Theorem nmblolbii 9799

Description: A lower bound for the norm of a bounded linear operator.

Hypotheses

Ref	Expression
nmblolbi.1	|- X = (BaseSet` U)
nmblolbi.4	|- L = (norm` U)
nmblolbi.5	|- M = (norm` W)
nmblolbi.6	|- N = (UnormOpW)
nmblolbi.7	|- B = (U BLnOp W)
nmblolbi.u	|- U e. NrmCVec
nmblolbi.w	|- W e. NrmCVec
nmblolbii.b	|- T e. B

Assertion

Ref	Expression
nmblolbii	|- (A e. X -> (M` (T` A)) <_ ((N` T) x. (L` A)))

Proof of Theorem nmblolbii

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (A = (0v` U) -> (T` A) = (T` (0v` U)))
2	1	fveq2d 4685	. . 3 |- (A = (0v` U) -> (M` (T` A)) = (M` (T` (0v` U))))
3		fveq2 4681	. . . 4 |- (A = (0v` U) -> (L` A) = (L` (0v` U)))
4	3	opreq2d 4898	. . 3 |- (A = (0v` U) -> ((N` T) x. (L` A)) = ((N` T) x. (L` (0v` U))))
5	2, 4	breq12d 3351	. 2 |- (A = (0v` U) -> ((M` (T` A)) <_ ((N` T) x. (L` A)) <-> (M` (T` (0v` U))) <_ ((N` T) x. (L` (0v` U)))))
6		nmblolbi.u	. . . . . . . . 9 |- U e. NrmCVec
7		nmblolbi.1	. . . . . . . . . 10 |- X = (BaseSet` U)
8		nmblolbi.4	. . . . . . . . . 10 |- L = (norm` U)
9	7, 8	nvcl 9619	. . . . . . . . 9 |- ((U e. NrmCVec /\ A e. X) -> (L` A) e. RR)
10	6, 9	mpan 759	. . . . . . . 8 |- (A e. X -> (L` A) e. RR)
11	10	adantr 425	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> (L` A) e. RR)
12		eqid 1884	. . . . . . . . . . 11 |- (0v` U) = (0v` U)
13	7, 12, 8	nvz 9629	. . . . . . . . . 10 |- ((U e. NrmCVec /\ A e. X) -> ((L` A) = 0 <-> A = (0v` U)))
14	6, 13	mpan 759	. . . . . . . . 9 |- (A e. X -> ((L` A) = 0 <-> A = (0v` U)))
15	14	necon3bid 2035	. . . . . . . 8 |- (A e. X -> ((L` A) =/= 0 <-> A =/= (0v` U)))
16	15	biimpar 461	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> (L` A) =/= 0)
17		rereccl 6981	. . . . . . 7 |- (((L` A) e. RR /\ (L` A) =/= 0) -> (1 / (L` A)) e. RR)
18	11, 16, 17	syl11anc 524	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (1 / (L` A)) e. RR)
19	7, 12, 8	nvgt0 9635	. . . . . . . . . 10 |- ((U e. NrmCVec /\ A e. X) -> (A =/= (0v` U) <-> 0 < (L` A)))
20	6, 19	mpan 759	. . . . . . . . 9 |- (A e. X -> (A =/= (0v` U) <-> 0 < (L` A)))
21	20	biimpa 460	. . . . . . . 8 |- ((A e. X /\ A =/= (0v` U)) -> 0 < (L` A))
22		recgt0 7043	. . . . . . . 8 |- (((L` A) e. RR /\ 0 < (L` A)) -> 0 < (1 / (L` A)))
23	11, 21, 22	syl11anc 524	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> 0 < (1 / (L` A)))
24		0re 6603	. . . . . . . . 9 |- 0 e. RR
25		ltle 6690	. . . . . . . . 9 |- ((0 e. RR /\ (1 / (L` A)) e. RR) -> (0 < (1 / (L` A)) -> 0 <_ (1 / (L` A))))
26	24, 25	mpan 759	. . . . . . . 8 |- ((1 / (L` A)) e. RR -> (0 < (1 / (L` A)) -> 0 <_ (1 / (L` A))))
