Theorem nmcoplbi 11595

Description: A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99.

Hypotheses

Ref	Expression
nmcopex.1	|- T e. LinOp
nmcopex.2	|- T e. ConOp

Assertion

Ref	Expression
nmcoplbi	|- (A e. ~H -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))

Proof of Theorem nmcoplbi

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . 7 |- (A = 0h -> (T` A) = (T` 0h))
2		nmcopex.1	. . . . . . . 8 |- T e. LinOp
3	2	lnop0i 11531	. . . . . . 7 |- (T` 0h) = 0h
4	1, 3	syl6eq 1944	. . . . . 6 |- (A = 0h -> (T` A) = 0h)
5	4	fveq2d 4685	. . . . 5 |- (A = 0h -> (normh` (T` A)) = (normh` 0h))
6		norm0 10628	. . . . 5 |- (normh` 0h) = 0
7	5, 6	syl6eq 1944	. . . 4 |- (A = 0h -> (normh` (T` A)) = 0)
8		fveq2 4681	. . . . . . . 8 |- (A = 0h -> (normh` A) = (normh` 0h))
9	8, 6	syl6eq 1944	. . . . . . 7 |- (A = 0h -> (normh` A) = 0)
10	9	opreq2d 4898	. . . . . 6 |- (A = 0h -> ((normop` T) x. (normh` A)) = ((normop` T) x. 0))
11		nmcopex.2	. . . . . . . . 9 |- T e. ConOp
12	2, 11	nmcopexi 11594	. . . . . . . 8 |- (normop` T) e. RR
13	12	recni 6467	. . . . . . 7 |- (normop` T) e. CC
14	13	mul01i 6594	. . . . . 6 |- ((normop` T) x. 0) = 0
15	10, 14	syl6req 1945	. . . . 5 |- (A = 0h -> 0 = ((normop` T) x. (normh` A)))
16		0re 6603	. . . . . 6 |- 0 e. RR
17	16	leidi 6790	. . . . 5 |- 0 <_ 0
18	15, 17	syl5breq 3372	. . . 4 |- (A = 0h -> 0 <_ ((normop` T) x. (normh` A)))
19	7, 18	eqbrtrd 3357	. . 3 |- (A = 0h -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
20	19	adantl 424	. 2 |- ((A e. ~H /\ A = 0h) -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
21	2	lnopfi 11530	. . . . . . . . . 10 |- T:~H-->~H
22	21	ffvelrni 4788	. . . . . . . . 9 |- (A e. ~H -> (T` A) e. ~H)
23		normcl 10624	. . . . . . . . 9 |- ((T` A) e. ~H -> (normh` (T` A)) e. RR)
24	22, 23	syl 12	. . . . . . . 8 |- (A e. ~H -> (normh` (T` A)) e. RR)
25	24	adantr 425	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (normh` (T` A)) e. RR)
26	25	recnd 6468	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` (T` A)) e. CC)
27		normcl 10624	. . . . . . . 8 |- (A e. ~H -> (normh` A) e. RR)
28	27	adantr 425	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (normh` A) e. RR)
29	28	recnd 6468	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` A) e. CC)
30		norm-i 10629	. . . . . . . . 9 |- (A e. ~H -> ((normh` A) = 0 <-> A = 0h))
31	30	notbid 673	. . . . . . . 8 |- (A e. ~H -> (-. (normh` A) = 0 <-> -. A = 0h))
32	31	biimpar 461	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> -. (normh` A) = 0)
33		df-ne 2019	. . . . . . 7 |- ((normh` A) =/= 0 <-> -. (normh` A) = 0)
34	32, 33	sylibr 217	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` A) =/= 0)
35		divrec2 6923	. . . . . 6 |- (((normh` (T` A)) e. CC /\ (normh` A) e. CC /\ (normh` A) =/= 0) -> ((normh` (T` A)) / (normh` A)) = ((1 / (normh` A)) x. (normh` (T` A))))
36	26, 29, 34, 35	syl111anc 1100	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> ((normh` (T` A)) / (normh` A)) = ((1 / (normh` A)) x. (normh` (T` A))))
37		rereccl 6981	. . . . . . . . . 10 |- (((normh` A) e. RR /\ (normh` A) =/= 0) -> (1 / (normh` A)) e. RR)
38	28, 34, 37	syl11anc 524	. . . . . . . . 9 |- ((A e. ~H /\ -. A = 0h) -> (1 / (normh` A)) e. RR)
39	38	recnd 6468	. . . . . . . 8 |- ((A e. ~H /\ -. A = 0h) -> (1 / (normh` A)) e. CC)
40		simpl 346	. . . . . . . 8 |- ((A e. ~H /\ -. A = 0h) -> A e. ~H)
41	2	lnopmuli 11533	. . . . . . . 8 |- (((1 / (normh` A)) e. CC /\ A e. ~H) -> (T` ((1 / (normh` A)) .h A)) = ((1 / (normh` A)) .h (T` A)))
42	39, 40, 41	syl11anc 524	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (T` ((1 / (normh` A)) .h A)) = ((1 / (normh` A)) .h (T` A)))
43	42	fveq2d 4685	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` (T` ((1 / (normh` A)) .h A))) = (normh` ((1 / (normh` A)) .h (T` A))))
44	22	adantr 425	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (T` A) e. ~H)
45		norm-iii 10640	. . . . . . 7 |- (((1 / (normh` A)) e. CC /\ (T` A) e. ~H) -> (normh` ((1 / (normh` A)) .h (T` A))) = ((abs` (1 / (normh` A))) x. (normh` (T` A))))
46	39, 44, 45	syl11anc 524	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h (T` A))) = ((abs` (1 / (normh` A))) x. (normh` (T` A))))
