Theorem nmcfnlbi 27265

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 5803	. . . . . 6 |- ( A = 0h -> ( T ` A ) = ( T ` 0h ) )
2		nmcfnex.1	. . . . . . 7 |- T e. LinFn
3	2	lnfn0i 27255	. . . . . 6 |- ( T ` 0h ) = 0
4	1, 3	syl6eq 2457	. . . . 5 |- ( A = 0h -> ( T ` A ) = 0 )
5	4	abs00bd 13178	. . . 4 |- ( A = 0h -> ( abs ` ( T ` A ) ) = 0 )
6		0le0 10584	. . . . 5 |- 0 <_ 0
7		fveq2 5803	. . . . . . . 8 |- ( A = 0h -> ( normh ` A ) = ( normh ` 0h ) )
8		norm0 26340	. . . . . . . 8 |- ( normh ` 0h ) = 0
9	7, 8	syl6eq 2457	. . . . . . 7 |- ( A = 0h -> ( normh ` A ) = 0 )
10	9	oveq2d 6248	. . . . . 6 |- ( A = 0h -> ( ( normfn ` T ) x. ( normh ` A ) ) = ( ( normfn ` T ) x. 0 ) )
11		nmcfnex.2	. . . . . . . . 9 |- T e. ConFn
12	2, 11	nmcfnexi 27264	. . . . . . . 8 |- ( normfn ` T ) e. RR
13	12	recni 9556	. . . . . . 7 |- ( normfn ` T ) e. CC
14	13	mul01i 9722	. . . . . 6 |- ( ( normfn ` T ) x. 0 ) = 0
15	10, 14	syl6req 2458	. . . . 5 |- ( A = 0h -> 0 = ( ( normfn ` T ) x. ( normh ` A ) ) )
16	6, 15	syl5breq 4427	. . . 4 |- ( A = 0h -> 0 <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
17	5, 16	eqbrtrd 4412	. . 3 |- ( A = 0h -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
18	17	adantl 464	. 2 |- ( ( A e. ~H /\ A = 0h ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
19	2	lnfnfi 27254	. . . . . . . . . 10 |- T : ~H --> CC
20	19	ffvelrni 5962	. . . . . . . . 9 |- ( A e. ~H -> ( T ` A ) e. CC )
21	20	abscld 13321	. . . . . . . 8 |- ( A e. ~H -> ( abs ` ( T ` A ) ) e. RR )
22	21	adantr 463	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. RR )
23	22	recnd 9570	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. CC )
24		normcl 26337	. . . . . . . 8 |- ( A e. ~H -> ( normh ` A ) e. RR )
25	24	adantr 463	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. RR )
26	25	recnd 9570	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. CC )
27		norm-i 26341	. . . . . . . . 9 |- ( A e. ~H -> ( ( normh ` A ) = 0 <-> A = 0h ) )
28	27	notbid 292	. . . . . . . 8 |- ( A e. ~H -> ( -. ( normh ` A ) = 0 <-> -. A = 0h ) )
29	28	biimpar 483	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> -. ( normh ` A ) = 0 )
30	29	neqned 2604	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) =/= 0 )
31	23, 26, 30	divrec2d 10283	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) )
32	25, 30	rereccld 10330	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. RR )
33	32	recnd 9570	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. CC )
34		simpl 455	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> A e. ~H )
35	2	lnfnmuli 27257	. . . . . . . 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 659	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) = ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) )
37	36	fveq2d 5807	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) = ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) )
38	20	adantr 463	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( T ` A ) e. CC )
39	33, 38	absmuld 13339	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( abs ` ( T ` A ) ) ) )
40		df-ne 2598	. . . . . . . . . . . 12 |- ( A =/= 0h <-> -. A = 0h )
41		normgt0 26339	. . . . . . . . . . . 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 482	. . . . . . . . . 10 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( normh ` A ) )
44	25, 43	recgt0d 10438	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( 1 / ( normh ` A ) ) )
45		0re 9544	. . . . . . . . . 10 |- 0 e. RR
46		ltle 9622	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( 1 / ( normh ` A ) ) e. RR ) -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
47	45, 46	mpan 668	. . . . . . . . 9 |- ( ( 1 / ( normh ` A ) ) e. RR -> ( 0 < ( 1 / ( normh ` A ) ) -> 0 <_ ( 1 / ( normh ` A ) ) ) )
48	32, 44, 47	sylc 59	. . . . . . . 8 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 <_ ( 1 / ( normh ` A ) ) )
49	32, 48	absidd 13308	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( 1 / ( normh ` A ) ) ) = ( 1 / ( normh ` A ) ) )
50	49	oveq1d 6247	. . . . . 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 2446	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
52	31, 51	eqtrd 2441	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
53		hvmulcl 26225	. . . . . 6 |- ( ( ( 1 / ( normh ` A ) ) e. CC /\ A e. ~H ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
54	33, 34, 53	syl2anc 659	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) .h A ) e. ~H )
55		normcl 26337	. . . . . . 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 26462	. . . . . . 7 |- ( ( A e. ~H /\ A =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
58	40, 57	sylan2br 474	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) = 1 )
59		eqle 9636	. . . . . 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 659	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` ( ( 1 / ( normh ` A ) ) .h A ) ) <_ 1 )
61		nmfnlb 27137	. . . . . 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 1311	. . . . 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 659	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) <_ ( normfn ` T ) )
64	52, 63	eqbrtrd 4412	. . 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 10380	. . . 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 1232	. . 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 790	1 |- ( A e. ~H -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1403   e. wcel 1840   =/= wne 2596   class class class wbr 4392   -->wf 5519   ` cfv 5523  (class class class)co 6232   CCcc 9438   RRcr 9439   0cc0 9440   1c1 9441   x. cmul 9445   < clt 9576   <_ cle 9577   / cdiv 10165   abscabs 13121   ~Hchil 26131   .h csm 26133   normhcno 26135   0hc0v 26136   normfncnmf 26163   ConFnccnfn 26165   LinFnclf 26166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-hilex 26211  ax-hv0cl 26215  ax-hvaddid 26216  ax-hfvmul 26217  ax-hvmulid 26218  ax-hvmulass 26219  ax-hvmul0 26222  ax-hfi 26291  ax-his1 26294  ax-his3 26296  ax-his4 26297

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-seq 12060  df-exp 12119  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-hnorm 26180  df-hvsub 26183  df-nmfn 27058  df-cnfn 27060  df-lnfn 27061

This theorem is referenced by:  nmcfnlb  27267