Theorem nmcfnlbi 26949

Description: A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

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 5856	. . . . . 6 |- ( A = 0h -> ( T ` A ) = ( T ` 0h ) )
2		nmcfnex.1	. . . . . . 7 |- T e. LinFn
3	2	lnfn0i 26939	. . . . . 6 |- ( T ` 0h ) = 0
4	1, 3	syl6eq 2500	. . . . 5 |- ( A = 0h -> ( T ` A ) = 0 )
5	4	abs00bd 13106	. . . 4 |- ( A = 0h -> ( abs ` ( T ` A ) ) = 0 )
6		0le0 10632	. . . . 5 |- 0 <_ 0
7		fveq2 5856	. . . . . . . 8 |- ( A = 0h -> ( normh ` A ) = ( normh ` 0h ) )
8		norm0 26023	. . . . . . . 8 |- ( normh ` 0h ) = 0
9	7, 8	syl6eq 2500	. . . . . . 7 |- ( A = 0h -> ( normh ` A ) = 0 )
10	9	oveq2d 6297	. . . . . 6 |- ( A = 0h -> ( ( normfn ` T ) x. ( normh ` A ) ) = ( ( normfn ` T ) x. 0 ) )
11		nmcfnex.2	. . . . . . . . 9 |- T e. ConFn
12	2, 11	nmcfnexi 26948	. . . . . . . 8 |- ( normfn ` T ) e. RR
13	12	recni 9611	. . . . . . 7 |- ( normfn ` T ) e. CC
14	13	mul01i 9773	. . . . . 6 |- ( ( normfn ` T ) x. 0 ) = 0
15	10, 14	syl6req 2501	. . . . 5 |- ( A = 0h -> 0 = ( ( normfn ` T ) x. ( normh ` A ) ) )
16	6, 15	syl5breq 4472	. . . 4 |- ( A = 0h -> 0 <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
17	5, 16	eqbrtrd 4457	. . 3 |- ( A = 0h -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
18	17	adantl 466	. 2 |- ( ( A e. ~H /\ A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
19	2	lnfnfi 26938	. . . . . . . . . 10 |- T : ~H --> CC
20	19	ffvelrni 6015	. . . . . . . . 9 |- ( A e. ~H -> ( T ` A ) e. CC )
21	20	abscld 13249	. . . . . . . 8 |- ( A e. ~H -> ( abs ` ( T ` A ) ) e. RR )
22	21	adantr 465	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. RR )
23	22	recnd 9625	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. CC )
24		normcl 26020	. . . . . . . 8 |- ( A e. ~H -> ( normh ` A ) e. RR )
25	24	adantr 465	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. RR )
26	25	recnd 9625	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. CC )
27		norm-i 26024	. . . . . . . . 9 |- ( A e. ~H -> ( ( normh ` A ) = 0 <-> A = 0h ) )
28	27	notbid 294	. . . . . . . 8 |- ( A e. ~H -> ( -. ( normh ` A ) = 0 <-> -. A = 0h ) )
29	28	biimpar 485	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> -. ( normh ` A ) = 0 )
30	29	neqned 2646	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) =/= 0 )
31	23, 26, 30	divrec2d 10331	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) )
32	25, 30	rereccld 10378	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. RR )
33	32	recnd 9625	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. CC )
34		simpl 457	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> A e. ~H )
35	2	lnfnmuli 26941	. . . . . . . 8 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) )
36	33, 34, 35	syl2anc 661	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) )
37	36	fveq2d 5860	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) = ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) )
38	20	adantr 465	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` A ) e. CC )
39	33, 38	absmuld 13267	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( abs ` ( T ` A ) ) ) )
40		df-ne 2640	. . . . . . . . . . . 12 |- ( A =/= 0h <-> -. A = 0h )
41		normgt0 26022	. . . . . . . . . . . 12 |- ( A e. ~H -> ( A =/= 0h <-> 0 < ( normh ` A ) ) )
42	40, 41	syl5bbr 259	. . . . . . . . . . 11 |- ( A e. ~H -> ( -. A = 0h <-> 0 < ( normh ` A ) ) )
43	42	biimpa 484	. . . . . . . . . 10 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( normh ` A ) )
44	25, 43	recgt0d 10487	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( 1 / ( normh ` A ) ) )
45		0re 9599	. . . . . . . . . 10 |- 0 e. RR
46		ltle 9676	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( 1 / ( normh ` A ) ) e. RR ) -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
47	45, 46	mpan 670	. . . . . . . . 9 |- ( ( 1 / ( normh ` A ) ) e. RR -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
48	32, 44, 47	sylc 60	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 <_ ( 1 / ( normh ` A ) ) )
49	32, 48	absidd 13236	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( 1 / ( normh ` A ) ) ) = ( 1 / ( normh ` A ) ) )
50	49	oveq1d 6296	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( abs ` ( T ` A ) ) ) = ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) )
51	37, 39, 50	3eqtrrd 2489	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
52	31, 51	eqtrd 2484	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
53		hvmulcl 25908	. . . . . 6 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
54	33, 34, 53	syl2anc 661	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
55		normcl 26020	. . . . . . 7 |- ( ( ( 1 / ( normh ` A ) ) .h A ) e. ~H -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR )
56	54, 55	syl 16	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR )
57		norm1 26145	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
58	40, 57	sylan2br 476	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
59		eqle 9690	. . . . . 6 |- ( ( ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR /\ ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
60	56, 58, 59	syl2anc 661	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
61		nmfnlb 26821	. . . . . 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 ) )
62	19, 61	mp3an1 1312	. . . . 5 |- ( ( ( ( 1 / ( normh ` A ) ) .h A ) e. ~H /\ ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normfn ` T ) )
63	54, 60, 62	syl2anc 661	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normfn ` T ) )
64	52, 63	eqbrtrd 4457	. . 3 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) <_ ( normfn ` T ) )
65	12	a1i 11	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normfn ` T ) e. RR )
66		ledivmul2 10429	. . . 4 |- ( ( ( abs ` ( T ` A ) ) e. RR /\ ( normfn ` T ) e. RR /\ ( ( normh ` A ) e. RR /\ 0 < ( normh ` A ) ) ) -> ( ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) <_ ( normfn ` T ) <-> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) ) )
67	22, 65, 25, 43, 66	syl112anc 1233	. . 3 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) <_ ( normfn ` T ) <-> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) ) )
68	64, 67	mpbid 210	. 2 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
69	18, 68	pm2.61dan 791	1 |- ( A e. ~H -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   class class class wbr 4437   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   x. cmul 9500   < clt 9631   <_ cle 9632   / cdiv 10213   abscabs 13049   ~Hchil 25814   .h csm 25816   normhcno 25818   0hc0v 25819   normfncnmf 25846   ConFnccnfn 25848   LinFnclf 25849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-hilex 25894  ax-hv0cl 25898  ax-hvaddid 25899  ax-hfvmul 25900  ax-hvmulid 25901  ax-hvmulass 25902  ax-hvmul0 25905  ax-hfi 25974  ax-his1 25977  ax-his3 25979  ax-his4 25980

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-seq 12090  df-exp 12149  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-hnorm 25863  df-hvsub 25866  df-nmfn 26742  df-cnfn 26744  df-lnfn 26745

This theorem is referenced by:  nmcfnlb  26951