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Theorem nmcoplbi 25351
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 10407 . . . . 5  |-  0  <_  0
21a1i 11 . . . 4  |-  ( A  =  0h  ->  0  <_  0 )
3 fveq2 5688 . . . . . . 7  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
4 nmcopex.1 . . . . . . . 8  |-  T  e. 
LinOp
54lnop0i 25293 . . . . . . 7  |-  ( T `
 0h )  =  0h
63, 5syl6eq 2489 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  0h )
76fveq2d 5692 . . . . 5  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  ( normh `  0h )
)
8 norm0 24449 . . . . 5  |-  ( normh `  0h )  =  0
97, 8syl6eq 2489 . . . 4  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  =  0 )
10 fveq2 5688 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
1110, 8syl6eq 2489 . . . . . 6  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
1211oveq2d 6106 . . . . 5  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  ( (
normop `  T )  x.  0 ) )
13 nmcopex.2 . . . . . . . 8  |-  T  e. 
ConOp
144, 13nmcopexi 25350 . . . . . . 7  |-  ( normop `  T )  e.  RR
1514recni 9394 . . . . . 6  |-  ( normop `  T )  e.  CC
1615mul01i 9555 . . . . 5  |-  ( (
normop `  T )  x.  0 )  =  0
1712, 16syl6eq 2489 . . . 4  |-  ( A  =  0h  ->  (
( normop `  T )  x.  ( normh `  A )
)  =  0 )
182, 9, 173brtr4d 4319 . . 3  |-  ( A  =  0h  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
1918adantl 463 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
20 normcl 24446 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2120adantr 462 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
22 normne0 24451 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
2322biimpar 482 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
2421, 23rereccld 10154 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
25 normgt0 24448 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
2625biimpa 481 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
2721, 26recgt0d 10263 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( 1  /  ( normh `  A
) ) )
28 0re 9382 . . . . . . . . 9  |-  0  e.  RR
29 ltle 9459 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
3028, 29mpan 665 . . . . . . . 8  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
3124, 27, 30sylc 60 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
3224, 31absidd 12905 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
3332oveq1d 6105 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( normh `  ( T `  A )
) ) )
3424recnd 9408 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
35 simpl 454 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
364lnopmuli 25295 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  .h  ( T `
 A ) ) )
3734, 35, 36syl2anc 656 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) )
3837fveq2d 5692 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  (
normh `  ( ( 1  /  ( normh `  A
) )  .h  ( T `  A )
) ) )
394lnopfi 25292 . . . . . . . . 9  |-  T : ~H
--> ~H
4039ffvelrni 5839 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
4140adantr 462 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  A
)  e.  ~H )
42 norm-iii 24461 . . . . . . 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 656 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  ( T `  A
) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
4438, 43eqtrd 2473 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  ( T `  A )
) ) )
45 normcl 24446 . . . . . . . . 9  |-  ( ( T `  A )  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
4640, 45syl 16 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  e.  RR )
4746adantr 462 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  RR )
4847recnd 9408 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  e.  CC )
4921recnd 9408 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
5048, 49, 23divrec2d 10107 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( normh `  ( T `  A
) ) ) )
5133, 44, 503eqtr4rd 2484 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) ) )
52 hvmulcl 24334 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5334, 35, 52syl2anc 656 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H )
54 normcl 24446 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5553, 54syl 16 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
56 norm1 24571 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
57 eqle 9473 . . . . . 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 656 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  <_ 
1 )
59 nmoplb 25230 . . . . . 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 1296 . . . . 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 656 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) ) )  <_  ( normop `  T ) )
6251, 61eqbrtrd 4309 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( normh `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normop `  T )
)
6314a1i 11 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normop `  T )  e.  RR )
64 ledivmul2 10205 . . . 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 1217 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( ( normh `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normop `  T
)  <->  ( normh `  ( T `  A )
)  <_  ( ( normop `  T )  x.  ( normh `  A ) ) ) )
6662, 65mpbid 210 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
6719, 66pm2.61dane 2687 1  |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_ 
( ( normop `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414    <_ cle 9415    / cdiv 9989   abscabs 12719   ~Hchil 24240    .h csm 24242   normhcno 24244   0hc0v 24245   normopcnop 24266   ConOpccop 24267   LinOpclo 24268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-hilex 24320  ax-hfvadd 24321  ax-hvcom 24322  ax-hvass 24323  ax-hv0cl 24324  ax-hvaddid 24325  ax-hfvmul 24326  ax-hvmulid 24327  ax-hvmulass 24328  ax-hvdistr1 24329  ax-hvdistr2 24330  ax-hvmul0 24331  ax-hfi 24400  ax-his1 24403  ax-his2 24404  ax-his3 24405  ax-his4 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-grpo 23597  df-gid 23598  df-ablo 23688  df-vc 23843  df-nv 23889  df-va 23892  df-ba 23893  df-sm 23894  df-0v 23895  df-nmcv 23897  df-hnorm 24289  df-hba 24290  df-hvsub 24292  df-nmop 25162  df-cnop 25163  df-lnop 25164
This theorem is referenced by:  nmcoplb  25353  cnlnadjlem2  25391  cnlnadjlem7  25396
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