27	18, 26	syl 12	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> (0 < (1 / (L` A)) -> 0 <_ (1 / (L` A))))
28	23, 27	mpd 29	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> 0 <_ (1 / (L` A)))
29		nmblolbi.w	. . . . . . . . 9 |- W e. NrmCVec
30		nmblolbii.b	. . . . . . . . 9 |- T e. B
31		eqid 1884	. . . . . . . . . 10 |- (BaseSet` W) = (BaseSet` W)
32		nmblolbi.7	. . . . . . . . . 10 |- B = (U BLnOp W)
33	7, 31, 32	blof 9785	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> T:X-->(BaseSet` W))
34	6, 29, 30, 33	mp3an 1191	. . . . . . . 8 |- T:X-->(BaseSet` W)
35	34	ffvelrni 4788	. . . . . . 7 |- (A e. X -> (T` A) e. (BaseSet` W))
36	35	adantr 425	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (T` A) e. (BaseSet` W))
37		eqid 1884	. . . . . . . 8 |- (.s` W) = (.s` W)
38		nmblolbi.5	. . . . . . . 8 |- M = (norm` W)
39	31, 37, 38	nvsge0 9623	. . . . . . 7 |- ((W e. NrmCVec /\ ((1 / (L` A)) e. RR /\ 0 <_ (1 / (L` A))) /\ (T` A) e. (BaseSet` W)) -> (M` ((1 / (L` A))(.s` W)(T` A))) = ((1 / (L` A)) x. (M` (T` A))))
40	29, 39	mp3an1 1178	. . . . . 6 |- ((((1 / (L` A)) e. RR /\ 0 <_ (1 / (L` A))) /\ (T` A) e. (BaseSet` W)) -> (M` ((1 / (L` A))(.s` W)(T` A))) = ((1 / (L` A)) x. (M` (T` A))))
41	18, 28, 36, 40	syl21anc 1099	. . . . 5 |- ((A e. X /\ A =/= (0v` U)) -> (M` ((1 / (L` A))(.s` W)(T` A))) = ((1 / (L` A)) x. (M` (T` A))))
42	18	recnd 6468	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> (1 / (L` A)) e. CC)
43		simpl 346	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> A e. X)
44		eqid 1884	. . . . . . . . . . 11 |- (U LnOp W) = (U LnOp W)
45	44, 32	bloln 9784	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> T e. (U LnOp W))
46	6, 29, 30, 45	mp3an 1191	. . . . . . . . 9 |- T e. (U LnOp W)
47	6, 29, 46	3pm3.2i 1048	. . . . . . . 8 |- (U e. NrmCVec /\ W e. NrmCVec /\ T e. (U LnOp W))
48		eqid 1884	. . . . . . . . 9 |- (.s` U) = (.s` U)
49	7, 48, 37, 44	lnomul 9760	. . . . . . . 8 |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. (U LnOp W)) /\ ((1 / (L` A)) e. CC /\ A e. X)) -> (T` ((1 / (L` A))(.s` U)A)) = ((1 / (L` A))(.s` W)(T` A)))
50	47, 49	mpan 759	. . . . . . 7 |- (((1 / (L` A)) e. CC /\ A e. X) -> (T` ((1 / (L` A))(.s` U)A)) = ((1 / (L` A))(.s` W)(T` A)))
51	42, 43, 50	syl11anc 524	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (T` ((1 / (L` A))(.s` U)A)) = ((1 / (L` A))(.s` W)(T` A)))
52	51	fveq2d 4685	. . . . 5 |- ((A e. X /\ A =/= (0v` U)) -> (M` (T` ((1 / (L` A))(.s` U)A))) = (M` ((1 / (L` A))(.s` W)(T` A))))
53	31, 38	nvcl 9619	. . . . . . . . . 10 |- ((W e. NrmCVec /\ (T` A) e. (BaseSet` W)) -> (M` (T` A)) e. RR)