47		normgt0OLD 10626	. . . . . . . . . . 11 |- (A e. ~H -> (-. A = 0h <-> 0 < (normh` A)))
48	47	biimpa 460	. . . . . . . . . 10 |- ((A e. ~H /\ -. A = 0h) -> 0 < (normh` A))
49		recgt0 7043	. . . . . . . . . 10 |- (((normh` A) e. RR /\ 0 < (normh` A)) -> 0 < (1 / (normh` A)))
50	28, 48, 49	syl11anc 524	. . . . . . . . 9 |- ((A e. ~H /\ -. A = 0h) -> 0 < (1 / (normh` A)))
51		ltle 6690	. . . . . . . . . 10 |- ((0 e. RR /\ (1 / (normh` A)) e. RR) -> (0 < (1 / (normh` A)) -> 0 <_ (1 / (normh` A))))
52	16, 51	mpan 759	. . . . . . . . 9 |- ((1 / (normh` A)) e. RR -> (0 < (1 / (normh` A)) -> 0 <_ (1 / (normh` A))))
53	38, 50, 52	sylc 83	. . . . . . . 8 |- ((A e. ~H /\ -. A = 0h) -> 0 <_ (1 / (normh` A)))
54		absid 8113	. . . . . . . 8 |- (((1 / (normh` A)) e. RR /\ 0 <_ (1 / (normh` A))) -> (abs` (1 / (normh` A))) = (1 / (normh` A)))
55	38, 53, 54	syl11anc 524	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (abs` (1 / (normh` A))) = (1 / (normh` A)))
56	55	opreq1d 4897	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> ((abs` (1 / (normh` A))) x. (normh` (T` A))) = ((1 / (normh` A)) x. (normh` (T` A))))
57	43, 46, 56	3eqtrrd 1930	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> ((1 / (normh` A)) x. (normh` (T` A))) = (normh` (T` ((1 / (normh` A)) .h A))))
58	36, 57	eqtrd 1925	. . . 4 |- ((A e. ~H /\ -. A = 0h) -> ((normh` (T` A)) / (normh` A)) = (normh` (T` ((1 / (normh` A)) .h A))))
59		hvmulcl 10515	. . . . . 6 |- (((1 / (normh` A)) e. CC /\ A e. ~H) -> ((1 / (normh` A)) .h A) e. ~H)
60	39, 40, 59	syl11anc 524	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> ((1 / (normh` A)) .h A) e. ~H)
61		normcl 10624	. . . . . . 7 |- (((1 / (normh` A)) .h A) e. ~H -> (normh` ((1 / (normh` A)) .h A)) e. RR)
62	60, 61	syl 12	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h A)) e. RR)
63		norm1 10754	. . . . . . 7 |- ((A e. ~H /\ A =/= 0h) -> (normh` ((1 / (normh` A)) .h A)) = 1)
64		df-ne 2019	. . . . . . 7 |- (A =/= 0h <-> -. A = 0h)
65	63, 64	sylan2br 502	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h A)) = 1)
66		eqle 6746	. . . . . 6 |- (((normh` ((1 / (normh` A)) .h A)) e. RR /\ (normh` ((1 / (normh` A)) .h A)) = 1) -> (normh` ((1 / (normh` A)) .h A)) <_ 1)
67	62, 65, 66	syl11anc 524	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h A)) <_ 1)
68		nmoplb 11468	. . . . . 6 |- ((T:~H-->~H /\ ((1 / (normh` A)) .h A) e. ~H /\ (normh` ((1 / (normh` A)) .h A)) <_ 1) -> (normh` (T` ((1 / (normh` A)) .h A))) <_ (normop` T))
69	21, 68	mp3an1 1178	. . . . 5 |- ((((1 / (normh` A)) .h A) e. ~H /\ (normh` ((1 / (normh` A)) .h A)) <_ 1) -> (normh` (T` ((1 / (normh` A)) .h A))) <_ (normop` T))
70	60, 67, 69	syl11anc 524	. . . 4 |- ((A e. ~H /\ -. A = 0h) -> (normh` (T` ((1 / (normh` A)) .h A))) <_ (normop` T))
71	58, 70	eqbrtrd 3357	. . 3 |- ((A e. ~H /\ -. A = 0h) -> ((normh` (T` A)) / (normh` A)) <_ (normop` T))
72	12	a1i 8	. . . 4 |- ((A e. ~H /\ -. A = 0h) -> (normop` T) e. RR)
73		ledivmul2OLD 7057	. . . 4 |- ((((normh` (T` A)) e. RR /\ (normh` A) e. RR /\ (normop` T) e. RR) /\ 0 < (normh` A)) -> (((normh` (T` A)) / (normh` A)) <_ (normop` T) <-> (normh` (T` A)) <_ ((normop` T) x. (normh` A))))
74	25, 28, 72, 48, 73	syl31anc 1103	. . 3 |- ((A e. ~H /\ -. A = 0h) -> (((normh` (T` A)) / (normh` A)) <_ (normop` T) <-> (normh` (T` A)) <_ ((normop` T) x. (normh` A))))
75	71, 74	mpbid 212	. 2 |- ((A e. ~H /\ -. A = 0h) -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
76	20, 75	pm2.61dan 535	1 |- (A e. ~H -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = 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  abscabs 8000  ~Hchil 10420   .h csm 10422  0hc0v 10423  normhcno 10426  norm_opcnop 10446  ConOpcco 10447  LinOpclo 10448

This theorem is referenced by:  nmcoplb 11597  cnlnadjlem2 11638  cnlnadjlem7 11643

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  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585

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-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-exp 7812  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-hnorm 10469  df-hvsub 10472  df-nmop 11402  df-cnop 11403  df-lnop 11404

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