X -> ((L` x) e. RR /\ 0 <_ (L` x)))
54	44, 50, 53	syl2an 503	. . . . . . . 8 |- (((r e. RR /\ 0 < r) /\ x e. X) -> 0 <_ ((2 / r) x. (L` x)))
55	43, 54	jca 310	. . . . . . 7 |- (((r e. RR /\ 0 < r) /\ x e. X) -> (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x))))
56	55	adantrr 431	. . . . . 6 |- (((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)) -> (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x))))
57	56	ad2antll 443	. . . . 5 |- (((k e. NN /\ n e. NN) /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x))))
58	29, 35, 57	3jca 1050	. . . 4 |- (((k e. NN /\ n e. NN) /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((N` ((T` n)` (QMp))) e. RR /\ (2 x. k) e. RR /\ (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x)))))
59	58	adantrr 431	. . 3 |- (((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)` (QMp))) e. RR /\ (2 x. k) e. RR /\ (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x)))))
60		ubthlem10.4	. . . 4 |- O = (UnormOpW)
61		ubthlem10.9	. . . 4 |- D = (IndMet` U)
62		ubthlem10.11	. . . 4 |- A = {<.j, y>. | (j e. NN /\ y = {z e. X | A.h e. NN (N` ((T` h)` z)) <_ j})}
63	6, 7, 25, 60, 8, 2, 4, 5, 61, 12, 13, 14, 15, 16, 62, 17	ubthlem11 9882	. . 3 |- (((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)` (QMp))) <_ (2 x. k))
64		lemul2a 7021	. . 3 |- ((((N` ((T` n)` (QMp))) e. RR /\ (2 x. k) e. RR /\ (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x)))) /\ (N` ((T` n)` (QMp))) <_ (2 x. k)) -> (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))) <_ (((2 / r) x. (L` x)) x. (2 x. k)))
65	59, 63, 64	syl11anc 524	. 2 |- (((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))) -> (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))) <_ (((2 / r) x. (L` x)) x. (2 x. k)))
66		eqid 1884	. . . . . 6 |- (.s` W) = (.s` W)
67	6, 4, 12, 13, 14, 15, 16, 17, 8, 2, 5, 66	ubthlem9 9880	. . . . 5 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> ((T` n)` x) = (((2 / r) x. (L` x))(.s` W)((T` n)` (QMp))))
68	67	fveq2d 4685	. . . 4 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (N` ((T` n)` x)) = (N` (((2 / r) x. (L` x))(.s` W)((T` n)` (QMp)))))
69	56	ad2antll 443	. . . . 5 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x))))
70	7, 66, 25	nvsge0 9623	. . . . . 6 |- ((W e. NrmCVec /\ (((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x))) /\ ((T` n)` (QMp)) e. Y) -> (N` (((2 / r) x. (L` x))(.s` W)((T` n)` (QMp)))) = (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))))
71	5, 70	mp3an1 1178	. . . . 5 |- (((((2 / r) x. (L` x)) e. RR /\ 0 <_ ((2 / r) x. (L` x))) /\ ((T` n)` (QMp)) e. Y) -> (N` (((2 / r) x. (L` x))(.s` W)((T` n)` (QMp)))) = (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))))
72	69, 24, 71	syl11anc 524	. . . 4 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (N` (((2 / r) x. (L` x))(.s` W)((T` n)` (QMp)))) = (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))))
73	68, 72	eqtr2d 1926	. . 3 |- ((n e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))) = (N` ((T` n)` x)))
74	73	ad2ant2lr 446	. 2 |- (((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))) -> (((2 / r) x. (L` x)) x. (N` ((T` n)` (QMp)))) = (N` ((T` n)` x)))
75		mul23 6580	. . . . . . . 8 |- (((2 / r) e. CC /\ (L` x) e. CC /\ (2 x. k) e. CC) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
76	40	recnd 6468	. . . . . . . 8 |- ((r e. RR /\ 0 < r) -> (2 / r) e. CC)
77	42	recnd 6468	. . . . . . . 8 |- (x e. X -> (L` x) e. CC)
78	34	recnd 6468	. . . . . . . 8 |- (k e. NN -> (2 x. k) e. CC)
79	75, 76, 77, 78	syl3an 1139	. . . . . . 7 |- (((r e. RR /\ 0 < r) /\ x e. X /\ k e. NN) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
80	79	3comr 1076	. . . . . 6 |- ((k e. NN /\ (r e. RR /\ 0 < r) /\ x e. X) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
81	80	3expb 1068	. . . . 5 |- ((k e. NN /\ ((r e. RR /\ 0 < r) /\ x e. X)) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
82	81	adantrrr 439	. . . 4 |- ((k e. NN /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z))) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
83	82	adantrl 430	. . 3 |- ((k e. NN /\ (p e. X /\ ((r e. RR /\ 0 < r) /\ (x e. X /\ x =/= Z)))) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
84	83	ad2ant2r 445	. 2 |- (((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))) -> (((2 / r) x. (L` x)) x. (2 x. k)) = (((2 / r) x. (2 x. k)) x. (L` x)))
85	65, 74, 84	3brtr3d 3366	1 |- (((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)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = 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  CCcc 6384  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  normOpcnmo 9741   BLnOp cblo 9742

This theorem is referenced by:  ubthlem13 9885  ubthlem13OLD 9886

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-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-blo 9746

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