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Theorem nmcoplbi 22624
Description: A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcopex.1  |-  T  e. 
LinOp
nmcopex.2  |-  T  e. 
ConOp
Assertion
Ref Expression
nmcoplbi  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmcoplbi
StepHypRef Expression
1 0le0 9843 . . . . 5  |-  0  <_  0
21a1i 10 . . . 4  |-  ( A  =  0h  ->  0  <_  0 )
3 fveq2 5541 . . . . . . 7  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
4 nmcopex.1 . . . . . . . 8  |-  T  e. 
LinOp
54lnop0i 22566 . . . . . . 7  |-  ( T `
 0h )  =  0h
63, 5syl6eq 2344 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  0h )
76fveq2d 5545 . . . . 5  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  ( normh `  0h )
)
8 norm0 21723 . . . . 5  |-  ( normh `  0h )  =  0
97, 8syl6eq 2344 . . . 4  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  0 )
10 fveq2 5541 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
1110, 8syl6eq 2344 . . . . . 6  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
1211oveq2d 5890 . . . . 5  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  ( (
normop `  T )  x.  0 ) )
13 nmcopex.2 . . . . . . . 8  |-  T  e. 
ConOp
144, 13nmcopexi 22623 . . . . . . 7  |-  ( normop `  T )  e.  RR
1514recni 8865 . . . . . 6  |-  ( normop `  T )  e.  CC
1615mul01i 9018 . . . . 5  |-  ( (
normop `  T )  x.  0 )  =  0
1712, 16syl6eq 2344 . . . 4  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  0 )
182, 9, 173brtr4d 4069 . . 3  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
1918adantl 452 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
20 normcl 21720 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2120adantr 451 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
22 normne0 21725 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
2322biimpar 471 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
2421, 23rereccld 9603 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
25 normgt0 21722 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
2625biimpa 470 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
2721, 26recgt0d 9707 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( 1  /  ( normh `  A
) ) )
28 0re 8854 . . . . . . . . 9  |-  0  e.  RR
29 ltle 8926 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
3028, 29mpan 651 . . . . . . . 8  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
3124, 27, 30sylc 56 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
3224, 31absidd 11921 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
3332oveq1d 5889 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( normh `  ( T `  A )
) ) )
3424recnd 8877 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
35 simpl 443 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
364lnopmuli 22568 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  .h  ( T `
 A ) ) )
3734, 35, 36syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) )
3837fveq2d 5545 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  (
normh `  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) ) )
394lnopfi 22565 . . . . . . . . 9  |-  T : ~H
--> ~H
4039ffvelrni 5680 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
4140adantr 451 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  A
)  e.  ~H )
42 norm-iii 21735 . . . . . . 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 )
) ) )
4334, 41, 42syl2anc 642 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  ( T `  A
) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
4438, 43eqtrd 2328 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
45 normcl 21720 . . . . . . . . 9  |-  ( ( T `  A )  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
4640, 45syl 15 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
4746adantr 451 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  RR )
4847recnd 8877 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  CC )
4921recnd 8877 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
5048, 49, 23divrec2d 9556 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( normh `  ( T `  A
) ) ) )
5133, 44, 503eqtr4rd 2339 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) ) )
52 hvmulcl 21609 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5334, 35, 52syl2anc 642 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H )
54 normcl 21720 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5553, 54syl 15 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
56 norm1 21844 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
57 eqle 8939 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
5855, 56, 57syl2anc 642 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  <_ 
1 )
59 nmoplb 22503 . . . . . 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
) )
6039, 59mp3an1 1264 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( normh `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normop `  T
) )
6153, 58, 60syl2anc 642 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  <_  ( normop `  T ) )
6251, 61eqbrtrd 4059 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normop `  T )
)
6314a1i 10 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normop `  T )  e.  RR )
64 ledivmul2 9649 . . . 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 ) ) ) )
6547, 63, 21, 26, 64syl112anc 1186 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( ( normh `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normop `  T
)  <->  ( normh `  ( T `  A )
)  <_  ( ( normop `  T )  x.  ( normh `  A ) ) ) )
6662, 65mpbid 201 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
6719, 66pm2.61dane 2537 1  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   abscabs 11735   ~Hchil 21515    .h csm 21517   normhcno 21519   0hc0v 21520   normopcnop 21541   ConOpccop 21542   LinOpclo 21543
This theorem is referenced by:  nmcoplb  22626  cnlnadjlem2  22664  cnlnadjlem7  22669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-grpo 20874  df-gid 20875  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-nmop 22435  df-cnop 22436  df-lnop 22437
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