Theorem nmcfnlbi 11624

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

Hypotheses

Ref	Expression
nmcfnex.1	|- T e. LinFn
nmcfnex.2	|- T e. ConFn

Assertion

Ref	Expression
nmcfnlbi	|- (A e. ~H -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))

Proof of Theorem nmcfnlbi

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . 7 |- (A = 0h -> (T` A) = (T` 0h))
2		nmcfnex.1	. . . . . . . 8 |- T e. LinFn
3	2	lnfn0i 11608	. . . . . . 7 |- (T` 0h) = 0
4	1, 3	syl6eq 1944	. . . . . 6 |- (A = 0h -> (T` A) = 0)
5	4	fveq2d 4685	. . . . 5 |- (A = 0h -> (abs` (T` A)) = (abs` 0))
6		abs0 8129	. . . . 5 |- (abs` 0) = 0
7	5, 6	syl6eq 1944	. . . 4 |- (A = 0h -> (abs` (T` A)) = 0)
8		fveq2 4681	. . . . . . . 8 |- (A = 0h -> (normh` A) = (normh` 0h))
9		norm0 10628	. . . . . . . 8 |- (normh` 0h) = 0
10	8, 9	syl6eq 1944	. . . . . . 7 |- (A = 0h -> (normh` A) = 0)
11	10	opreq2d 4898	. . . . . 6 |- (A = 0h -> ((normfn` T) x. (normh` A)) = ((normfn` T) x. 0))
12		nmcfnex.2	. . . . . . . . 9 |- T e. ConFn
13	2, 12	nmcfnexi 11623	. . . . . . . 8 |- (normfn` T) e. RR
14	13	recni 6467	. . . . . . 7 |- (normfn` T) e. CC
15	14	mul01i 6594	. . . . . 6 |- ((normfn` T) x. 0) = 0
16	11, 15	syl6req 1945	. . . . 5 |- (A = 0h -> 0 = ((normfn` T) x. (normh` A)))
17		0re 6603	. . . . . 6 |- 0 e. RR
18	17	leidi 6790	. . . . 5 |- 0 <_ 0
19	16, 18	syl5breq 3372	. . . 4 |- (A = 0h -> 0 <_ ((normfn` T) x. (normh` A)))
20	7, 19	eqbrtrd 3357	. . 3 |- (A = 0h -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
21	20	adantl 424	. 2 |- ((A e. ~H /\ A = 0h) -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
22	2	lnfnfi 11607	. . . . . . . . . 10 |- T:~H-->CC
23	22	ffvelrni 4788	. . . . . . . . 9 |- (A e. ~H -> (T` A) e. CC)
24		abscl 8084	. . . . . . . . 9 |- ((T` A) e. CC -> (abs` (T` A)) e. RR)
25	23, 24	syl 12	. . . . . . . 8 |- (A e. ~H -> (abs` (T` A)) e. RR)
26	25	adantr 425	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (abs` (T` A)) e. RR)
27	26	recnd 6468	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (abs` (T` A)) e. CC)
28		normcl 10624	. . . . . . . 8 |- (A e. ~H -> (normh` A) e. RR)
29	28	adantr 425	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (normh` A) e. RR)
30	29	recnd 6468	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` A) e. CC)
31		norm-i 10629	. . . . . . . . 9 |- (A e. ~H -> ((normh` A) = 0 <-> A = 0h))
32	31	notbid 673	. . . . . . . 8 |- (A e. ~H -> (-. (normh` A) = 0 <-> -. A = 0h))
33	32	biimpar 461	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> -. (normh` A) = 0)
34		df-ne 2019	. . . . . . 7 |- ((normh` A) =/= 0 <-> -. (normh` A) = 0)
35	33, 34	sylibr 217	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` A) =/= 0)
36		divrec2 6923	. . . . . 6 |- (((abs` (T` A)) e. CC /\ (normh` A) e. CC /\ (normh` A) =/= 0) -> ((abs` (T` A)) / (normh` A)) = ((1 / (normh` A)) x. (abs` (T` A))))
37	27, 30, 35, 36	syl111anc 1100	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> ((abs` (T` A)) / (normh` A)) = ((1 / (normh` A)) x. (abs` (T` A))))
38		rereccl 6981	. . . . . . . . . 10 |- (((normh` A) e. RR /\ (normh` A) =/= 0) -> (1 / (normh` A)) e. RR)
39	29, 35, 38	syl11anc 524	. . . . . . . . 9 |- ((A e. ~H /\ -. A = 0h) -> (1 / (normh` A)) e. RR)
40	39	recnd 6468	. . . . . . . 8 |- ((A e. ~H /\ -. A = 0h) -> (1 / (normh` A)) e. CC)
41		simpl 346	. . . . . . . 8 |- ((A e. ~H /\ -. A = 0h) -> A e. ~H)
42	2	lnfnmuli 11610	. . . . . . . 8 |- (((1 / (normh` A)) e. CC /\ A e. ~H) -> (T` ((1 / (normh` A)) .h A)) = ((1 / (normh` A)) x. (T` A)))
43	40, 41, 42	syl11anc 524	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (T` ((1 / (normh` A)) .h A)) = ((1 / (normh` A)) x. (T` A)))
44	43	fveq2d 4685	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (abs` (T` ((1 / (normh` A)) .h A))) = (abs` ((1 / (normh` A)) x. (T` A))))
45	23	adantr 425	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (T` A) e. CC)
46		absmul 8109	. . . . . . 7 |- (((1 / (normh` A)) e. CC /\ (T` A) e. CC) -> (abs` ((1 / (normh` A)) x. (T` A))) = ((abs` (1 / (normh` A))) x. (abs` (T` A))))
