Theorem nmcfnlbi 27640

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 5818	. . . . . 6 |- ( A = 0h -> ( T ` A ) = ( T ` 0h ) )
2		nmcfnex.1	. . . . . . 7 |- T e. LinFn
3	2	lnfn0i 27630	. . . . . 6 |- ( T ` 0h ) = 0
4	1, 3	syl6eq 2472	. . . . 5 |- ( A = 0h -> ( T ` A ) = 0 )
5	4	abs00bd 13291	. . . 4 |- ( A = 0h -> ( abs ` ( T ` A ) ) = 0 )
6		0le0 10643	. . . . 5 |- 0 <_ 0
7		fveq2 5818	. . . . . . . 8 |- ( A = 0h -> ( normh ` A ) = ( normh ` 0h ) )
8		norm0 26716	. . . . . . . 8 |- ( normh ` 0h ) = 0
9	7, 8	syl6eq 2472	. . . . . . 7 |- ( A = 0h -> ( normh ` A ) = 0 )
10	9	oveq2d 6258	. . . . . 6 |- ( A = 0h -> ( ( normfn ` T ) x. ( normh ` A ) ) = ( ( normfn ` T ) x. 0 ) )
11		nmcfnex.2	. . . . . . . . 9 |- T e. ConFn
12	2, 11	nmcfnexi 27639	. . . . . . . 8 |- ( normfn ` T ) e. RR
13	12	recni 9599	. . . . . . 7 |- ( normfn ` T ) e. CC
14	13	mul01i 9767	. . . . . 6 |- ( ( normfn ` T ) x. 0 ) = 0
15	10, 14	syl6req 2473	. . . . 5 |- ( A = 0h -> 0 = ( ( normfn ` T ) x. ( normh ` A ) ) )
16	6, 15	syl5breq 4395	. . . 4 |- ( A = 0h -> 0 <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
17	5, 16	eqbrtrd 4380	. . 3 |- ( A = 0h -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
18	17	adantl 467	. 2 |- ( ( A e. ~H /\ A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
19	2	lnfnfi 27629	. . . . . . . . . 10 |- T : ~H --> CC
20	19	ffvelrni 5973	. . . . . . . . 9 |- ( A e. ~H -> ( T ` A ) e. CC )
21	20	abscld 13434	. . . . . . . 8 |- ( A e. ~H -> ( abs ` ( T ` A ) ) e. RR )
22	21	adantr 466	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. RR )
23	22	recnd 9613	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. CC )
24		normcl 26713	. . . . . . . 8 |- ( A e. ~H -> ( normh ` A ) e. RR )
25	24	adantr 466	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. RR )
26	25	recnd 9613	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. CC )
27		norm-i 26717	. . . . . . . . 9 |- ( A e. ~H -> ( ( normh ` A ) = 0 <-> A = 0h ) )
28	27	notbid 295	. . . . . . . 8 |- ( A e. ~H -> ( -. ( normh ` A ) = 0 <-> -. A = 0h ) )
29	28	biimpar 487	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> -. ( normh ` A ) = 0 )
30	29	neqned 2602	. . . . . 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 9613	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. CC )
34		simpl 458	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> A e. ~H )
35	2	lnfnmuli 27632	. . . . . . . 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 665	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) )
37	36	fveq2d 5822	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) = ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) )
38	20	adantr 466	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` A ) e. CC )
39	33, 38	absmuld 13452	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( abs ` ( T ` A ) ) ) )
40		df-ne 2595	. . . . . . . . . . . 12 |- ( A =/= 0h <-> -. A = 0h )
41		normgt0 26715	. . . . . . . . . . . 12 |- ( A e. ~H -> ( A =/= 0h <-> 0 < ( normh ` A ) ) )
42	40, 41	syl5bbr 262	. . . . . . . . . . 11 |- ( A e. ~H -> ( -. A = 0h <-> 0 < ( normh ` A ) ) )
43	42	biimpa 486	. . . . . . . . . 10 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( normh ` A ) )
44	25, 43	recgt0d 10485	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( 1 / ( normh ` A ) ) )
45		0re 9587	. . . . . . . . . 10 |- 0 e. RR
46		ltle 9666	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( 1 / ( normh ` A ) ) e. RR ) -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
47	45, 46	mpan 674	. . . . . . . . 9 |- ( ( 1 / ( normh ` A ) ) e. RR -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
48	32, 44, 47	sylc 62	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 <_ ( 1 / ( normh ` A ) ) )
49	32, 48	absidd 13421	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( 1 / ( normh ` A ) ) ) = ( 1 / ( normh ` A ) ) )
50	49	oveq1d 6257	. . . . . 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 2461	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
52	31, 51	eqtrd 2456	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
53		hvmulcl 26601	. . . . . 6 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
54	33, 34, 53	syl2anc 665	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
55		normcl 26713	. . . . . . 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 26837	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
58	40, 57	sylan2br 478	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
59		eqle 9680	. . . . . 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 665	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
61		nmfnlb 27512	. . . . . 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 1347	. . . . 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 665	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normfn ` T ) )
64	52, 63	eqbrtrd 4380	. . 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 10428	. . . 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 1268	. . 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 213	. 2 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
69	18, 68	pm2.61dan 798	1 |- ( A e. ~H -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2593   class class class wbr 4359   -->wf 5533   ` cfv 5537  (class class class)co 6242   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484   x. cmul 9488   < clt 9619   <_ cle 9620   / cdiv 10213   abscabs 13234   ~Hchil 26507   .h csm 26509   normhcno 26511   0hc0v 26512   normfncnmf 26539   ConFnccnfn 26541   LinFnclf 26542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-hilex 26587  ax-hv0cl 26591  ax-hvaddid 26592  ax-hfvmul 26593  ax-hvmulid 26594  ax-hvmulass 26595  ax-hvmul0 26598  ax-hfi 26667  ax-his1 26670  ax-his3 26672  ax-his4 26673

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-om 6644  df-2nd 6745  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-er 7311  df-map 7422  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-div 10214  df-nn 10554  df-2 10612  df-3 10613  df-n0 10814  df-z 10882  df-uz 11104  df-rp 11247  df-seq 12157  df-exp 12216  df-cj 13099  df-re 13100  df-im 13101  df-sqrt 13235  df-abs 13236  df-hnorm 26556  df-hvsub 26559  df-nmfn 27433  df-cnfn 27435  df-lnfn 27436

This theorem is referenced by:  nmcfnlb  27642