Theorem nmcfnlbi 25601

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 5792	. . . . . 6 |- ( A = 0h -> ( T ` A ) = ( T ` 0h ) )
2		nmcfnex.1	. . . . . . 7 |- T e. LinFn
3	2	lnfn0i 25591	. . . . . 6 |- ( T ` 0h ) = 0
4	1, 3	syl6eq 2508	. . . . 5 |- ( A = 0h -> ( T ` A ) = 0 )
5	4	abs00bd 12891	. . . 4 |- ( A = 0h -> ( abs ` ( T ` A ) ) = 0 )
6		0le0 10515	. . . . 5 |- 0 <_ 0
7		fveq2 5792	. . . . . . . 8 |- ( A = 0h -> ( normh ` A ) = ( normh ` 0h ) )
8		norm0 24675	. . . . . . . 8 |- ( normh ` 0h ) = 0
9	7, 8	syl6eq 2508	. . . . . . 7 |- ( A = 0h -> ( normh ` A ) = 0 )
10	9	oveq2d 6209	. . . . . 6 |- ( A = 0h -> ( ( normfn ` T ) x. ( normh ` A ) ) = ( ( normfn ` T ) x. 0 ) )
11		nmcfnex.2	. . . . . . . . 9 |- T e. ConFn
12	2, 11	nmcfnexi 25600	. . . . . . . 8 |- ( normfn ` T ) e. RR
13	12	recni 9502	. . . . . . 7 |- ( normfn ` T ) e. CC
14	13	mul01i 9663	. . . . . 6 |- ( ( normfn ` T ) x. 0 ) = 0
15	10, 14	syl6req 2509	. . . . 5 |- ( A = 0h -> 0 = ( ( normfn ` T ) x. ( normh ` A ) ) )
16	6, 15	syl5breq 4428	. . . 4 |- ( A = 0h -> 0 <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
17	5, 16	eqbrtrd 4413	. . 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 25590	. . . . . . . . . 10 |- T : ~H --> CC
20	19	ffvelrni 5944	. . . . . . . . 9 |- ( A e. ~H -> ( T ` A ) e. CC )
21	20	abscld 13033	. . . . . . . 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 9516	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( T ` A ) ) e. CC )
24		normcl 24672	. . . . . . . 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 9516	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) e. CC )
27		norm-i 24676	. . . . . . . . 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	neneqad 2652	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( normh ` A ) =/= 0 )
31	23, 26, 30	divrec2d 10215	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) )
32	25, 30	rereccld 10262	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> ( 1 / ( normh ` A ) ) e. RR )
33	32	recnd 9516	. . . . . . . 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 25593	. . . . . . . 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 5796	. . . . . 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 13051	. . . . . 6 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( ( 1 / ( normh ` A ) ) x. ( T ` A ) ) ) = ( ( abs ` ( 1 / ( normh ` A ) ) ) x. ( abs ` ( T ` A ) ) ) )
40		df-ne 2646	. . . . . . . . . . . 12 |- ( A =/= 0h <-> -. A = 0h )
41		normgt0 24674	. . . . . . . . . . . 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 10371	. . . . . . . . 9 |- ( ( A e. ~H /\ -. A = 0h ) -> 0 < ( 1 / ( normh ` A ) ) )
45		0re 9490	. . . . . . . . . 10 |- 0 e. RR
46		ltle 9567	. . . . . . . . . 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 13020	. . . . . . 7 |- ( ( A e. ~H /\ -. A = 0h ) -> ( abs ` ( 1 / ( normh ` A ) ) ) = ( 1 / ( normh ` A ) ) )
50	49	oveq1d 6208	. . . . . 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 2497	. . . . 5 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( 1 / ( normh ` A ) ) x. ( abs ` ( T ` A ) ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
52	31, 51	eqtrd 2492	. . . 4 |- ( ( A e. ~H /\ -. A = 0h ) -> ( ( abs ` ( T ` A ) ) / ( normh ` A ) ) = ( abs ` ( T ` ( ( 1 / ( normh ` A ) ) .h A ) ) ) )
53		hvmulcl 24560	. . . . . 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 24672	. . . . . . 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 24797	. . . . . . 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 9581	. . . . . 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 25473	. . . . . 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 1302	. . . . 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 4413	. . 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 10313	. . . 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 1223	. . 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 789	1 |- ( A e. ~H -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4393   -->wf 5515   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387   x. cmul 9391   < clt 9522   <_ cle 9523   / cdiv 10097   abscabs 12834   ~Hchil 24466   .h csm 24468   normhcno 24470   0hc0v 24471   normfncnmf 24498   ConFnccnfn 24500   LinFnclf 24501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-hilex 24546  ax-hv0cl 24550  ax-hvaddid 24551  ax-hfvmul 24552  ax-hvmulid 24553  ax-hvmulass 24554  ax-hvmul0 24557  ax-hfi 24626  ax-his1 24629  ax-his3 24631  ax-his4 24632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-hnorm 24515  df-hvsub 24518  df-nmfn 25394  df-cnfn 25396  df-lnfn 25397

This theorem is referenced by:  nmcfnlb  25603