54	29, 53	mpan 759	. . . . . . . . 9 |- ((T` A) e. (BaseSet` W) -> (M` (T` A)) e. RR)
55	35, 54	syl 12	. . . . . . . 8 |- (A e. X -> (M` (T` A)) e. RR)
56	55	adantr 425	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> (M` (T` A)) e. RR)
57	56	recnd 6468	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (M` (T` A)) e. CC)
58	11	recnd 6468	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (L` A) e. CC)
59		divrec2 6923	. . . . . 6 |- (((M` (T` A)) e. CC /\ (L` A) e. CC /\ (L` A) =/= 0) -> ((M` (T` A)) / (L` A)) = ((1 / (L` A)) x. (M` (T` A))))
60	57, 58, 16, 59	syl111anc 1100	. . . . 5 |- ((A e. X /\ A =/= (0v` U)) -> ((M` (T` A)) / (L` A)) = ((1 / (L` A)) x. (M` (T` A))))
61	41, 52, 60	3eqtr4rd 1939	. . . 4 |- ((A e. X /\ A =/= (0v` U)) -> ((M` (T` A)) / (L` A)) = (M` (T` ((1 / (L` A))(.s` U)A))))
62	7, 48	nvscl 9579	. . . . . . 7 |- ((U e. NrmCVec /\ (1 / (L` A)) e. CC /\ A e. X) -> ((1 / (L` A))(.s` U)A) e. X)
63	6, 62	mp3an1 1178	. . . . . 6 |- (((1 / (L` A)) e. CC /\ A e. X) -> ((1 / (L` A))(.s` U)A) e. X)
64	42, 43, 63	syl11anc 524	. . . . 5 |- ((A e. X /\ A =/= (0v` U)) -> ((1 / (L` A))(.s` U)A) e. X)
65	63	ancoms 484	. . . . . . . 8 |- ((A e. X /\ (1 / (L` A)) e. CC) -> ((1 / (L` A))(.s` U)A) e. X)
66	42, 65	syldan 516	. . . . . . 7 |- ((A e. X /\ A =/= (0v` U)) -> ((1 / (L` A))(.s` U)A) e. X)
67	7, 8	nvcl 9619	. . . . . . . 8 |- ((U e. NrmCVec /\ ((1 / (L` A))(.s` U)A) e. X) -> (L` ((1 / (L` A))(.s` U)A)) e. RR)
68	6, 67	mpan 759	. . . . . . 7 |- (((1 / (L` A))(.s` U)A) e. X -> (L` ((1 / (L` A))(.s` U)A)) e. RR)
69	66, 68	syl 12	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (L` ((1 / (L` A))(.s` U)A)) e. RR)
70	7, 48, 12, 8	nv1 9636	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ A =/= (0v` U)) -> (L` ((1 / (L` A))(.s` U)A)) = 1)
71	6, 70	mp3an1 1178	. . . . . 6 |- ((A e. X /\ A =/= (0v` U)) -> (L` ((1 / (L` A))(.s` U)A)) = 1)
72		eqle 6746	. . . . . 6 |- (((L` ((1 / (L` A))(.s` U)A)) e. RR /\ (L` ((1 / (L` A))(.s` U)A)) = 1) -> (L` ((1 / (L` A))(.s` U)A)) <_ 1)
73	69, 71, 72	syl11anc 524	. . . . 5 |- ((A e. X /\ A =/= (0v` U)) -> (L` ((1 / (L` A))(.s` U)A)) <_ 1)
74	6, 29, 34	3pm3.2i 1048	. . . . . 6 |- (U e. NrmCVec /\ W e. NrmCVec /\ T:X-->(BaseSet` W))
75		nmblolbi.6	. . . . . . 7 |- N = (UnormOpW)
76	7, 31, 8, 38, 75	nmolb 9773	. . . . . 6 |- (((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->(BaseSet` W)) /\ (((1 / (L` A))(.s` U)A) e. X /\ (L` ((1 / (L` A))(.s` U)A)) <_ 1)) -> (M` (T` ((1 / (L` A))(.s` U)A))) <_ (N` T))
77	74, 76	mpan 759	. . . . 5 |- ((((1 / (L` A))(.s` U)A) e. X /\ (L` ((1 / (L` A))(.s` U)A)) <_ 1) -> (M` (T` ((1 / (L` A))(.s` U)A))) <_ (N` T))
