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Theorem nmbdoplbi 23519
Description: A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
nmbdoplb.1  |-  T  e.  BndLinOp
Assertion
Ref Expression
nmbdoplbi  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmbdoplbi
StepHypRef Expression
1 fveq2 5720 . . . 4  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
21fveq2d 5724 . . 3  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  ( normh `  ( T `  0h ) ) )
3 fveq2 5720 . . . 4  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
43oveq2d 6089 . . 3  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  ( (
normop `  T )  x.  ( normh `  0h )
) )
52, 4breq12d 4217 . 2  |-  ( A  =  0h  ->  (
( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) )  <->  ( normh `  ( T `  0h ) )  <_  (
( normop `  T )  x.  ( normh `  0h )
) ) )
6 nmbdoplb.1 . . . . . . . . . . . 12  |-  T  e.  BndLinOp
7 bdopln 23356 . . . . . . . . . . . 12  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )
86, 7ax-mp 8 . . . . . . . . . . 11  |-  T  e. 
LinOp
98lnopfi 23464 . . . . . . . . . 10  |-  T : ~H
--> ~H
109ffvelrni 5861 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
11 normcl 22619 . . . . . . . . 9  |-  ( ( T `  A )  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
1210, 11syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
1312adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  RR )
1413recnd 9106 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  CC )
15 normcl 22619 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
1615adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
1716recnd 9106 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
18 normne0 22624 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
1918biimpar 472 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
2014, 17, 19divrec2d 9786 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( normh `  ( T `  A
) ) ) )
2116, 19rereccld 9833 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
2221recnd 9106 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
23 simpl 444 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
248lnopmuli 23467 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  .h  ( T `
 A ) ) )
2522, 23, 24syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) )
2625fveq2d 5724 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  (
normh `  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) ) )
2710adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  A
)  e.  ~H )
28 norm-iii 22634 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  ( T `  A )  e.  ~H )  ->  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
2922, 27, 28syl2anc 643 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  ( T `  A
) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
30 normgt0 22621 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
3130biimpa 471 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
3216, 31recgt0d 9937 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( 1  /  ( normh `  A
) ) )
33 0re 9083 . . . . . . . . . 10  |-  0  e.  RR
34 ltle 9155 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
3533, 34mpan 652 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
3621, 32, 35sylc 58 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
3721, 36absidd 12217 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
3837oveq1d 6088 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( normh `  ( T `  A )
) ) )
3926, 29, 383eqtrrd 2472 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  x.  ( normh `  ( T `  A
) ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) ) )
4020, 39eqtrd 2467 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) ) )
41 hvmulcl 22508 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
4222, 23, 41syl2anc 643 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H )
43 normcl 22619 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
4442, 43syl 16 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
45 norm1 22743 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
46 eqle 9168 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
4744, 45, 46syl2anc 643 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  <_ 
1 )
48 nmoplb 23402 . . . . . 6  |-  ( ( T : ~H --> ~H  /\  ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( normh `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normop `  T
) )
499, 48mp3an1 1266 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( normh `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normop `  T
) )
5042, 47, 49syl2anc 643 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  <_  ( normop `  T ) )
5140, 50eqbrtrd 4224 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normop `  T )
)
52 nmopre 23365 . . . . . 6  |-  ( T  e.  BndLinOp  ->  ( normop `  T
)  e.  RR )
536, 52ax-mp 8 . . . . 5  |-  ( normop `  T )  e.  RR
5453a1i 11 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normop `  T )  e.  RR )
55 ledivmul2 9879 . . . 4  |-  ( ( ( normh `  ( T `  A ) )  e.  RR  /\  ( normop `  T )  e.  RR  /\  ( ( normh `  A
)  e.  RR  /\  0  <  ( normh `  A
) ) )  -> 
( ( ( normh `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normop `  T
)  <->  ( normh `  ( T `  A )
)  <_  ( ( normop `  T )  x.  ( normh `  A ) ) ) )
5613, 54, 16, 31, 55syl112anc 1188 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( ( normh `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normop `  T
)  <->  ( normh `  ( T `  A )
)  <_  ( ( normop `  T )  x.  ( normh `  A ) ) ) )
5751, 56mpbid 202 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
58 0le0 10073 . . . 4  |-  0  <_  0
598lnop0i 23465 . . . . . 6  |-  ( T `
 0h )  =  0h
6059fveq2i 5723 . . . . 5  |-  ( normh `  ( T `  0h ) )  =  (
normh `  0h )
61 norm0 22622 . . . . 5  |-  ( normh `  0h )  =  0
6260, 61eqtri 2455 . . . 4  |-  ( normh `  ( T `  0h ) )  =  0
6361oveq2i 6084 . . . . 5  |-  ( (
normop `  T )  x.  ( normh `  0h )
)  =  ( (
normop `  T )  x.  0 )
6453recni 9094 . . . . . 6  |-  ( normop `  T )  e.  CC
6564mul01i 9248 . . . . 5  |-  ( (
normop `  T )  x.  0 )  =  0
6663, 65eqtri 2455 . . . 4  |-  ( (
normop `  T )  x.  ( normh `  0h )
)  =  0
6758, 62, 663brtr4i 4232 . . 3  |-  ( normh `  ( T `  0h ) )  <_  (
( normop `  T )  x.  ( normh `  0h )
)
6867a1i 11 . 2  |-  ( A  e.  ~H  ->  ( normh `  ( T `  0h ) )  <_  (
( normop `  T )  x.  ( normh `  0h )
) )
695, 57, 68pm2.61ne 2673 1  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    <_ cle 9113    / cdiv 9669   abscabs 12031   ~Hchil 22414    .h csm 22416   normhcno 22418   0hc0v 22419   normopcnop 22440   LinOpclo 22442   BndLinOpcbo 22443
This theorem is referenced by:  nmbdoplb  23520  nmopcoadji  23596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-grpo 21771  df-gid 21772  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071  df-hnorm 22463  df-hba 22464  df-hvsub 22466  df-nmop 23334  df-lnop 23336  df-bdop 23337
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