Theorem nmcfnlbi 27786

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 5879	. . . . . 6 |- ( A = 0h -> ( T ` A ) = ( T ` 0h ) )
2		nmcfnex.1	. . . . . . 7 |- T e. LinFn
3	2	lnfn0i 27776	. . . . . 6 |- ( T ` 0h ) = 0
4	1, 3	syl6eq 2521	. . . . 5 |- ( A = 0h -> ( T ` A ) = 0 )
5	4	abs00bd 13431	. . . 4 |- ( A = 0h -> ( abs ` ( T ` A ) ) = 0 )
6		0le0 10721	. . . . 5 |- 0 <_ 0
7		fveq2 5879	. . . . . . . 8 |- ( A = 0h -> ( normh ` A ) = ( normh ` 0h ) )
8		norm0 26862	. . . . . . . 8 |- ( normh ` 0h ) = 0
9	7, 8	syl6eq 2521	. . . . . . 7 |- ( A = 0h -> ( normh ` A ) = 0 )
10	9	oveq2d 6324	. . . . . 6 |- ( A = 0h -> ( ( normfn ` T ) x. ( normh ` A ) ) = ( ( normfn ` T ) x. 0 ) )
11		nmcfnex.2	. . . . . . . . 9 |- T e. ConFn
12	2, 11	nmcfnexi 27785	. . . . . . . 8 |- ( normfn ` T ) e. RR
13	12	recni 9673	. . . . . . 7 |- ( normfn ` T ) e. CC
14	13	mul01i 9841	. . . . . 6 |- ( ( normfn ` T ) x. 0 ) = 0
15	10, 14	syl6req 2522	. . . . 5 |- ( A = 0h -> 0 = ( ( normfn ` T ) x. ( normh ` A ) ) )
16	6, 15	syl5breq 4431	. . . 4 |- ( A = 0h -> 0 <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
17	5, 16	eqbrtrd 4416	. . 3 |- ( A = 0h -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
18	17	adantl 473	. 2 |- ( ( A e. ~H /\ A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
19	2	lnfnfi 27775	. . . . . . . . . 10 |- T : ~H --> CC
20	19	ffvelrni 6036	. . . . . . . . 9 |- ( A e. ~H -> ( T ` A ) e. CC )
21	20	abscld 13575	. . . . . . . 8 |- ( A e. ~H -> ( abs ` ( T ` A ) ) e. RR )
22	21	adantr 472	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. RR )
23	22	recnd 9687	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. CC )
24		normcl 26859	. . . . . . . 8 |- ( A e. ~H -> ( normh ` A ) e. RR )
25	24	adantr 472	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. RR )
26	25	recnd 9687	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. CC )
27		norm-i 26863	. . . . . . . . 9 |- ( A e. ~H -> ( ( normh ` A ) = 0 <-> A = 0h ) )
28	27	notbid 301	. . . . . . . 8 |- ( A e. ~H -> ( -. ( normh ` A ) = 0 <-> -. A = 0h ) )
29	28	biimpar 493	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> -. ( normh ` A ) = 0 )
30	29	neqned 2650	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) =/= 0 )
31	23, 26, 30	divrec2d 10409	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) )
32	25, 30	rereccld 10456	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. RR )
33	32	recnd 9687	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. CC )
34		simpl 464	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> A e. ~H )
35	2	lnfnmuli 27778	. . . . . . . 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 673	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) )
37	36	fveq2d 5883	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) = ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) )
38	20	adantr 472	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` A ) e. CC )
39	33, 38	absmuld 13593	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( abs ` ( T ` A ) ) ) )
40		df-ne 2643	. . . . . . . . . . . 12 |- ( A =/= 0h <-> -. A = 0h )
41		normgt0 26861	. . . . . . . . . . . 12 |- ( A e. ~H -> ( A =/= 0h <-> 0 < ( normh ` A ) ) )
42	40, 41	syl5bbr 267	. . . . . . . . . . 11 |- ( A e. ~H -> ( -. A = 0h <-> 0 < ( normh ` A ) ) )
43	42	biimpa 492	. . . . . . . . . 10 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( normh ` A ) )
44	25, 43	recgt0d 10563	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( 1 / ( normh ` A ) ) )
45		0re 9661	. . . . . . . . . 10 |- 0 e. RR
46		ltle 9740	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( 1 / ( normh ` A ) ) e. RR ) -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
47	45, 46	mpan 684	. . . . . . . . 9 |- ( ( 1 / ( normh ` A ) ) e. RR -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
48	32, 44, 47	sylc 61	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 <_ ( 1 / ( normh ` A ) ) )
49	32, 48	absidd 13561	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( 1 / ( normh ` A ) ) ) = ( 1 / ( normh ` A ) ) )
50	49	oveq1d 6323	. . . . . 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 2510	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
52	31, 51	eqtrd 2505	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
53		hvmulcl 26747	. . . . . 6 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
54	33, 34, 53	syl2anc 673	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
55		normcl 26859	. . . . . . 7 |- ( ( ( 1 / ( normh ` A ) ) .h A ) e. ~H -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR )
56	54, 55	syl 17	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) e. RR )
57		norm1 26983	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
58	40, 57	sylan2br 484	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
59		eqle 9754	. . . . . 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 673	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
61		nmfnlb 27658	. . . . . 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 1377	. . . . 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 673	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normfn ` T ) )
64	52, 63	eqbrtrd 4416	. . 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 10506	. . . 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 1296	. . 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 215	. 2 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
69	18, 68	pm2.61dan 808	1 |- ( A e. ~H -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   class class class wbr 4395   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   x. cmul 9562   < clt 9693   <_ cle 9694   / cdiv 10291   abscabs 13374   ~Hchil 26653   .h csm 26655   normhcno 26657   0hc0v 26658   normfncnmf 26685   ConFnccnfn 26687   LinFnclf 26688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-hilex 26733  ax-hv0cl 26737  ax-hvaddid 26738  ax-hfvmul 26739  ax-hvmulid 26740  ax-hvmulass 26741  ax-hvmul0 26744  ax-hfi 26813  ax-his1 26816  ax-his3 26818  ax-his4 26819

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-hnorm 26702  df-hvsub 26705  df-nmfn 27579  df-cnfn 27581  df-lnfn 27582

This theorem is referenced by:  nmcfnlb  27788