47	40, 45, 46	syl11anc 524	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (abs` ((1 / (normh` A)) x. (T` A))) = ((abs` (1 / (normh` A))) x. (abs` (T` A))))
48		normgt0OLD 10626	. . . . . . . . . . 11 |- (A e. ~H -> (-. A = 0h <-> 0 < (normh` A)))
49	48	biimpa 460	. . . . . . . . . 10 |- ((A e. ~H /\ -. A = 0h) -> 0 < (normh` A))
50		recgt0 7043	. . . . . . . . . 10 |- (((normh` A) e. RR /\ 0 < (normh` A)) -> 0 < (1 / (normh` A)))
51	29, 49, 50	syl11anc 524	. . . . . . . . 9 |- ((A e. ~H /\ -. A = 0h) -> 0 < (1 / (normh` A)))
52		ltle 6690	. . . . . . . . . 10 |- ((0 e. RR /\ (1 / (normh` A)) e. RR) -> (0 < (1 / (normh` A)) -> 0 <_ (1 / (normh` A))))
53	17, 52	mpan 759	. . . . . . . . 9 |- ((1 / (normh` A)) e. RR -> (0 < (1 / (normh` A)) -> 0 <_ (1 / (normh` A))))
54	39, 51, 53	sylc 83	. . . . . . . 8 |- ((A e. ~H /\ -. A = 0h) -> 0 <_ (1 / (normh` A)))
55		absid 8113	. . . . . . . 8 |- (((1 / (normh` A)) e. RR /\ 0 <_ (1 / (normh` A))) -> (abs` (1 / (normh` A))) = (1 / (normh` A)))
56	39, 54, 55	syl11anc 524	. . . . . . 7 |- ((A e. ~H /\ -. A = 0h) -> (abs` (1 / (normh` A))) = (1 / (normh` A)))
57	56	opreq1d 4897	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> ((abs` (1 / (normh` A))) x. (abs` (T` A))) = ((1 / (normh` A)) x. (abs` (T` A))))
58	44, 47, 57	3eqtrrd 1930	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> ((1 / (normh` A)) x. (abs` (T` A))) = (abs` (T` ((1 / (normh` A)) .h A))))
59	37, 58	eqtrd 1925	. . . 4 |- ((A e. ~H /\ -. A = 0h) -> ((abs` (T` A)) / (normh` A)) = (abs` (T` ((1 / (normh` A)) .h A))))
60		hvmulcl 10515	. . . . . 6 |- (((1 / (normh` A)) e. CC /\ A e. ~H) -> ((1 / (normh` A)) .h A) e. ~H)
61	40, 41, 60	syl11anc 524	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> ((1 / (normh` A)) .h A) e. ~H)
62		normcl 10624	. . . . . . 7 |- (((1 / (normh` A)) .h A) e. ~H -> (normh` ((1 / (normh` A)) .h A)) e. RR)
63	61, 62	syl 12	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h A)) e. RR)
64		norm1 10754	. . . . . . 7 |- ((A e. ~H /\ A =/= 0h) -> (normh` ((1 / (normh` A)) .h A)) = 1)
65		df-ne 2019	. . . . . . 7 |- (A =/= 0h <-> -. A = 0h)
66	64, 65	sylan2br 502	. . . . . 6 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h A)) = 1)
67		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)
68	63, 66, 67	syl11anc 524	. . . . 5 |- ((A e. ~H /\ -. A = 0h) -> (normh` ((1 / (normh` A)) .h A)) <_ 1)
69		nmfnlb 11485	. . . . . 6 |- ((T:~H-->CC /\ ((1 / (normh` A)) .h A) e. ~H /\ (normh` ((1 / (normh` A)) .h A)) <_ 1) -> (abs` (T` ((1 / (normh` A)) .h A))) <_ (normfn` T))
70	22, 69	mp3an1 1178	. . . . 5 |- ((((1 / (normh` A)) .h A) e. ~H /\ (normh` ((1 / (normh` A)) .h A)) <_ 1) -> (abs` (T` ((1 / (normh` A)) .h A))) <_ (normfn` T))
71	61, 68, 70	syl11anc 524	. . . 4 |- ((A e. ~H /\ -. A = 0h) -> (abs` (T` ((1 / (normh` A)) .h A))) <_ (normfn` T))
72	59, 71	eqbrtrd 3357	. . 3 |- ((A e. ~H /\ -. A = 0h) -> ((abs` (T` A)) / (normh` A)) <_ (normfn` T))
73	13	a1i 8	. . . 4 |- ((A e. ~H /\ -. A = 0h) -> (normfn` T) e. RR)
74		ledivmul2OLD 7057	. . . 4 |- ((((abs` (T` A)) e. RR /\ (normh` A) e. RR /\ (normfn` T) e. RR) /\ 0 < (normh` A)) -> (((abs` (T` A)) / (normh` A)) <_ (normfn` T) <-> (abs` (T` A)) <_ ((normfn` T) x. (normh` A))))
75	26, 29, 73, 49, 74	syl31anc 1103	. . 3 |- ((A e. ~H /\ -. A = 0h) -> (((abs` (T` A)) / (normh` A)) <_ (normfn` T) <-> (abs` (T` A)) <_ ((normfn` T) x. (normh` A))))
76	72, 75	mpbid 212	. 2 |- ((A e. ~H /\ -. A = 0h) -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
77	21, 76	pm2.61dan 535	1 |- (A e. ~H -> (abs` (T` A)) <_ ((normfn` 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_fncnmf 10452  ConFnccnf 10454  LinFnclf 10455

This theorem is referenced by:  nmcfnlb 11626

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-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  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-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-hnorm 10469  df-hvsub 10472  df-nmfn 11408  df-cnfn 11410  df-lnfn 11411

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