78	64, 73, 77	syl11anc 524	. . . 4 |- ((A e. X /\ A =/= (0v` U)) -> (M` (T` ((1 / (L` A))(.s` U)A))) <_ (N` T))
79	61, 78	eqbrtrd 3357	. . 3 |- ((A e. X /\ A =/= (0v` U)) -> ((M` (T` A)) / (L` A)) <_ (N` T))
80	7, 31, 75, 32	nmblore 9786	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> (N` T) e. RR)
81	6, 29, 30, 80	mp3an 1191	. . . . . . 7 |- (N` T) e. RR
82	81	a1i 8	. . . . . 6 |- (A e. X -> (N` T) e. RR)
83	55, 10, 82	3jca 1050	. . . . 5 |- (A e. X -> ((M` (T` A)) e. RR /\ (L` A) e. RR /\ (N` T) e. RR))
84	83	adantr 425	. . . 4 |- ((A e. X /\ A =/= (0v` U)) -> ((M` (T` A)) e. RR /\ (L` A) e. RR /\ (N` T) e. RR))
85		ledivmul2OLD 7057	. . . 4 |- ((((M` (T` A)) e. RR /\ (L` A) e. RR /\ (N` T) e. RR) /\ 0 < (L` A)) -> (((M` (T` A)) / (L` A)) <_ (N` T) <-> (M` (T` A)) <_ ((N` T) x. (L` A))))
86	84, 21, 85	syl11anc 524	. . 3 |- ((A e. X /\ A =/= (0v` U)) -> (((M` (T` A)) / (L` A)) <_ (N` T) <-> (M` (T` A)) <_ ((N` T) x. (L` A))))
87	79, 86	mpbid 212	. 2 |- ((A e. X /\ A =/= (0v` U)) -> (M` (T` A)) <_ ((N` T) x. (L` A)))
88	24	leidi 6790	. . . 4 |- 0 <_ 0
89		eqid 1884	. . . . . . . 8 |- (0v` W) = (0v` W)
90	7, 31, 12, 89, 44	lno0 9756	. . . . . . 7 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. (U LnOp W)) -> (T` (0v` U)) = (0v` W))
91	6, 29, 46, 90	mp3an 1191	. . . . . 6 |- (T` (0v` U)) = (0v` W)
92	91	fveq2i 4684	. . . . 5 |- (M` (T` (0v` U))) = (M` (0v` W))
93	89, 38	nvz0 9628	. . . . . 6 |- (W e. NrmCVec -> (M` (0v` W)) = 0)
94	29, 93	ax-mp 7	. . . . 5 |- (M` (0v` W)) = 0
95	92, 94	eqtri 1908	. . . 4 |- (M` (T` (0v` U))) = 0
96	12, 8	nvz0 9628	. . . . . . 7 |- (U e. NrmCVec -> (L` (0v` U)) = 0)
97	6, 96	ax-mp 7	. . . . . 6 |- (L` (0v` U)) = 0
98	97	opreq2i 4893	. . . . 5 |- ((N` T) x. (L` (0v` U))) = ((N` T) x. 0)
99	81	recni 6467	. . . . . 6 |- (N` T) e. CC
100	99	mul01i 6594	. . . . 5 |- ((N` T) x. 0) = 0
101	98, 100	eqtri 1908	. . . 4 |- ((N` T) x. (L` (0v` U))) = 0
102	88, 95, 101	3brtr4i 3365	. . 3 |- (M` (T` (0v` U))) <_ ((N` T) x. (L` (0v` U)))
103	102	a1i 8	. 2 |- (A e. X -> (M` (T` (0v` U))) <_ ((N` T) x. (L` (0v` U))))
104	5, 87, 103	pm2.61ne 2087	1 |- (A e. X -> (M` (T` A)) <_ ((N` T) x. (L` A)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447   <_ cle 6448   < clt 6653  NrmCVeccnv 9535  BaseSetcba 9537  .scns 9538  0vcn0v 9539  normcnm 9541   LnOp clno 9740  normOpcnmo 9741   BLnOp cblo 9742

This theorem is referenced by:  nmblolbi 9800

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-2 7154  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-lno 9744  df-nmo 9745  df-blo 9746

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